Abstract group 1152.133925: $D_6^2.C_2^3$

• Label: 1152.133925
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $96$
• Rank: $5$

Group information

Description:	$D_6^2.C_2^3$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_2^7\times C_6^2).(D_4\times S_4).C_2$, of order \(1769472\)\(\medspace = 2^{16} \cdot 3^{3} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	175	8	464	248	256	1152
Conjugacy classes	1	31	2	48	30	32	144
Divisions	1	31	2	28	30	16	108
Autjugacy classes	1	6	1	5	3	2	18

Dimension	1	2	4	8
Irr. complex chars.	64	16	64	0	144
Irr. rational chars.	32	24	36	16	108

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$96$
Rank:	$5$
Inequivalent generating 5-tuples:	$341644800$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{4}=d^{6}=e^{12}=[a,b]=[a,c]=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(3,6)(7,8,10,9), (1,2)(3,4)(5,6), (7,9,10,8)(11,12)(13,14), (7,10)(8,9) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_4$ $\, \times\, $ $\SOPlus(4,2)$
Semidirect product:	$(C_{12}:D_6)$ $\,\rtimes\,$ $D_4$	$(C_4\times D_6^2)$ $\,\rtimes\,$ $C_2$	$(D_6^2:C_4)$ $\,\rtimes\,$ $C_2$	$(C_6^2:C_4)$ $\,\rtimes\,$ $D_4$	all 31
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6^2$ . $C_2^3$	$(S_3\times D_6)$ . $C_2^4$	$(C_6:S_3)$ . $C_2^5$	$(C_6:D_6)$ . $C_2^4$	all 32
Aut. group:	$\Aut(S_3^2:C_{16})$	$\Aut(C_8.\SOPlus(4,2))$	$\Aut(S_3^2:C_{20})$	$\Aut(C_5\times C_3^2:\SD_{16})$

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{4} \times C_{4} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 22388 subgroups in 3890 conjugacy classes, 799 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $\SOPlus(4,2)$
Commutator:	$G' \simeq$ $C_3:S_3$	$G/G' \simeq$ $C_2^4\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $D_6^2.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^3$
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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Subgroup information

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Classes of subgroups up to conjugation

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

Order 576: $D_6^2.C_2^2$ x 24, $C_4\times D_6^2$ x 2, $D_6^2:C_4$ x 2, $D_6^2:C_2^2$, $C_6^2:C_4^2$, $C_2\times C_6^2.D_4$

Order 288: $C_4\times \SOPlus(4,2)$ x 64, $C_2.D_6^2$ x 36, $C_6^2:D_4$ x 28, $C_6^2.D_4$ x 24, $C_2\times C_3^2:C_4^2$ x 12, $C_6^2.D_4$ x 12, $C_6^2.C_2^3$ x 2, $C_2.D_6^2$ x 2, $C_2.D_6^2$ x 2, $C_2\times D_6^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2.C_2^3$

Order 192: $C_{12}:C_2^4$ x 2

Order 144: $S_3^2:C_2^2$ x 112, $C_4\times S_3^2$ x 80, $D_6^2$ x 42, $S_3^2:C_4$ x 32, $C_6^2:C_4$ x 24, $C_{12}\times D_6$ x 24, $D_6.D_6$ x 24, $C_2^2.S_3^2$ x 18, $C_3^2:C_4^2$ x 16, $C_2.\SOPlus(4,2)$ x 16, $C_{12}:D_6$ x 12, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$, $C_2\times C_6\times C_{12}$, $C_6^2:C_4$

Order 128: $C_4^2:C_2^3$

Order 96: $C_{12}:C_2^3$ x 56, $C_2^3\times D_6$ x 2, $C_2^3\times C_{12}$ x 2, $C_6.C_2^4$ x 2

Order 72: $S_3\times D_6$ x 140, $\SOPlus(4,2)$ x 64, $C_2\times C_3^2:C_4$ x 44, $S_3\times C_{12}$ x 32, $C_6.D_6$ x 32, $C_6\times D_6$ x 28, $C_6.D_6$ x 20, $C_{12}:S_3$ x 16, $C_6:D_6$ x 14, $C_6:C_{12}$ x 12, $C_6\times C_{12}$ x 6, $C_6.D_6$ x 6, $C_2\times C_6^2$

Order 64: $C_4^2:C_2^2$ x 24, $C_2^4\times C_4$ x 2, $C_2^4:C_4$ x 2, $D_4\times C_2^3$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 48: $C_4\times D_6$ x 224, $C_2^2\times D_6$ x 60, $C_2^2\times C_{12}$ x 28, $C_6.C_2^3$ x 28, $C_2^3\times C_6$ x 2

Order 36: $S_3^2$ x 72, $C_6\times S_3$ x 56, $C_6:S_3$ x 28, $C_3^2:C_4$ x 12, $C_3:C_{12}$ x 8, $C_6^2$ x 7, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4

Order 32: $C_4\times D_4$ x 64, $C_2^3\times C_4$ x 45, $C_2^2\times D_4$ x 28, $C_2^3:C_4$ x 24, $C_2^2.D_4$ x 12, $C_2\times C_4^2$ x 12, $C_2^5$ x 2

Order 24: $C_2\times D_6$ x 280, $C_4\times S_3$ x 128, $C_2\times C_{12}$ x 56, $C_6:C_4$ x 56, $C_2^2\times C_6$ x 30

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

Order 16: $C_2^2\times C_4$ x 186, $C_2\times D_4$ x 112, $C_2^4$ x 45, $C_2^2:C_4$ x 32, $C_4:C_4$ x 16, $C_4^2$ x 16

Order 12: $D_6$ x 240, $C_2\times C_6$ x 70, $C_{12}$ x 16, $C_3:C_4$ x 16

Order 9: $C_3^2$

Order 8: $C_2^3$ x 183, $C_2\times C_4$ x 168, $D_4$ x 64

Order 6: $S_3$ x 32, $C_6$ x 30

Order 4: $C_2^2$ x 163, $C_4$ x 28

Order 3: $C_3$ x 2

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 576: $D_6^2:C_2^2$, $C_4\times D_6^2$, $C_6^2:C_4^2$, $C_2\times C_6^2.D_4$, $D_6^2:C_4$, $D_6^2.C_2^2$

Order 288: $C_2.D_6^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$, $C_2.D_6^2$, $C_2.D_6^2$, $C_2\times C_3^2:C_4^2$, $C_6^2.D_4$, $C_6^2.D_4$, $C_4\times \SOPlus(4,2)$, $C_2\times D_6^2$, $C_6^2.C_2^3$

Order 192: $C_{12}:C_2^4$

Order 144: $D_6^2$ x 4, $C_6^2:C_4$ x 4, $S_3^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_2^2.S_3^2$ x 2, $C_4\times S_3^2$ x 2, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_2\times C_6\times C_{12}$, $C_6^2:C_4$, $C_6^2:C_4$, $C_{12}\times D_6$, $D_6.D_6$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$, $S_3^2:C_4$

Order 128: $C_4^2:C_2^3$

Order 96: $C_{12}:C_2^3$ x 6, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 72: $S_3\times D_6$ x 5, $C_6:D_6$ x 4, $C_2\times C_3^2:C_4$ x 4, $C_6\times D_6$ x 2, $C_{12}:S_3$ x 2, $C_6.D_6$ x 2, $C_2\times C_6^2$, $\SOPlus(4,2)$, $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$, $S_3\times C_{12}$, $C_6.D_6$

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

Order 48: $C_4\times D_6$ x 8, $C_2^2\times D_6$ x 6, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_2^3\times C_6$

Order 36: $C_6:S_3$ x 5, $S_3^2$ x 3, $C_3^2:C_4$ x 2, $C_6^2$ x 2, $C_6\times S_3$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$

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

Order 24: $C_2\times D_6$ x 11, $C_4\times S_3$ x 6, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_2^2\times C_6$ x 3

Order 18: $C_3:S_3$ x 3, $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $C_2^2\times C_4$ x 15, $C_2^4$ x 6, $C_2\times D_4$ x 2, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$

Order 12: $D_6$ x 10, $C_2\times C_6$ x 4, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 12, $D_4$

Order 6: $S_3$ x 4, $C_6$ x 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 1152: $D_6^2.C_2^3$ ($C_1$)

Order 576: $D_6^2.C_2^2$ ($C_2$) x 24, $C_4\times D_6^2$ ($C_2$) x 2, $D_6^2:C_4$ ($C_2$) x 2, $D_6^2:C_2^2$ ($C_2$), $C_6^2:C_4^2$ ($C_2$), $C_2\times C_6^2.D_4$ ($C_2$)

Order 288: $C_4\times \SOPlus(4,2)$ ($C_2^2$) x 64, $C_2.D_6^2$ ($C_2^2$) x 24, $C_6^2.D_4$ ($C_2^2$) x 24, $C_6^2:D_4$ ($C_4$) x 16, $C_2\times C_3^2:C_4^2$ ($C_2^2$) x 12, $C_6^2.D_4$ ($C_2^2$) x 12, $C_6^2:D_4$ ($C_2^2$) x 12, $C_2.D_6^2$ ($C_2^2$) x 2, $C_2\times D_6^2$ ($C_2^2$) x 2, $C_2\times C_6^2:C_4$ ($C_2^2$) x 2, $C_6^2.C_2^3$ ($C_2^2$)

Order 144: $S_3^2:C_2^2$ ($C_2\times C_4$) x 96, $C_4\times S_3^2$ ($C_2^3$) x 32, $S_3^2:C_4$ ($C_2^3$) x 32, $D_6^2$ ($C_2\times C_4$) x 16, $S_3^2:C_2^2$ ($C_2^3$) x 16, $C_3^2:C_4^2$ ($C_2^3$) x 16, $C_2.\SOPlus(4,2)$ ($C_2^3$) x 16, $D_6^2$ ($C_2^3$) x 12, $C_6^2:C_4$ ($C_2^3$) x 12, $C_2^2.S_3^2$ ($C_2^3$) x 12, $C_6^2:C_4$ ($C_2\times C_4$) x 8, $C_{12}:D_6$ ($C_2^3$) x 6, $C_{12}:D_6$ ($D_4$) x 6, $C_6^2:C_2^2$ ($C_2^3$), $C_2\times C_6\times C_{12}$ ($D_4$), $C_6^2:C_4$ ($D_4$)

Order 72: $\SOPlus(4,2)$ ($C_2^2\times C_4$) x 64, $S_3\times D_6$ ($C_2^2\times C_4$) x 48, $C_2\times C_3^2:C_4$ ($C_2^2\times C_4$) x 24, $C_{12}:S_3$ ($C_2\times D_4$) x 12, $S_3\times D_6$ ($C_2^4$) x 8, $C_2\times C_3^2:C_4$ ($C_2^4$) x 8, $C_6.D_6$ ($C_2^4$) x 8, $C_6\times C_{12}$ ($C_2\times D_4$) x 6, $C_6.D_6$ ($C_2\times D_4$) x 6, $C_6:D_6$ ($D_4:C_2$) x 4, $C_6:D_6$ ($C_2^2\times C_4$) x 4, $C_{12}:S_3$ ($C_2^4$) x 4, $C_6:D_6$ ($C_2^4$) x 3, $C_6:D_6$ ($C_2\times D_4$) x 3, $C_2\times C_6^2$ ($C_2\times D_4$)

Order 36: $S_3^2$ ($C_2^3\times C_4$) x 16, $C_6:S_3$ ($C_4\times D_4$) x 12, $C_3^2:C_4$ ($C_2^3\times C_4$) x 8, $C_6:S_3$ ($D_4:C_2^2$) x 6, $C_6:S_3$ ($C_2^3\times C_4$) x 6, $C_3\times C_{12}$ ($C_2^2\times D_4$) x 4, $C_3^2:C_4$ ($C_2^2\times D_4$) x 4, $C_6^2$ ($C_4\times D_4$) x 4, $C_6^2$ ($C_2^2\times D_4$) x 3, $C_6:S_3$ ($C_2^2\times D_4$) x 3, $C_6:S_3$ ($C_2^5$)

Order 18: $C_3\times C_6$ ($C_4^2:C_2^2$) x 6, $C_3:S_3$ ($C_4^2:C_2^2$) x 6, $C_3\times C_6$ ($D_4\times C_2^3$), $C_3:S_3$ ($D_4:C_2^3$), $C_3:S_3$ ($C_2^4\times C_4$)

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

Order 9: $C_3^2$ ($C_4^2:C_2^3$)

Order 8: $C_2\times C_4$ ($S_3^2:C_2^2$) x 6, $C_2^3$ ($S_3^2:C_2^2$)

Order 4: $C_2^2$ ($C_4\times \SOPlus(4,2)$) x 4, $C_4$ ($C_6^2:D_4$) x 4, $C_2^2$ ($C_6^2:D_4$) x 3

Order 2: $C_2$ ($D_6^2.C_2^2$) x 6, $C_2$ ($D_6^2:C_2^2$)

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 576: $D_6^2:C_2^2$ ($C_2$), $C_4\times D_6^2$ ($C_2$), $C_6^2:C_4^2$ ($C_2$), $C_2\times C_6^2.D_4$ ($C_2$), $D_6^2:C_4$ ($C_2$), $D_6^2.C_2^2$ ($C_2$)

Order 288: $C_2\times C_6^2:C_4$ ($C_2^2$) x 2, $C_2.D_6^2$ ($C_2^2$), $C_2.D_6^2$ ($C_2^2$), $C_2\times C_3^2:C_4^2$ ($C_2^2$), $C_6^2.D_4$ ($C_2^2$), $C_6^2.D_4$ ($C_2^2$), $C_4\times \SOPlus(4,2)$ ($C_2^2$), $C_2\times D_6^2$ ($C_2^2$), $C_6^2:D_4$ ($C_2^2$), $C_6^2:D_4$ ($C_4$), $C_6^2.C_2^3$ ($C_2^2$)

Order 144: $C_6^2:C_4$ ($C_2^3$) x 2, $C_6^2:C_2^2$ ($C_2^3$), $D_6^2$ ($C_2^3$), $D_6^2$ ($C_2\times C_4$), $C_6^2:C_4$ ($C_2\times C_4$), $S_3^2:C_2^2$ ($C_2^3$), $S_3^2:C_2^2$ ($C_2\times C_4$), $C_2\times C_6\times C_{12}$ ($D_4$), $C_6^2:C_4$ ($D_4$), $C_{12}:D_6$ ($C_2^3$), $C_{12}:D_6$ ($D_4$), $C_2^2.S_3^2$ ($C_2^3$), $C_4\times S_3^2$ ($C_2^3$), $C_3^2:C_4^2$ ($C_2^3$), $C_2.\SOPlus(4,2)$ ($C_2^3$), $S_3^2:C_4$ ($C_2^3$)

Order 72: $C_2\times C_3^2:C_4$ ($C_2^4$) x 2, $C_2\times C_6^2$ ($C_2\times D_4$), $C_6:D_6$ ($C_2^4$), $C_6:D_6$ ($D_4:C_2$), $C_6:D_6$ ($C_2\times D_4$), $C_6:D_6$ ($C_2^2\times C_4$), $S_3\times D_6$ ($C_2^4$), $S_3\times D_6$ ($C_2^2\times C_4$), $C_2\times C_3^2:C_4$ ($C_2^2\times C_4$), $\SOPlus(4,2)$ ($C_2^2\times C_4$), $C_6\times C_{12}$ ($C_2\times D_4$), $C_6.D_6$ ($C_2\times D_4$), $C_{12}:S_3$ ($C_2^4$), $C_{12}:S_3$ ($C_2\times D_4$), $C_6.D_6$ ($C_2^4$)

Order 36: $C_3^2:C_4$ ($C_2^3\times C_4$), $C_3\times C_{12}$ ($C_2^2\times D_4$), $C_3^2:C_4$ ($C_2^2\times D_4$), $C_6^2$ ($C_2^2\times D_4$), $C_6^2$ ($C_4\times D_4$), $C_6:S_3$ ($C_2^5$), $C_6:S_3$ ($D_4:C_2^2$), $C_6:S_3$ ($C_2^2\times D_4$), $C_6:S_3$ ($C_2^3\times C_4$), $C_6:S_3$ ($C_4\times D_4$), $S_3^2$ ($C_2^3\times C_4$)

Order 18: $C_3\times C_6$ ($D_4\times C_2^3$), $C_3\times C_6$ ($C_4^2:C_2^2$), $C_3:S_3$ ($D_4:C_2^3$), $C_3:S_3$ ($C_2^4\times C_4$), $C_3:S_3$ ($C_4^2:C_2^2$)

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

Order 9: $C_3^2$ ($C_4^2:C_2^3$)

Order 8: $C_2^3$ ($S_3^2:C_2^2$), $C_2\times C_4$ ($S_3^2:C_2^2$)

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

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

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

Series

Derived series

$D_6^2.C_2^3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$D_6^2.C_2^3$

$\rhd$

$C_4\times D_6^2$

$\rhd$

$C_6^2.C_2^3$

$\rhd$

$C_6^2:C_2^2$

$\rhd$

$C_6:D_6$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$D_6^2.C_2^3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

See the $144 \times 144$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $108 \times 108$ rational character table.