Abstract group $C_{32}.D_{128}$

• Label: 8192.vf
• Order: \( 2^{13} \)
• Exponent: \( 2^{8} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{18} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $512$
• Trans deg.: $512$
• Rank: $2$

Group information

Description:	$C_{32}.D_{128}$
Order:	\(8192\)\(\medspace = 2^{13} \)
Exponent:	\(256\)\(\medspace = 2^{8} \)
Automorphism group:	Group of order \(262144\)\(\medspace = 2^{18} \)
Outer automorphisms:	Group of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 13
Nilpotency class:	$8$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	32	64	128	256
Elements	1	131	140	304	704	1280	2560	1024	2048	8192
Conjugacy classes	1	3	8	28	104	144	272	512	1024	2096
Divisions	1	3	5	9	15	11	10	9	8	71
Autjugacy classes	1	3	4	6	8	9	6	5	1	43

Minimal Presentations

Permutation degree:	$512$
Transitive degree:	$512$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{256}=c^{16}=[b,c]=1, b^{a}=b^{183}c^{15}, c^{a}=b^{208}c^{9} \rangle$
Permutation group:	Degree $512$ $\langle(1,2,7,17,37,57,67,47,60,85,101,100,115,123,102,116,141,157,156,171,179,158,172,197,213,212,227,235,214,228,253,269,268,283,291,270,284,309,325,324,339,347,326,340,365,381,380,395,403,382,396,421,437,436,451,459,438,452,477,492,491,503,507,493,504,496,505,508,495,494,485,462,442,454,461,441,440,429,406,386,398,405,385,384,373,350,330,342,349,329,328,317,294,274,286,293,273,272,261,238,218,230,237,217,216,205,182,162,174,181,161,160,149,126,106,118,125,105,104,93,70,50,62,69,49,32,14,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 200 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 131 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{257})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{128}.C_{32})$ $\,\rtimes\,$ $C_2$	$(C_{16}\times C_{256})$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_{256}$ . $C_{32}$	$D_{128}$ . $C_{32}$	$C_{32}$ . $D_{128}$	$C_{32}$ . $\SD_{256}$	all 66

Elements of the group are displayed as matrices in $\GL_{2}(\F_{257})$.

Homology

Abelianization:	$C_{2} \times C_{32} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 2110 subgroups in 230 conjugacy classes, 72 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{32}$	$G/Z \simeq$ $D_{128}$
Commutator:	$G' \simeq$ $C_{128}$	$G/G' \simeq$ $C_2\times C_{32}$
Frattini:	$\Phi \simeq$ $C_{16}\times C_{128}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{32}.D_{128}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{32}.D_{128}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_{64}.(C_2\times C_{32})$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{32}.D_{128}$

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 8192: $C_{32}.D_{128}$

Order 4096: $D_{128}:C_{16}$, $C_{16}\times C_{256}$, $C_{128}.C_{32}$

Order 2048: $C_{128}.C_{16}$, $D_{128}:C_8$, $C_{16}\times C_{128}$, $C_8\times C_{256}$, $C_{16}.D_{64}$, $D_{64}:C_{16}$, $C_{64}.C_{32}$

Order 1024: $C_8\times C_{128}$ x 2, $C_4\times C_{256}$ x 2, $C_{128}.C_8$, $D_{128}:C_4$, $C_8.D_{64}$, $C_{16}.D_{32}$, $C_{16}\times C_{64}$, $C_{64}.C_{16}$, $D_{64}:C_8$, $C_{32}.C_{32}$, $D_{32}:C_{16}$

Order 512: $C_2\times C_{256}$ x 4, $C_4\times C_{128}$ x 3, $C_8\times C_{64}$ x 2, $C_{16}^2.C_2$, $C_{16}\times C_{32}$, $C_{64}.C_2^3$, $C_8.(C_4\times D_8)$, $C_8.(C_8\times D_4)$, $(C_8\times C_{32}).C_2$, $C_{32}.C_4:C_4$, $C_{16}.D_4:C_4$, $C_{16}.C_4:C_8$, $(C_8\times C_{32}).C_2$, $C_8.D_4:C_8$, $C_8.C_4:C_{16}$

Order 256: $C_{256}$ x 8, $C_2\times C_{128}$ x 5, $C_4\times C_{64}$ x 3, $C_8\times C_{32}$ x 2, $D_{64}:C_2$, $D_{32}:C_4$, $D_8.C_{16}$, $Q_{256}$, $\SD_{256}$, $D_{128}$, $C_{64}.C_4$, $C_8.C_{32}$, $C_4.D_{32}$, $D_{16}:C_8$, $C_{32}.C_8$, $C_{16}.C_{16}$, $C_{16}^2$, $C_8.D_{16}$, $C_{16}.D_8$

Order 128: $C_{128}$ x 9, $C_2\times C_{64}$ x 5, $C_4\times C_{32}$ x 4, $C_8\times C_{16}$ x 2, $D_{32}:C_2$, $\OD_{64}:C_2$, $D_{16}:C_4$, $C_{16}.D_4$, $C_8\wr C_2$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $\OD_{128}$, $C_{32}.C_4$, $C_8.C_{16}$, $C_4.D_{16}$, $\OD_{64}:C_2$, $C_{16}.C_8$

Order 64: $C_{64}$ x 10, $C_2\times C_{32}$ x 7, $C_4\times C_{16}$ x 4, $\OD_{64}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_{16}.C_4$, $C_8.C_8$, $C_4.D_8$, $\OD_{32}:C_2$, $C_8^2$, $D_{16}:C_2$, $\OD_{32}:C_2$, $D_8:C_4$

Order 32: $C_{32}$ x 11, $C_2\times C_{16}$ x 9, $C_4\times C_8$ x 2, $\OD_{32}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 15, $C_2\times C_8$ x 5, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

Order 8: $C_8$ x 9, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 8192: $C_{32}.D_{128}$

Order 4096: $D_{128}:C_{16}$, $C_{16}\times C_{256}$, $C_{128}.C_{32}$

Order 2048: $C_{128}.C_{16}$, $D_{128}:C_8$, $C_{16}\times C_{128}$, $C_8\times C_{256}$, $C_{16}.D_{64}$, $D_{64}:C_{16}$, $C_{64}.C_{32}$

Order 1024: $C_8\times C_{128}$ x 2, $C_{128}.C_8$, $D_{128}:C_4$, $C_4\times C_{256}$, $C_8.D_{64}$, $C_{16}.D_{32}$, $C_{16}\times C_{64}$, $C_{64}.C_{16}$, $D_{64}:C_8$, $C_{32}.C_{32}$, $D_{32}:C_{16}$

Order 512: $C_4\times C_{128}$ x 3, $C_8\times C_{64}$ x 2, $C_{16}^2.C_2$, $C_{16}\times C_{32}$, $C_{64}.C_2^3$, $C_8.(C_4\times D_8)$, $C_8.(C_8\times D_4)$, $(C_8\times C_{32}).C_2$, $C_2\times C_{256}$, $C_{32}.C_4:C_4$, $C_{16}.D_4:C_4$, $C_{16}.C_4:C_8$, $(C_8\times C_{32}).C_2$, $C_8.D_4:C_8$, $C_8.C_4:C_{16}$

Order 256: $C_2\times C_{128}$ x 4, $C_4\times C_{64}$ x 3, $C_8\times C_{32}$ x 2, $D_{64}:C_2$, $D_{32}:C_4$, $D_8.C_{16}$, $Q_{256}$, $\SD_{256}$, $D_{128}$, $C_{64}.C_4$, $C_8.C_{32}$, $C_4.D_{32}$, $D_{16}:C_8$, $C_{32}.C_8$, $C_{16}.C_{16}$, $C_{16}^2$, $C_8.D_{16}$, $C_{16}.D_8$, $C_{256}$

Order 128: $C_{128}$ x 5, $C_2\times C_{64}$ x 4, $C_4\times C_{32}$ x 4, $C_8\times C_{16}$ x 2, $D_{32}:C_2$, $\OD_{64}:C_2$, $D_{16}:C_4$, $C_{16}.D_4$, $C_8\wr C_2$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $\OD_{128}$, $C_{32}.C_4$, $C_8.C_{16}$, $C_4.D_{16}$, $\OD_{64}:C_2$, $C_{16}.C_8$

Order 64: $C_2\times C_{32}$ x 7, $C_{64}$ x 6, $C_4\times C_{16}$ x 3, $\OD_{64}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_{16}.C_4$, $C_8.C_8$, $C_4.D_8$, $\OD_{32}:C_2$, $C_8^2$, $D_{16}:C_2$, $\OD_{32}:C_2$, $D_8:C_4$

Order 32: $C_{32}$ x 9, $C_2\times C_{16}$ x 6, $C_4\times C_8$ x 2, $\OD_{32}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 8, $C_2\times C_8$ x 4, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

Order 8: $C_8$ x 6, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 8192: $C_{32}.D_{128}$ ($C_1$)

Order 4096: $D_{128}:C_{16}$ ($C_2$), $C_{16}\times C_{256}$ ($C_2$), $C_{128}.C_{32}$ ($C_2$)

Order 2048: $C_{128}.C_{16}$ ($C_4$), $D_{128}:C_8$ ($C_4$), $C_{16}\times C_{128}$ ($C_2^2$)

Order 1024: $C_8\times C_{128}$ ($D_4$), $C_8\times C_{128}$ ($C_2\times C_4$), $C_{128}.C_8$ ($C_8$), $D_{128}:C_4$ ($C_8$), $C_{16}\times C_{64}$ ($D_4$)

Order 512: $C_{16}\times C_{32}$ ($D_8$), $C_{64}.C_2^3$ ($C_{16}$), $C_{32}.C_4:C_4$ ($C_{16}$), $C_4\times C_{128}$ ($\OD_{16}$), $C_4\times C_{128}$ ($C_2\times C_8$), $C_8\times C_{64}$ ($\SD_{16}$), $C_8\times C_{64}$ ($C_2^2:C_4$)

Order 256: $Q_{256}$ ($C_{32}$), $D_{128}$ ($C_{32}$), $C_2\times C_{128}$ ($\OD_{32}$), $C_2\times C_{128}$ ($C_2\times C_{16}$), $C_4\times C_{64}$ ($C_2^2:C_8$), $C_4\times C_{64}$ ($C_4\wr C_2$), $C_{16}^2$ ($\SD_{32}$), $C_8\times C_{32}$ ($D_4:C_4$), $C_8\times C_{32}$ ($D_{16}$)

Order 128: $C_8\times C_{16}$ ($D_8:C_4$), $C_2\times C_{64}$ ($\OD_{32}:C_2$), $C_2\times C_{64}$ ($C_2^2:C_{16}$), $C_4\times C_{32}$ ($D_4:C_8$), $C_4\times C_{32}$ ($\SD_{64}$), $C_4\times C_{32}$ ($D_{32}$), $C_4\times C_{32}$ ($C_4.D_8$), $C_{128}$ ($\OD_{64}$), $C_{128}$ ($C_2\times C_{32}$)

Order 64: $C_2\times C_{32}$ ($C_8\wr C_2$), $C_2\times C_{32}$ ($D_4:C_{16}$), $C_2\times C_{32}$ ($\SD_{128}$), $C_2\times C_{32}$ ($D_{64}$), $C_4\times C_{16}$ ($D_8:C_8$), $C_4\times C_{16}$ ($D_{16}:C_4$), $C_8^2$ ($C_4.D_{16}$), $C_{64}$ ($\OD_{64}:C_2$), $C_{64}$ ($C_2^2:C_{32}$)

Order 32: $C_4\times C_8$ ($D_{16}:C_8$), $C_2\times C_{16}$ ($D_8:C_{16}$), $C_2\times C_{16}$ ($C_4.D_{32}$), $C_2\times C_{16}$ ($D_{32}:C_4$), $C_2\times C_{16}$ ($C_8.D_{16}$), $C_{32}$ ($\SD_{256}$), $C_{32}$ ($D_{128}$), $C_{32}$ ($C_{16}.D_8$), $C_{32}$ ($D_4:C_{32}$)

Order 16: $C_2\times C_8$ ($C_8.C_2^2:C_{16}$), $C_2\times C_8$ ($C_8.D_4:C_8$), $C_4^2$ ($C_8.D_4:C_8$), $C_{16}$ ($C_{16}^2.C_2$), $C_{16}$ ($C_8.(C_2\times C_{32})$), $C_{16}$ ($C_{16}.D_4:C_4$), $C_{16}$ ($C_{16}.D_4:C_4$)

Order 8: $C_2\times C_4$ ($C_{32}.(C_2\times C_{16})$), $C_8$ ($C_{16}.D_4:C_8$), $C_8$ ($C_{16}^2.C_2^2$), $C_8$ ($C_8.D_{64}$), $C_8$ ($C_{16}.D_{32}$)

Order 4: $C_2^2$ ($C_{16}.D_{64}$), $C_4$ ($C_{64}.(C_2\times C_{16})$), $C_4$ ($C_{32}.(C_2\times C_{32})$)

Order 2: $C_2$ ($C_{64}.(C_2\times C_{32})$)

Order 1: $C_1$ ($C_{32}.D_{128}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8192: $C_{32}.D_{128}$ ($C_1$)

Order 4096: $D_{128}:C_{16}$ ($C_2$), $C_{16}\times C_{256}$ ($C_2$), $C_{128}.C_{32}$ ($C_2$)

Order 2048: $C_{128}.C_{16}$ ($C_4$), $D_{128}:C_8$ ($C_4$), $C_{16}\times C_{128}$ ($C_2^2$)

Order 1024: $C_8\times C_{128}$ ($D_4$), $C_8\times C_{128}$ ($C_2\times C_4$), $C_{128}.C_8$ ($C_8$), $D_{128}:C_4$ ($C_8$), $C_{16}\times C_{64}$ ($D_4$)

Order 512: $C_{16}\times C_{32}$ ($D_8$), $C_{64}.C_2^3$ ($C_{16}$), $C_{32}.C_4:C_4$ ($C_{16}$), $C_4\times C_{128}$ ($\OD_{16}$), $C_4\times C_{128}$ ($C_2\times C_8$), $C_8\times C_{64}$ ($\SD_{16}$), $C_8\times C_{64}$ ($C_2^2:C_4$)

Order 256: $Q_{256}$ ($C_{32}$), $D_{128}$ ($C_{32}$), $C_2\times C_{128}$ ($\OD_{32}$), $C_2\times C_{128}$ ($C_2\times C_{16}$), $C_4\times C_{64}$ ($C_2^2:C_8$), $C_4\times C_{64}$ ($C_4\wr C_2$), $C_{16}^2$ ($\SD_{32}$), $C_8\times C_{32}$ ($D_4:C_4$), $C_8\times C_{32}$ ($D_{16}$)

Order 128: $C_8\times C_{16}$ ($D_8:C_4$), $C_2\times C_{64}$ ($\OD_{32}:C_2$), $C_2\times C_{64}$ ($C_2^2:C_{16}$), $C_4\times C_{32}$ ($D_4:C_8$), $C_4\times C_{32}$ ($\SD_{64}$), $C_4\times C_{32}$ ($D_{32}$), $C_4\times C_{32}$ ($C_4.D_8$), $C_{128}$ ($\OD_{64}$), $C_{128}$ ($C_2\times C_{32}$)

Order 64: $C_2\times C_{32}$ ($C_8\wr C_2$), $C_2\times C_{32}$ ($D_4:C_{16}$), $C_2\times C_{32}$ ($\SD_{128}$), $C_2\times C_{32}$ ($D_{64}$), $C_4\times C_{16}$ ($D_8:C_8$), $C_4\times C_{16}$ ($D_{16}:C_4$), $C_8^2$ ($C_4.D_{16}$), $C_{64}$ ($\OD_{64}:C_2$), $C_{64}$ ($C_2^2:C_{32}$)

Order 32: $C_4\times C_8$ ($D_{16}:C_8$), $C_2\times C_{16}$ ($D_8:C_{16}$), $C_2\times C_{16}$ ($C_4.D_{32}$), $C_2\times C_{16}$ ($D_{32}:C_4$), $C_2\times C_{16}$ ($C_8.D_{16}$), $C_{32}$ ($\SD_{256}$), $C_{32}$ ($D_{128}$), $C_{32}$ ($C_{16}.D_8$), $C_{32}$ ($D_4:C_{32}$)

Order 16: $C_2\times C_8$ ($C_8.C_2^2:C_{16}$), $C_2\times C_8$ ($C_8.D_4:C_8$), $C_4^2$ ($C_8.D_4:C_8$), $C_{16}$ ($C_{16}^2.C_2$), $C_{16}$ ($C_8.(C_2\times C_{32})$), $C_{16}$ ($C_{16}.D_4:C_4$), $C_{16}$ ($C_{16}.D_4:C_4$)

Order 8: $C_2\times C_4$ ($C_{32}.(C_2\times C_{16})$), $C_8$ ($C_{16}.D_4:C_8$), $C_8$ ($C_{16}^2.C_2^2$), $C_8$ ($C_8.D_{64}$), $C_8$ ($C_{16}.D_{32}$)

Order 4: $C_2^2$ ($C_{16}.D_{64}$), $C_4$ ($C_{64}.(C_2\times C_{16})$), $C_4$ ($C_{32}.(C_2\times C_{32})$)

Order 2: $C_2$ ($C_{64}.(C_2\times C_{32})$)

Order 1: $C_1$ ($C_{32}.D_{128}$)

Series

Derived series

$C_{32}.D_{128}$

$\rhd$

$C_{128}$

$\rhd$

$C_1$

Chief series

$C_{32}.D_{128}$

$\rhd$

$C_{16}\times C_{256}$

$\rhd$

$C_{16}\times C_{128}$

$\rhd$

$C_8\times C_{128}$

$\rhd$

$C_4\times C_{128}$

$\rhd$

$C_2\times C_{128}$

$\rhd$

$C_{128}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{32}.D_{128}$

$\rhd$

$C_{128}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{32}$

$\lhd$

$C_2\times C_{32}$

$\lhd$

$C_4\times C_{32}$

$\lhd$

$C_8\times C_{32}$

$\lhd$

$C_{16}\times C_{32}$

$\lhd$

$C_{16}\times C_{64}$

$\lhd$

$C_{16}\times C_{128}$

$\lhd$

$C_{32}.D_{128}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $2096 \times 2096$ character table is not available for this group.

Rational character table

The $71 \times 71$ rational character table is not available for this group.