Abstract group 96.220: $C_2^3\times C_{12}$

• Label: 96.220
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 7 \)
• Perm deg.: $13$
• Trans deg.: $96$
• Rank: $4$

Group information

Description:	$C_2^3\times C_{12}$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2.C_2^7:\GL(3,2)$, of order \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).

Group statistics

Order	1	2	3	4	6	12
Elements	1	15	2	16	30	32	96
Conjugacy classes	1	15	2	16	30	32	96
Divisions	1	15	1	8	15	8	48
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	4
Irr. complex chars.	96	0	0	96
Irr. rational chars.	16	24	8	48

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$96$
Rank:	$4$
Inequivalent generating quadruples:	$600$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{12}=1 \rangle$
Permutation group:	Degree $13$ $\langle(7,10,8,9), (1,2), (3,4), (5,6), (11,13,12), (7,8)(9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha^{2} & 1 & 0 & \alpha^{2} \\ 1 & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ \alpha & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & \alpha \\ 0 & \alpha & 0 & 0 \\ \alpha & \alpha & \alpha & 1 \\ \alpha & 0 & 0 & 0 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $C_4$ $\, \times\, $ $C_3$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_6$	$C_6$ . $C_2^4$	$(C_2^3\times C_6)$ . $C_2$	$(C_2\times C_6)$ . $C_2^3$	all 8
Aut. group:	$\Aut(C_{280})$	$\Aut(C_{312})$	$\Aut(C_{336})$	$\Aut(C_{360})$	all 5

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

Homology

Primary decomposition:	$C_{2}^{3} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$0$

Subgroups

There are 236 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3\times C_{12}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2^3\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3\times C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_4$
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

Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 96: $C_2^3\times C_{12}$

Order 48: $C_2^2\times C_{12}$ x 14, $C_2^3\times C_6$

Order 32: $C_2^3\times C_4$

Order 24: $C_2\times C_{12}$ x 28, $C_2^2\times C_6$ x 15

Order 16: $C_2^2\times C_4$ x 14, $C_2^4$

Order 12: $C_2\times C_6$ x 35, $C_{12}$ x 8

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 15

Order 6: $C_6$ x 15

Order 4: $C_2^2$ x 35, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $C_2^3\times C_{12}$

Order 48: $C_2^3\times C_6$, $C_2^2\times C_{12}$

Order 32: $C_2^3\times C_4$

Order 24: $C_2^2\times C_6$ x 2, $C_2\times C_{12}$

Order 16: $C_2^4$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 2, $C_{12}$

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $C_2^3\times C_{12}$ ($C_1$)

Order 48: $C_2^2\times C_{12}$ ($C_2$) x 14, $C_2^3\times C_6$ ($C_2$)

Order 32: $C_2^3\times C_4$ ($C_3$)

Order 24: $C_2\times C_{12}$ ($C_2^2$) x 28, $C_2^2\times C_6$ ($C_4$) x 8, $C_2^2\times C_6$ ($C_2^2$) x 7

Order 16: $C_2^2\times C_4$ ($C_6$) x 14, $C_2^4$ ($C_6$)

Order 12: $C_2\times C_6$ ($C_2\times C_4$) x 28, $C_{12}$ ($C_2^3$) x 8, $C_2\times C_6$ ($C_2^3$) x 7

Order 8: $C_2\times C_4$ ($C_2\times C_6$) x 28, $C_2^3$ ($C_{12}$) x 8, $C_2^3$ ($C_2\times C_6$) x 7

Order 6: $C_6$ ($C_2^2\times C_4$) x 14, $C_6$ ($C_2^4$)

Order 4: $C_2^2$ ($C_2\times C_{12}$) x 28, $C_4$ ($C_2^2\times C_6$) x 8, $C_2^2$ ($C_2^2\times C_6$) x 7

Order 3: $C_3$ ($C_2^3\times C_4$)

Order 2: $C_2$ ($C_2^2\times C_{12}$) x 14, $C_2$ ($C_2^3\times C_6$)

Order 1: $C_1$ ($C_2^3\times C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $C_2^3\times C_{12}$ ($C_1$)

Order 48: $C_2^3\times C_6$ ($C_2$), $C_2^2\times C_{12}$ ($C_2$)

Order 32: $C_2^3\times C_4$ ($C_3$)

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

Order 16: $C_2^4$ ($C_6$), $C_2^2\times C_4$ ($C_6$)

Order 12: $C_2\times C_6$ ($C_2^3$), $C_2\times C_6$ ($C_2\times C_4$), $C_{12}$ ($C_2^3$)

Order 8: $C_2^3$ ($C_2\times C_6$), $C_2^3$ ($C_{12}$), $C_2\times C_4$ ($C_2\times C_6$)

Order 6: $C_6$ ($C_2^4$), $C_6$ ($C_2^2\times C_4$)

Order 4: $C_2^2$ ($C_2\times C_{12}$), $C_2^2$ ($C_2^2\times C_6$), $C_4$ ($C_2^2\times C_6$)

Order 3: $C_3$ ($C_2^3\times C_4$)

Order 2: $C_2$ ($C_2^3\times C_6$), $C_2$ ($C_2^2\times C_{12}$)

Order 1: $C_1$ ($C_2^3\times C_{12}$)

Series

Derived series

$C_2^3\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_2^3\times C_{12}$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^3\times C_{12}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_{12}$

Supergroups

This group is a maximal subgroup of 90 larger groups in the database.

This group is a maximal quotient of 52 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.