Abstract group $(C_2\times C_4^3):D_8$

• Label: 2048.cok
• Order: \( 2^{11} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$(C_2\times C_4^3):D_8$
Order:	\(2048\)\(\medspace = 2^{11} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$(C_5^3\times C_{10}).Q_8$, of order \(32768\)\(\medspace = 2^{15} \) (generators)
Outer automorphisms:	$C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:	$C_2$ x 11
Nilpotency class:	$7$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	271	752	1024	2048
Conjugacy classes	1	15	25	12	53
Divisions	1	15	25	10	51
Autjugacy classes	1	10	11	3	25

Dimension	1	2	4	8	16
Irr. complex chars.	8	10	13	20	2	53
Irr. rational chars.	8	6	15	20	2	51

Minimal Presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$86016$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{8}=c^{4}=d^{2}=e^{4}=f^{4}=[a,f]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,7,13,10,2,8,14,9)(3,5,16,12)(4,6,15,11), (1,10,3,11,2,9,4,12)(5,13,8,15,6,14,7,16), (1,3)(2,4)(5,10,7,11,6,9,8,12)\rangle$
Transitive group:	16T1446	16T1488			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\times C_4^3)$ $\,\rtimes\,$ $D_8$	$(C_4^2:C_2^3)$ $\,\rtimes\,$ $D_8$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^5:D_4)$ . $D_4$	$C_2^5$ . $(D_4:D_4)$	$C_2^4$ . $(C_2\wr D_4)$	$(C_2^5.C_2^3)$ . $D_4$	all 19

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 40632 subgroups in 3320 conjugacy classes, 40 normal (24 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2^2\times C_4^2):D_8$
Commutator:	$G' \simeq$ $C_2^5.C_2^3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^5.C_2^3$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5.C_2{\rm wrC}_2^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^5.C_2{\rm wrC}_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $(C_2^2\times C_4^2):D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2{\rm wrC}_2^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 2048: $C_2^5.C_2{\rm wrC}_2^2$

Order 1024: $C_2^4.C_2{\rm wrC}_4$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^4.C_2^3:C_8$, $C_2^5.Q_8:C_2^2$, $C_2^4.C_2^4.C_2^2$

Order 512: $C_2^4.C_2^3.C_2^2$ x 2, $C_2^3.C_2{\rm wrC}_4$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^5.D_4.C_2$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^4.C_2^2:C_8$ x 2, $C_2^5.C_4^2$, $C_2^4.C_2^4.C_2$, $C_2^3.D_4^2$, $C_2^5.D_4.C_2$, $C_2^5.C_2^4$, $C_2^5.C_4.C_2^2$, $C_2^3.C_2^5.C_2$

Order 256: $C_4.D_4^2$ x 6, $D_4^2:C_4$ x 4, $C_2^5.D_4$ x 4, $D_4^2:C_2^2$ x 4, $C_4^3:C_2^2$ x 4, $C_4^3.C_2^2$ x 4, $C_2^4.C_4^2$ x 3, $C_2^5:D_4$ x 3, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^4:D_8$ x 2, $C_2^4.D_8$ x 2, $C_2^5:D_4$ x 2, $C_2^5:C_8$ x 2, $C_2^5.D_4$ x 2, $C_2^5.C_2^3$ x 2, $D_4^2:C_2^2$, $C_4^2.C_2^4$, $C_4^2:C_2^4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_2\times C_4^3):C_2$, $C_4^3:C_2^2$, $C_2^3.C_2^5$, $C_2^5:D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $(C_4\times \OD_{16}):C_2^2$, $C_4^3.C_2^2$

Order 128: $C_4^2:D_4$ x 24, $C_4^2:C_2^3$ x 22, $C_2^4.D_4$ x 10, $C_4^2:D_4$ x 8, $C_2^4.D_4$ x 8, $D_4^2:C_2$ x 8, $C_2^5:C_4$ x 7, $C_4^2:C_2^3$ x 6, $C_2^4:D_4$ x 6, $D_4^2:C_2$ x 5, $C_4^2.D_4$ x 4, $C_2\wr D_4$ x 4, $C_2^4.D_4$ x 4, $\OD_{16}:D_4$ x 4, $C_2^4.D_4$ x 4, $C_4^2:D_4$ x 4, $D_4:C_4^2$ x 4, $C_4^2.C_2^3$ x 4, $D_4^2:C_2$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_2^4:D_4$ x 4, $C_2^4.D_4$ x 4, $C_4^3:C_2$ x 4, $C_2^4:D_4$ x 3, $C_2^5.C_4$ x 3, $\OD_{16}:C_2^3$ x 3, $C_2^5:C_4$ x 2, $C_2^3.D_8$ x 2, $(C_2\times C_8):D_4$ x 2, $(C_2\times C_4):D_8$ x 2, $C_4^2:Q_8$ x 2, $C_4^3:C_2$ x 2, $C_2^4.D_4$ x 2, $C_2^4:C_8$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^3:D_8$ x 2, $C_8:C_2^4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $\OD_{16}:D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2\times C_4^3$, $C_2^3.C_4^2$, $D_4:C_2^4$, $C_4^2:C_2^3$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $D_4:D_4$ x 52, $C_4^2:C_2^2$ x 51, $D_4^2$ x 32, $C_8:C_2^3$ x 28, $C_4^2:C_2^2$ x 26, $D_4:D_4$ x 24, $C_4^2:C_2^2$ x 24, $\OD_{16}:C_2^2$ x 22, $C_2^3:D_4$ x 17, $C_2^4:C_4$ x 16, $C_2^3:D_4$ x 16, $C_2\wr C_2^2$ x 16, $D_4:C_2^3$ x 15, $C_2^3.D_4$ x 14, $C_2\wr C_4$ x 12, $D_4\times C_2^3$ x 12, $C_4^2:C_2^2$ x 12, $Q_8:D_4$ x 12, $C_4^2.C_2^2$ x 12, $C_4^2:C_4$ x 12, $C_2^3:D_4$ x 10, $C_2^3.D_4$ x 10, $D_4:D_4$ x 10, $C_4^2.C_2^2$ x 9, $C_4^2:C_4$ x 8, $C_2^3.D_4$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2.C_2^2$ x 8, $Q_8:D_4$ x 8, $C_4:D_8$ x 8, $C_2^3.D_4$ x 8, $C_2^2:\SD_{16}$ x 8, $C_2^4:C_4$ x 5, $C_4^2:C_2^2$ x 5, $C_2^2.D_8$ x 4, $C_2^3.C_2^3$ x 4, $C_4^2:C_2^2$ x 4, $C_4^2.C_2^2$ x 4, $C_8:D_4$ x 4, $C_4^2.C_2^2$ x 4, $C_4.D_8$ x 4, $C_2^2.C_4^2$ x 3, $C_4^2.C_2^2$ x 3, $C_2^2\times C_4^2$ x 3, $C_2^2:\OD_{16}$ x 2, $C_4\times \OD_{16}$ x 2, $C_2^2.D_8$ x 2, $C_2^3.C_2^3$ x 2, $C_2^2:C_4^2$ x 2, $C_4^3$ x 2, $C_2^3:C_8$ x 2, $D_4:C_2^3$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_2^2\times D_8$ x 2, $C_2^2\times \OD_{16}$ x 2, $C_2^3:D_4$ x 2, $C_4:D_8$ x 2, $C_8:D_4$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $C_4:\OD_{16}$ x 2, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 32: $C_2^2\times D_4$ x 151, $C_4:D_4$ x 99, $C_4:D_4$ x 91, $C_2^2\wr C_2$ x 86, $D_4:C_2^2$ x 58, $D_8:C_2$ x 56, $C_4\times D_4$ x 44, $D_4:C_2^2$ x 42, $C_2^2.D_4$ x 38, $C_2^3:C_4$ x 37, $C_2^3:C_4$ x 32, $C_4\wr C_2$ x 32, $D_4:C_4$ x 26, $\OD_{16}:C_2$ x 24, $C_2\times D_8$ x 22, $C_2\times C_4^2$ x 22, $C_2\times \SD_{16}$ x 20, $C_4^2:C_2$ x 20, $C_2\times \OD_{16}$ x 18, $C_4^2:C_2$ x 14, $C_2^2:Q_8$ x 8, $C_4\times Q_8$ x 8, $C_2^5$ x 6, $C_2^2:C_8$ x 6, $C_2^3\times C_4$ x 6, $C_2.C_4^2$ x 6, $C_4^2:C_2$ x 4, $C_4.Q_8$ x 4, $C_4:C_8$ x 4, $C_4:Q_8$ x 3, $C_8:C_4$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$ x 2, $C_2^2.D_4$ x 2, $C_2^2\times Q_8$

Order 16: $C_2\times D_4$ x 408, $D_4:C_2$ x 96, $C_2^2:C_4$ x 88, $C_2^2\times C_4$ x 63, $C_2^4$ x 50, $C_4^2$ x 31, $D_8$ x 28, $C_4:C_4$ x 27, $\SD_{16}$ x 24, $\OD_{16}$ x 20, $C_2\times C_8$ x 14, $C_2\times Q_8$ x 11

Order 8: $D_4$ x 171, $C_2^3$ x 120, $C_2\times C_4$ x 107, $Q_8$ x 14, $C_8$ x 10

Order 4: $C_2^2$ x 79, $C_4$ x 25

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2048: $C_2^5.C_2{\rm wrC}_2^2$

Order 1024: $C_2^4.C_2^3:C_8$, $C_2^4.C_2{\rm wrC}_4$, $C_2^5.C_2^4.C_2$, $C_2^5.Q_8:C_2^2$, $C_2^4.C_2^4.C_2^2$

Order 512: $C_2^5.C_4^2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^3.C_2^2$, $C_2^3.D_4^2$, $C_2^3.C_2{\rm wrC}_4$, $C_2^5.D_4.C_2$, $C_2^5.C_2^3.C_2$, $C_2^5.C_2^4$, $C_2^5.D_4.C_2$, $C_2^5.C_4.C_2^2$, $C_2^4.(C_4\times D_4)$, $C_2^4.C_2^2:C_8$, $C_2^3.C_2^5.C_2$

Order 256: $C_4.D_4^2$ x 4, $C_2^5.D_4$ x 2, $C_2^4.C_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^5.C_2^3$ x 2, $C_2^5.D_4$, $D_4^2:C_4$, $C_2^5.D_4$, $D_4^2:C_2^2$, $C_4^2.C_2^4$, $C_4^2:C_2^4$, $C_2^4:D_8$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_2\times C_4^3):C_2$, $C_2^4.D_8$, $C_2^5:D_4$, $C_2^5:C_8$, $C_4^3:C_2^2$, $C_2^3.C_2^5$, $D_4^2:C_2^2$, $C_2^5:D_4$, $C_2^5.D_4$, $C_4^3:C_2^2$, $C_4^3.C_2^2$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_4\times \OD_{16}):C_2^2$, $C_4^3.C_2^2$

Order 128: $C_4^2:C_2^3$ x 6, $C_4^2:C_2^3$ x 6, $C_4^2:D_4$ x 5, $C_2^5:C_4$ x 5, $C_2^4:D_4$ x 5, $C_2^4.D_4$ x 4, $D_4^2:C_2$ x 4, $C_4^2:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_4^2:D_4$ x 2, $C_2^5.C_4$ x 2, $D_4^2:C_2$ x 2, $C_4^2.C_2^3$ x 2, $D_4^2:C_2$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $\OD_{16}:C_2^3$ x 2, $C_2\times C_4^3$, $C_4^2.D_4$, $C_2\wr D_4$, $C_2^4.D_4$, $C_2^5:C_4$, $C_2^3.D_8$, $(C_2\times C_8):D_4$, $(C_2\times C_4):D_8$, $C_4^2:Q_8$, $\OD_{16}:D_4$, $C_4^3:C_2$, $C_2^4.D_4$, $D_4:C_4^2$, $C_2^4:C_8$, $C_2^4.D_4$, $C_2^3.C_4^2$, $C_2^3.C_4^2$, $C_2^3:D_8$, $D_4:C_2^4$, $C_8:C_2^4$, $C_4^2:C_2^3$, $C_4^2.C_2^3$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $\OD_{16}:D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_2^4:D_4$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_4^3:C_2$, $C_2^3.C_4^2$, $C_2.D_4^2$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $C_4^2:C_2^2$ x 22, $D_4\times C_2^3$ x 12, $C_4^2:C_2^2$ x 10, $C_2^3:D_4$ x 10, $C_2^3:D_4$ x 9, $\OD_{16}:C_2^2$ x 8, $D_4:C_2^3$ x 8, $C_8:C_2^3$ x 8, $D_4^2$ x 8, $D_4:D_4$ x 8, $C_2^4:C_4$ x 7, $C_2^3.D_4$ x 7, $D_4:D_4$ x 6, $C_2^3:D_4$ x 5, $C_4^2:C_2^2$ x 5, $C_2^3.D_4$ x 4, $C_2^4:C_4$ x 4, $C_4^2.C_2^2$ x 4, $C_4^2:C_2^2$ x 4, $C_2\wr C_2^2$ x 4, $C_2\wr C_4$ x 3, $C_4^2:C_2^2$ x 3, $Q_8:D_4$ x 3, $C_4^2.C_2^2$ x 3, $C_4^2.C_2^2$ x 3, $C_4^2:C_4$ x 3, $C_2^2\times C_4^2$ x 3, $D_4:D_4$ x 3, $C_2^2.D_8$ x 2, $C_2^3.C_2^3$ x 2, $C_4^2:C_4$ x 2, $C_2^3.D_4$ x 2, $D_4:C_2^3$ x 2, $C_4^2.C_2^2$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $Q_8:D_4$ x 2, $C_4:D_8$ x 2, $C_2^3.D_4$ x 2, $C_2^2:\SD_{16}$ x 2, $C_2^2:\OD_{16}$, $C_4\times \OD_{16}$, $C_2^2.D_8$, $C_2^3.C_2^3$, $C_2^2:C_4^2$, $C_4^3$, $C_2^3:C_8$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_8:D_4$, $C_4:D_8$, $C_8:D_4$, $C_2^2.C_4^2$, $C_4.D_8$, $C_4^2.C_4$, $C_4^2.C_4$, $C_4:\OD_{16}$

Order 32: $C_2^2\times D_4$ x 73, $C_4:D_4$ x 29, $C_2^2\wr C_2$ x 26, $C_4:D_4$ x 25, $C_2^3:C_4$ x 21, $D_4:C_2^2$ x 16, $D_4:C_2^2$ x 16, $C_4\times D_4$ x 12, $C_2\times C_4^2$ x 12, $C_2^2.D_4$ x 11, $C_2^3:C_4$ x 9, $D_8:C_2$ x 8, $C_4^2:C_2$ x 8, $D_4:C_4$ x 7, $\OD_{16}:C_2$ x 7, $C_2\times \OD_{16}$ x 7, $C_2^5$ x 6, $C_2^3\times C_4$ x 6, $C_2\times D_8$ x 6, $C_4^2:C_2$ x 6, $C_2\times \SD_{16}$ x 5, $C_4\wr C_2$ x 4, $C_2^2:Q_8$ x 3, $C_4\times Q_8$ x 3, $C_2.C_4^2$ x 3, $C_2^2:C_8$ x 2, $C_4:Q_8$ x 2, $C_4.Q_8$ x 2, $C_4:C_8$ x 2, $C_2^2\times Q_8$, $C_8:C_4$, $C_2^2\times C_8$, $C_4^2:C_2$, $C_4\times C_8$, $C_2^2.D_4$

Order 16: $C_2\times D_4$ x 134, $C_2^4$ x 34, $C_2^2\times C_4$ x 34, $C_2^2:C_4$ x 29, $D_4:C_2$ x 19, $C_4^2$ x 15, $C_4:C_4$ x 10, $\OD_{16}$ x 6, $C_2\times Q_8$ x 6, $C_2\times C_8$ x 5, $D_8$ x 4, $\SD_{16}$ x 3

Order 8: $C_2^3$ x 70, $C_2\times C_4$ x 48, $D_4$ x 42, $Q_8$ x 4, $C_8$ x 3

Order 4: $C_2^2$ x 43, $C_4$ x 11

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2048: $C_2^5.C_2{\rm wrC}_2^2$ ($C_1$)

Order 1024: $C_2^4.C_2{\rm wrC}_4$ ($C_2$) x 2, $C_2^5.C_2^4.C_2$ ($C_2$) x 2, $C_2^4.C_2^3:C_8$ ($C_2$), $C_2^5.Q_8:C_2^2$ ($C_2$), $C_2^4.C_2^4.C_2^2$ ($C_2$)

Order 512: $C_2^5.C_2^3.C_2$ ($C_2^2$) x 2, $C_2^4.C_2^2:C_8$ ($C_2^2$) x 2, $C_2^5.C_4^2$ ($C_2^2$), $C_2^4.C_2^4.C_2$ ($C_2^2$), $C_2^5.C_2^4$ ($C_2^2$)

Order 256: $C_2^4.C_4^2$ ($D_4$) x 2, $C_2^5:D_4$ ($D_4$) x 2, $C_4^3:C_2^2$ ($D_4$), $C_2^5.C_2^3$ ($C_2^3$), $C_2^5.C_2^3$ ($D_4$)

Order 128: $C_2^5:C_4$ ($C_2\times D_4$) x 2, $C_2\times C_4^3$ ($D_8$), $C_4^2:C_2^3$ ($D_8$), $C_4^2:C_2^3$ ($C_2\times D_4$)

Order 64: $D_4\times C_2^3$ ($D_8:C_2$), $D_4\times C_2^3$ ($C_2^2\wr C_2$), $C_2^2\times C_4^2$ ($C_2\times D_8$)

Order 32: $C_2^5$ ($C_2\wr C_2^2$), $C_2^5$ ($D_4:D_4$), $C_2^3\times C_4$ ($D_4:D_4$)

Order 16: $C_2^4$ ($C_2\wr D_4$) x 2, $C_2^4$ ($C_2^3:D_8$)

Order 8: $C_2^3$ ($C_2^4:D_8$)

Order 4: $C_2^2$ ($C_2^5:D_8$)

Order 2: $C_2$ ($(C_2^2\times C_4^2):D_8$)

Order 1: $C_1$ ($(C_2\times C_4^3):D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2048: $C_2^5.C_2{\rm wrC}_2^2$ ($C_1$)

Order 1024: $C_2^4.C_2^3:C_8$ ($C_2$), $C_2^4.C_2{\rm wrC}_4$ ($C_2$), $C_2^5.C_2^4.C_2$ ($C_2$), $C_2^5.Q_8:C_2^2$ ($C_2$), $C_2^4.C_2^4.C_2^2$ ($C_2$)

Order 512: $C_2^5.C_4^2$ ($C_2^2$), $C_2^4.C_2^4.C_2$ ($C_2^2$), $C_2^5.C_2^3.C_2$ ($C_2^2$), $C_2^5.C_2^4$ ($C_2^2$), $C_2^4.C_2^2:C_8$ ($C_2^2$)

Order 256: $C_2^4.C_4^2$ ($D_4$), $C_4^3:C_2^2$ ($D_4$), $C_2^5:D_4$ ($D_4$), $C_2^5.C_2^3$ ($C_2^3$), $C_2^5.C_2^3$ ($D_4$)

Order 128: $C_2\times C_4^3$ ($D_8$), $C_2^5:C_4$ ($C_2\times D_4$), $C_4^2:C_2^3$ ($D_8$), $C_4^2:C_2^3$ ($C_2\times D_4$)

Order 64: $D_4\times C_2^3$ ($D_8:C_2$), $D_4\times C_2^3$ ($C_2^2\wr C_2$), $C_2^2\times C_4^2$ ($C_2\times D_8$)

Order 32: $C_2^5$ ($C_2\wr C_2^2$), $C_2^5$ ($D_4:D_4$), $C_2^3\times C_4$ ($D_4:D_4$)

Order 16: $C_2^4$ ($C_2\wr D_4$), $C_2^4$ ($C_2^3:D_8$)

Order 8: $C_2^3$ ($C_2^4:D_8$)

Order 4: $C_2^2$ ($C_2^5:D_8$)

Order 2: $C_2$ ($(C_2^2\times C_4^2):D_8$)

Order 1: $C_1$ ($(C_2\times C_4^3):D_8$)

Series

Derived series

$C_2^5.C_2{\rm wrC}_2^2$

$\rhd$

$C_2^5.C_2^3$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^5.C_2{\rm wrC}_2^2$

$\rhd$

$C_2^4.C_2^4.C_2^2$

$\rhd$

$C_2^4.C_2^4.C_2$

$\rhd$

$C_2^5.C_2^3$

$\rhd$

$C_2^5:C_4$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.C_2{\rm wrC}_2^2$

$\rhd$

$C_2^5.C_2^3$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^5$

$\lhd$

$D_4\times C_2^3$

$\lhd$

$C_2^5.C_2^3$

$\lhd$

$C_2^5.C_2{\rm wrC}_2^2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $53 \times 53$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $51 \times 51$ rational character table.