Abstract group 96.162: $C_2^3:C_{12}$

• Label: 96.162
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_2^3:C_{12}$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^7:D_4$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 5, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	15	2	16	30	32	96
Conjugacy classes	1	11	2	8	22	16	60
Divisions	1	11	1	4	11	4	32
Autjugacy classes	1	4	1	1	4	1	12

Dimension	1	2	4
Irr. complex chars.	48	12	0	60
Irr. rational chars.	8	16	8	32

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$273$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{6}=[a,b]=[a,c]=[a,d]=[b,d]=[c,d]=1, c^{b}=cd^{3} \rangle$
Permutation group:	Degree $13$ $\langle(1,2,3,4)(5,6,7,8)(9,10), (2,4)(5,7)(6,8)(9,10), (9,10), (11,13,12), (1,3)(2,4), (1,3)(2,4)(5,7)(6,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ \alpha & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha^{2} & 1 & 0 & \alpha^{2} \\ 1 & 1 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & 0 & 0 & 0 \\ \alpha^{2} & \alpha^{2} & 0 & \alpha^{2} \\ \alpha^{2} & 0 & \alpha^{2} & 1 \\ 0 & 0 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_2^2:C_4)$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $C_{12}$	$(C_2^2\times C_6)$ $\,\rtimes\,$ $C_4$	$(C_2^2\times C_4)$ $\,\rtimes\,$ $C_6$	$(C_2^2\times C_{12})$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_6$	$(C_2\times C_6)$ . $D_4$	$C_6$ . $(C_2\times D_4)$	$C_2$ . $(C_6\times D_4)$	all 14

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

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 188 subgroups in 132 conjugacy classes, 76 normal (12 characteristic).

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

Special subgroups

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

Order 48: $C_2^2:C_{12}$ x 4, $C_2^2\times C_{12}$ x 2, $C_2^3\times C_6$

Order 32: $C_2^3:C_4$

Order 24: $C_2^2\times C_6$ x 11, $C_2\times C_{12}$ x 8

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

Order 12: $C_2\times C_6$ x 23, $C_{12}$ x 4

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

Order 6: $C_6$ x 11

Order 4: $C_2^2$ x 23, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

Order 32: $C_2^3:C_4$

Order 24: $C_2^2\times C_6$ x 4, $C_2\times C_{12}$ x 2

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

Order 12: $C_2\times C_6$ x 6, $C_{12}$

Order 8: $C_2^3$ x 4, $C_2\times C_4$ x 2

Order 6: $C_6$ x 4

Order 4: $C_2^2$ x 6, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 48: $C_2^2:C_{12}$ ($C_2$) x 4, $C_2^2\times C_{12}$ ($C_2$) x 2, $C_2^3\times C_6$ ($C_2$)

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

Order 24: $C_2\times C_{12}$ ($C_2^2$) x 4, $C_2^2\times C_6$ ($C_4$) x 4, $C_2^2\times C_6$ ($C_2^2$) x 3

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

Order 12: $C_2\times C_6$ ($C_2\times C_4$) x 6, $C_2\times C_6$ ($D_4$) x 4, $C_2\times C_6$ ($C_2^3$)

Order 8: $C_2^3$ ($C_{12}$) x 4, $C_2\times C_4$ ($C_2\times C_6$) x 4, $C_2^3$ ($C_2\times C_6$) x 3

Order 6: $C_6$ ($C_2^2:C_4$) x 4, $C_6$ ($C_2\times D_4$) x 2, $C_6$ ($C_2^2\times C_4$)

Order 4: $C_2^2$ ($C_2\times C_{12}$) x 6, $C_2^2$ ($C_3\times D_4$) x 4, $C_2^2$ ($C_2^2\times C_6$)

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

Order 2: $C_2$ ($C_2^2:C_{12}$) x 4, $C_2$ ($C_6\times D_4$) x 2, $C_2$ ($C_2^2\times C_{12}$)

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

Normal subgroups up to automorphism (quotient in parentheses)

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

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

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

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

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

Order 12: $C_2\times C_6$ ($C_2\times C_4$) x 2, $C_2\times C_6$ ($C_2^3$), $C_2\times C_6$ ($D_4$)

Order 8: $C_2^3$ ($C_2\times C_6$) x 2, $C_2^3$ ($C_{12}$), $C_2\times C_4$ ($C_2\times C_6$)

Order 6: $C_6$ ($C_2^2:C_4$), $C_6$ ($C_2\times D_4$), $C_6$ ($C_2^2\times C_4$)

Order 4: $C_2^2$ ($C_2\times C_{12}$) x 2, $C_2^2$ ($C_2^2\times C_6$), $C_2^2$ ($C_3\times D_4$)

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

Order 2: $C_2$ ($C_6\times D_4$), $C_2$ ($C_2^2\times C_{12}$), $C_2$ ($C_2^2:C_{12}$)

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

Series

Derived series

$C_2^3:C_{12}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^3:C_{12}$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^3:C_{12}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_2^3:C_{12}$

Supergroups

This group is a maximal subgroup of 125 larger groups in the database.

This group is a maximal quotient of 73 larger groups in the database.

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.