Abstract group 1152.157863: $C_2^3:D_6^2$

• Label: 1152.157863
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• Perm deg.: $13$
• Trans deg.: $72$
• Rank: $5$

Group information

Description:	$C_2^3:D_6^2$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(S_3\times S_4).C_2^6.\PSL(2,7)$, of order \(1548288\)\(\medspace = 2^{13} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$3$

This group is nonabelian, monomial (hence solvable), and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	319	26	192	518	96	1152
Conjugacy classes	1	47	3	16	45	8	120
Divisions	1	47	3	16	45	8	120
Autjugacy classes	1	7	3	2	7	1	21

Dimension	1	2	3	4	6
Irr. complex chars.	32	32	32	8	16	120
Irr. rational chars.	32	32	32	8	16	120

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$72$
Rank:	$5$
Inequivalent generating 5-tuples:	$379440000$

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:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{6}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(1,2)(3,4)(5,6)(8,9), (5,6)(11,13), (1,2)(3,4)(5,6), (1,3)(2,4)(5,6), (5,6), (11,12,13), (7,8,9), (10,11)(12,13), (10,12)(11,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & -1 & -1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $S_3$ $\, \times\, $ $S_4$
Semidirect product:	$C_2^5$ $\,\rtimes\,$ $S_3^2$	$C_2^3$ $\,\rtimes\,$ $D_6^2$	$(S_3\times A_4)$ $\,\rtimes\,$ $C_2^4$	$(D_6\times C_2^4)$ $\,\rtimes\,$ $S_3$	all 36
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(\OD_{16}:D_6)$	$\Aut(\OD_{16}:D_6)$	$\Aut(C_3\times C_{12}.C_2^3)$	$\Aut(C_6^2.C_2^3)$	all 54

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{11}$
Commutator length:	$1$

Subgroups

There are 70814 subgroups in 10388 conjugacy classes, 607 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $S_3\times S_4$
Commutator:	$G' \simeq$ $C_3\times A_4$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^3:D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_6$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_2^3:D_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_6$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1152: $C_2^3:D_6^2$

Order 576: $C_2^2:D_6^2$ x 28, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$

Order 384: $C_2^4\times S_4$, $C_{12}:C_2^5$

Order 288: $D_6\times S_4$ x 112, $C_2\times A_4\times D_6$ x 14, $C_2\times C_6:S_4$ x 14, $C_2\times C_6\times S_4$ x 14, $C_2^3:C_6^2$, $C_2\times D_6^2$

Order 192: $C_{12}:C_2^4$ x 56, $C_2^3\times S_4$ x 31, $D_6\times C_2^4$ x 2, $D_6:C_2^4$ x 2, $A_4\times C_2^4$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$

Order 144: $S_3\times S_4$ x 64, $D_6^2$ x 28, $A_4\times D_6$ x 28, $C_6:S_4$ x 28, $C_6\times S_4$ x 28, $C_2^2:C_6^2$ x 7, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$

Order 128: $D_4\times C_2^4$

Order 96: $D_4\times D_6$ x 448, $C_2^2\times S_4$ x 154, $C_2^3\times D_6$ x 76, $C_2^3:D_6$ x 56, $C_{12}:C_2^3$ x 28, $C_2^2\times D_{12}$ x 28, $C_{12}:C_2^3$ x 28, $C_2^3\times A_4$ x 16, $C_2^4\times C_6$ x 2, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 72: $S_3\times D_6$ x 112, $C_6\times D_6$ x 28, $C_6:D_6$ x 14, $S_3\times A_4$ x 8, $C_3:S_4$ x 8, $C_3\times S_4$ x 8, $C_6\times A_4$ x 7, $C_2\times C_6^2$

Order 64: $D_4\times C_2^3$ x 60, $C_2^6$ x 2, $C_2^4\times C_4$

Order 48: $C_2^2\times D_6$ x 580, $S_3\times D_4$ x 512, $C_6:D_4$ x 224, $C_2\times S_4$ x 148, $C_6\times D_4$ x 112, $C_2\times D_{12}$ x 112, $C_4\times D_6$ x 112, $C_2^2\times A_4$ x 42, $C_2^3\times C_6$ x 38, $C_2^2\times C_{12}$ x 14, $C_6.C_2^3$ x 14

Order 36: $S_3^2$ x 64, $C_6\times S_3$ x 56, $C_6:S_3$ x 28, $C_6^2$ x 7, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 560, $C_2^5$ x 77, $C_2^3\times C_4$ x 30

Order 24: $C_2\times D_6$ x 1080, $C_2^2\times C_6$ x 150, $C_3:D_4$ x 128, $D_{12}$ x 64, $C_4\times S_3$ x 64, $C_3\times D_4$ x 64, $C_2\times C_{12}$ x 28, $C_6:C_4$ x 28, $S_4$ x 24, $C_2\times A_4$ x 22

Order 18: $C_3\times S_3$ x 16, $C_3:S_3$ x 8, $C_3\times C_6$ x 7

Order 16: $C_2\times D_4$ x 960, $C_2^4$ x 586, $C_2^2\times C_4$ x 140

Order 12: $D_6$ x 496, $C_2\times C_6$ x 156, $C_{12}$ x 8, $C_3:C_4$ x 8, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 1060, $D_4$ x 256, $C_2\times C_4$ x 120

Order 6: $S_3$ x 48, $C_6$ x 45

Order 4: $C_2^2$ x 462, $C_4$ x 16

Order 3: $C_3$ x 3

Order 2: $C_2$ x 47

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $C_2^3:D_6^2$

Order 576: $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $C_2^2:D_6^2$

Order 384: $C_2^4\times S_4$, $C_{12}:C_2^5$

Order 288: $C_2^3:C_6^2$, $C_2\times D_6^2$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $D_6\times S_4$

Order 192: $C_2^3\times S_4$ x 4, $D_6\times C_2^4$ x 2, $D_6:C_2^4$ x 2, $C_{12}:C_2^4$ x 2, $A_4\times C_2^4$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$

Order 144: $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $D_6^2$, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $S_3\times S_4$

Order 128: $D_4\times C_2^4$

Order 96: $C_2^3\times D_6$ x 10, $C_2^2\times S_4$ x 4, $C_2^3\times A_4$ x 3, $C_2^3:D_6$ x 3, $C_2^4\times C_6$ x 2, $C_{12}:C_2^3$ x 2, $D_4\times D_6$ x 2, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 72: $C_6\times D_6$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $C_6\times A_4$, $S_3\times D_6$, $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 64: $D_4\times C_2^3$ x 6, $C_2^6$ x 2, $C_2^4\times C_4$

Order 48: $C_2^2\times D_6$ x 20, $C_2^3\times C_6$ x 7, $C_2\times S_4$ x 4, $C_2^2\times A_4$ x 3, $C_6:D_4$ x 3, $C_6\times D_4$ x 2, $S_3\times D_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $C_6\times S_3$ x 2, $C_6^2$, $C_6:S_3$, $C_3\times A_4$, $S_3^2$

Order 32: $C_2^5$ x 11, $C_2^2\times D_4$ x 8, $C_2^3\times C_4$ x 3

Order 24: $C_2\times D_6$ x 20, $C_2^2\times C_6$ x 12, $C_3:D_4$ x 3, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2^4$ x 22, $C_2\times D_4$ x 8, $C_2^2\times C_4$ x 3

Order 12: $D_6$ x 15, $C_2\times C_6$ x 12, $A_4$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 25, $D_4$ x 6, $C_2\times C_4$ x 3

Order 6: $C_6$ x 7, $S_3$ x 6

Order 4: $C_2^2$ x 20, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $C_2^3:D_6^2$ ($C_1$)

Order 576: $C_2^2:D_6^2$ ($C_2$) x 28, $C_2^2\times A_4\times D_6$ ($C_2$), $C_2^2\times C_6:S_4$ ($C_2$), $C_2^2\times C_6\times S_4$ ($C_2$)

Order 288: $D_6\times S_4$ ($C_2^2$) x 112, $C_2\times A_4\times D_6$ ($C_2^2$) x 14, $C_2\times C_6:S_4$ ($C_2^2$) x 14, $C_2\times C_6\times S_4$ ($C_2^2$) x 14, $C_2^3:C_6^2$ ($C_2^2$)

Order 192: $D_6\times C_2^4$ ($S_3$), $C_2^3\times S_4$ ($S_3$)

Order 144: $S_3\times S_4$ ($C_2^3$) x 64, $A_4\times D_6$ ($C_2^3$) x 28, $C_6:S_4$ ($C_2^3$) x 28, $C_6\times S_4$ ($C_2^3$) x 28, $C_2^2:C_6^2$ ($C_2^3$) x 7

Order 96: $C_2^3\times D_6$ ($D_6$) x 14, $C_2^2\times S_4$ ($D_6$) x 14, $C_2^4\times C_6$ ($D_6$), $C_2^3\times A_4$ ($D_6$)

Order 72: $S_3\times A_4$ ($C_2^4$) x 8, $C_3:S_4$ ($C_2^4$) x 8, $C_3\times S_4$ ($C_2^4$) x 8, $C_6\times A_4$ ($C_2^4$) x 7

Order 48: $C_2^2\times D_6$ ($C_2\times D_6$) x 28, $C_2\times S_4$ ($C_2\times D_6$) x 28, $C_2^3\times C_6$ ($C_2\times D_6$) x 7, $C_2^2\times A_4$ ($C_2\times D_6$) x 7, $C_2^2\times D_6$ ($S_4$)

Order 36: $C_3\times A_4$ ($C_2^5$)

Order 32: $C_2^5$ ($S_3^2$)

Order 24: $C_2\times D_6$ ($C_2\times S_4$) x 14, $C_2\times D_6$ ($C_2^2\times D_6$) x 8, $S_4$ ($C_2^2\times D_6$) x 8, $C_2^2\times C_6$ ($C_2^2\times D_6$) x 7, $C_2\times A_4$ ($C_2^2\times D_6$) x 7, $C_2^2\times C_6$ ($C_2\times S_4$)

Order 16: $C_2^4$ ($S_3\times D_6$) x 7

Order 12: $D_6$ ($C_2^2\times S_4$) x 28, $C_2\times C_6$ ($C_2^2\times S_4$) x 7, $C_2\times C_6$ ($C_2^3\times D_6$), $A_4$ ($C_2^3\times D_6$)

Order 8: $C_2^3$ ($D_6^2$) x 7, $C_2^3$ ($S_3\times S_4$)

Order 6: $S_3$ ($C_2^3\times S_4$) x 8, $C_6$ ($C_2^3\times S_4$) x 7

Order 4: $C_2^2$ ($D_6\times S_4$) x 7, $C_2^2$ ($C_2\times D_6^2$)

Order 3: $C_3$ ($C_2^4\times S_4$)

Order 2: $C_2$ ($C_2^2:D_6^2$) x 7

Order 1: $C_1$ ($C_2^3:D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $C_2^3:D_6^2$ ($C_1$)

Order 576: $C_2^2\times A_4\times D_6$ ($C_2$), $C_2^2\times C_6:S_4$ ($C_2$), $C_2^2\times C_6\times S_4$ ($C_2$), $C_2^2:D_6^2$ ($C_2$)

Order 288: $C_2^3:C_6^2$ ($C_2^2$), $C_2\times A_4\times D_6$ ($C_2^2$), $C_2\times C_6:S_4$ ($C_2^2$), $C_2\times C_6\times S_4$ ($C_2^2$), $D_6\times S_4$ ($C_2^2$)

Order 192: $D_6\times C_2^4$ ($S_3$), $C_2^3\times S_4$ ($S_3$)

Order 144: $C_2^2:C_6^2$ ($C_2^3$), $A_4\times D_6$ ($C_2^3$), $C_6:S_4$ ($C_2^3$), $C_6\times S_4$ ($C_2^3$), $S_3\times S_4$ ($C_2^3$)

Order 96: $C_2^4\times C_6$ ($D_6$), $C_2^3\times D_6$ ($D_6$), $C_2^3\times A_4$ ($D_6$), $C_2^2\times S_4$ ($D_6$)

Order 72: $C_6\times A_4$ ($C_2^4$), $S_3\times A_4$ ($C_2^4$), $C_3:S_4$ ($C_2^4$), $C_3\times S_4$ ($C_2^4$)

Order 48: $C_2^3\times C_6$ ($C_2\times D_6$), $C_2^2\times D_6$ ($C_2\times D_6$), $C_2^2\times D_6$ ($S_4$), $C_2^2\times A_4$ ($C_2\times D_6$), $C_2\times S_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($C_2^5$)

Order 32: $C_2^5$ ($S_3^2$)

Order 24: $C_2^2\times C_6$ ($C_2^2\times D_6$), $C_2^2\times C_6$ ($C_2\times S_4$), $C_2\times D_6$ ($C_2^2\times D_6$), $C_2\times D_6$ ($C_2\times S_4$), $C_2\times A_4$ ($C_2^2\times D_6$), $S_4$ ($C_2^2\times D_6$)

Order 16: $C_2^4$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_2^3\times D_6$), $C_2\times C_6$ ($C_2^2\times S_4$), $D_6$ ($C_2^2\times S_4$), $A_4$ ($C_2^3\times D_6$)

Order 8: $C_2^3$ ($D_6^2$), $C_2^3$ ($S_3\times S_4$)

Order 6: $C_6$ ($C_2^3\times S_4$), $S_3$ ($C_2^3\times S_4$)

Order 4: $C_2^2$ ($C_2\times D_6^2$), $C_2^2$ ($D_6\times S_4$)

Order 3: $C_3$ ($C_2^4\times S_4$)

Order 2: $C_2$ ($C_2^2:D_6^2$)

Order 1: $C_1$ ($C_2^3:D_6^2$)

Series

Derived series

$C_2^3:D_6^2$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3:D_6^2$

$\rhd$

$C_2^2\times C_6\times S_4$

$\rhd$

$C_2\times C_6\times S_4$

$\rhd$

$C_6\times S_4$

$\rhd$

$C_3\times S_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^3:D_6^2$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 23 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $120 \times 120$ rational character table. Alternatively, you may search for characters of this group with desired properties.