Abstract group $(C_2\times C_4).D_{24}$

• Label: 384.32
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $19$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$(C_2\times C_4).D_{24}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	Group of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) (generators)
Outer automorphisms:	$C_2^5$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	55	2	72	14	128	48	64	384
Conjugacy classes	1	7	1	7	5	8	8	8	45
Divisions	1	7	1	5	5	3	4	1	27
Autjugacy classes	1	6	1	5	5	2	4	1	25

Dimension	1	2	4	8
Irr. complex chars.	8	26	9	2	45
Irr. rational chars.	4	6	10	7	27

Minimal Presentations

Permutation degree:	$19$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	6	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{8}=c^{2}=d^{12}=[a,c]=[c,d]=1, b^{a}=b^{7}cd^{9}, d^{a}=cd^{11}, c^{b}=cd^{6}, d^{b}=cd \rangle$
Permutation group:	Degree $19$ $\langle(1,2,5,7,3,8,4,6)(9,10)(11,13)(12,15)(14,16)(18,19), (1,3)(2,6)(4,5)(7,8) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^2.\OD_{16})$ $\,\rtimes\,$ $S_3$	$((C_2\times C_4).D_{12})$ $\,\rtimes\,$ $C_2$	$((C_2\times C_4):C_{24})$ $\,\rtimes\,$ $C_2$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{12})$ . $D_8$	$(C_2\times C_4)$ . $D_{24}$	$C_6$ . $(C_2\wr C_4)$	$(C_2^2.D_4)$ . $D_6$	all 29

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 934 subgroups in 168 conjugacy classes, 35 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2.D_{12}$
Commutator:	$G' \simeq$ $C_2^2\times C_{12}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2.D_4$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2\times C_4):C_{24}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2\times C_4).D_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2\times C_4).D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 384: $(C_2\times C_4).D_{24}$

Order 192: $(C_2\times C_4).D_{12}$, $(C_2\times C_4):D_{12}$, $(C_2\times C_4):C_{24}$

Order 128: $(C_2\times C_4).D_8$

Order 96: $C_2^2.D_{12}$ x 2, $C_2^2:C_{24}$, $C_6.C_4^2$, $C_2^2\times D_{12}$, $C_2\times C_4:C_{12}$, $C_6:\OD_{16}$

Order 64: $C_2^3.D_4$, $C_2^2.\OD_{16}$, $\OD_{16}:C_4$

Order 48: $D_6:C_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_2\times D_{12}$ x 3, $C_3:\OD_{16}$ x 3, $C_6:C_8$, $C_2^2\times D_6$, $C_6.C_2^3$, $C_2\times C_{24}$, $C_4:C_{12}$

Order 32: $C_2^3:C_4$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2\times \OD_{16}$, $C_2^2.D_4$, $C_2.C_4^2$

Order 24: $C_2\times C_{12}$ x 8, $C_2\times D_6$ x 8, $D_{12}$ x 4, $C_6:C_4$ x 3, $C_3:C_8$ x 2, $C_{24}$, $C_2^2\times C_6$

Order 16: $C_2^2:C_4$ x 4, $C_2^2\times C_4$ x 4, $\OD_{16}$ x 3, $C_2\times D_4$ x 3, $C_2\times C_8$ x 2, $C_4:C_4$, $C_2^4$

Order 12: $D_6$ x 10, $C_2\times C_6$ x 5, $C_{12}$ x 4, $C_3:C_4$

Order 8: $C_2\times C_4$ x 11, $C_2^3$ x 9, $D_4$ x 4, $C_8$ x 3

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$ x 15, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_2\times C_4).D_{24}$

Order 192: $(C_2\times C_4).D_{12}$, $(C_2\times C_4):D_{12}$, $(C_2\times C_4):C_{24}$

Order 128: $(C_2\times C_4).D_8$

Order 96: $C_2^2.D_{12}$ x 2, $C_2^2:C_{24}$, $C_6.C_4^2$, $C_2^2\times D_{12}$, $C_2\times C_4:C_{12}$, $C_6:\OD_{16}$

Order 64: $C_2^3.D_4$, $C_2^2.\OD_{16}$, $\OD_{16}:C_4$

Order 48: $C_2^2\times C_{12}$ x 3, $D_6:C_4$ x 3, $C_2\times D_{12}$ x 2, $C_3:\OD_{16}$ x 2, $C_6:C_8$, $C_2^2\times D_6$, $C_6.C_2^3$, $C_2\times C_{24}$, $C_4:C_{12}$

Order 32: $C_2^3:C_4$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2\times \OD_{16}$, $C_2^2.D_4$, $C_2.C_4^2$

Order 24: $C_2\times C_{12}$ x 8, $C_2\times D_6$ x 7, $C_6:C_4$ x 3, $D_{12}$ x 2, $C_{24}$, $C_2^2\times C_6$, $C_3:C_8$

Order 16: $C_2^2\times C_4$ x 4, $C_2^2:C_4$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$, $C_2^4$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 5, $C_{12}$ x 4, $C_3:C_4$

Order 8: $C_2\times C_4$ x 11, $C_2^3$ x 8, $D_4$ x 2, $C_8$ x 2

Order 6: $C_6$ x 5, $S_3$

Order 4: $C_2^2$ x 12, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_2\times C_4).D_{24}$ ($C_1$)

Order 192: $(C_2\times C_4).D_{12}$ ($C_2$), $(C_2\times C_4):D_{12}$ ($C_2$), $(C_2\times C_4):C_{24}$ ($C_2$)

Order 96: $C_6.C_4^2$ ($C_4$), $C_2^2\times D_{12}$ ($C_4$), $C_2\times C_4:C_{12}$ ($C_2^2$)

Order 64: $C_2^2.\OD_{16}$ ($S_3$)

Order 48: $C_2^2\times C_{12}$ ($D_4$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$)

Order 32: $C_2^2.D_4$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($\SD_{16}$), $C_2\times C_{12}$ ($D_8$), $C_2^2\times C_6$ ($C_2^2:C_4$)

Order 16: $C_2^2\times C_4$ ($C_3:D_4$), $C_2^2\times C_4$ ($D_{12}$), $C_2^2\times C_4$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($D_4:C_4$), $C_2\times C_6$ ($C_2^3:C_4$), $C_2\times C_6$ ($C_4\wr C_2$)

Order 8: $C_2^3$ ($D_6:C_4$), $C_2\times C_4$ ($D_{24}$), $C_2\times C_4$ ($C_{24}:C_2$)

Order 6: $C_6$ ($C_2^2.D_8$), $C_6$ ($C_4^2.C_4$), $C_6$ ($C_2\wr C_4$)

Order 4: $C_2^2$ ($D_{12}:C_4$), $C_2^2$ ($D_{12}:C_4$), $C_2^2$ ($C_2^2.D_{12}$)

Order 3: $C_3$ ($(C_2\times C_4).D_8$)

Order 2: $C_2$ ($(C_2\times C_4).D_{12}$), $C_2$ ($(C_2^2\times D_6):C_4$), $C_2$ ($C_2^2.D_{24}$)

Order 1: $C_1$ ($(C_2\times C_4).D_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_2\times C_4).D_{24}$ ($C_1$)

Order 192: $(C_2\times C_4).D_{12}$ ($C_2$), $(C_2\times C_4):D_{12}$ ($C_2$), $(C_2\times C_4):C_{24}$ ($C_2$)

Order 96: $C_6.C_4^2$ ($C_4$), $C_2^2\times D_{12}$ ($C_4$), $C_2\times C_4:C_{12}$ ($C_2^2$)

Order 64: $C_2^2.\OD_{16}$ ($S_3$)

Order 48: $C_2^2\times C_{12}$ ($D_4$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$)

Order 32: $C_2^2.D_4$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($\SD_{16}$), $C_2\times C_{12}$ ($D_8$), $C_2^2\times C_6$ ($C_2^2:C_4$)

Order 16: $C_2^2\times C_4$ ($C_3:D_4$), $C_2^2\times C_4$ ($D_{12}$), $C_2^2\times C_4$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($D_4:C_4$), $C_2\times C_6$ ($C_2^3:C_4$), $C_2\times C_6$ ($C_4\wr C_2$)

Order 8: $C_2^3$ ($D_6:C_4$), $C_2\times C_4$ ($D_{24}$), $C_2\times C_4$ ($C_{24}:C_2$)

Order 6: $C_6$ ($C_2^2.D_8$), $C_6$ ($C_4^2.C_4$), $C_6$ ($C_2\wr C_4$)

Order 4: $C_2^2$ ($D_{12}:C_4$), $C_2^2$ ($D_{12}:C_4$), $C_2^2$ ($C_2^2.D_{12}$)

Order 3: $C_3$ ($(C_2\times C_4).D_8$)

Order 2: $C_2$ ($(C_2\times C_4).D_{12}$), $C_2$ ($(C_2^2\times D_6):C_4$), $C_2$ ($C_2^2.D_{24}$)

Order 1: $C_1$ ($(C_2\times C_4).D_{24}$)

Series

Derived series

$(C_2\times C_4).D_{24}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_1$

Chief series

$(C_2\times C_4).D_{24}$

$\rhd$

$(C_2\times C_4):D_{12}$

$\rhd$

$C_2\times C_4:C_{12}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_4).D_{24}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2.D_4$

$\lhd$

$C_2^2.\OD_{16}$

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $27 \times 27$ rational character table.