Abstract group 1152.156064: $D_6.\GL(2,\mathbb{Z}/4)$

• Label: 1152.156064
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$D_6.\GL(2,\mathbb{Z}/4)$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^5.D_6^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	95	26	224	310	192	208	96	1152
Conjugacy classes	1	11	3	8	15	8	7	4	57
Divisions	1	11	3	8	12	4	7	2	48
Autjugacy classes	1	9	3	8	11	4	7	2	45

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	8	18	8	9	10	1	3	0	57
Irr. rational chars.	8	10	8	9	6	3	3	1	48

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$80640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	7	9	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{8}=c^{6}=d^{2}=e^{6}=[d,e]=1, a^{2}=b^{4}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(10,11), (1,2,4,7)(3,5,8,6)(13,15), (2,5)(3,8)(6,7), (1,3,4,8)(2,6,7,5), (1,4)(2,7)(3,8)(5,6), (13,14,15), (9,10,11), (12,13)(14,15), (12,14)(13,15)\rangle$
Direct product:	$S_3$ $\, \times\, $ $(A_4:\SD_{16})$
Semidirect product:	$(A_4:Q_8)$ $\,\rtimes\,$ $D_6$	$(A_4:C_8)$ $\,\rtimes\,$ $D_6$	$(S_3\times A_4)$ $\,\rtimes\,$ $\SD_{16}$ (2)	$A_4$ $\,\rtimes\,$ $(S_3\times \SD_{16})$	all 19
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(S_3\times D_4)$ . $S_4$	$D_4$ . $(S_3\times S_4)$	$C_4$ . $(D_6\times S_4)$	$(A_4\times D_6)$ . $D_4$	all 33

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 6886 subgroups in 734 conjugacy classes, 59 normal (53 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times \GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $C_{12}\times A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:C_2^3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $D_6.\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_6:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 1152: $D_6.\GL(2,\mathbb{Z}/4)$

Order 576: $S_3\times D_4\times A_4$, $S_3\times A_4:Q_8$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_8$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$, $S_3\times C_2^2:\SD_{16}$

Order 288: $C_2^3.C_6^2$, $A_4\times C_3:D_4$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $C_{12}.S_4$, $C_{12}.S_4$, $D_6.S_4$, $C_2^3.S_3^2$, $D_{12}.D_6$, $C_{12}.S_4$, $A_4:C_{24}$, $C_2\times A_4\times D_6$

Order 192: $A_4:\SD_{16}$ x 5, $D_6:\SD_{16}$ x 2, $D_6:\SD_{16}$ x 2, $D_6.D_8$ x 2, $D_6\times \SD_{16}$ x 2, $C_2\times A_4:C_8$, $C_3\times C_2^2:\SD_{16}$, $(C_2\times C_6):\SD_{16}$, $D_{12}.D_4$, $D_{12}.D_4$, $D_6.\OD_{16}$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4.D_6$, $S_3\times C_2^2:Q_8$

Order 144: $A_4\times D_6$ x 4, $C_{12}.D_6$, $C_{12}.D_6$, $C_6.D_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_2^2:C_6^2$, $C_{12}:D_6$, $C_{12}\times A_4$, $C_{12}.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$

Order 128: $C_2^3:\SD_{16}$

Order 96: $S_3\times \SD_{16}$ x 8, $D_4\times D_6$ x 6, $D_4\times A_4$ x 4, $A_4:Q_8$ x 4, $A_4:C_8$ x 3, $C_6:\SD_{16}$ x 3, $D_6.D_4$ x 2, $D_4:C_{12}$ x 2, $C_6.D_8$ x 2, $D_{12}:C_4$ x 2, $D_6:C_8$ x 2, $C_6\times \SD_{16}$ x 2, $D_{12}:C_4$ x 2, $Q_8:D_6$ x 2, $C_8:D_6$ x 2, $C_8\times D_6$ x 2, $D_6.D_4$, $C_2^3.D_6$, $C_2^2:C_{24}$, $C_{12}.D_4$, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^3:D_6$, $Q_8\times D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_{12}.D_4$, $D_6:Q_8$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:Q_8$

Order 72: $S_3\times A_4$ x 3, $C_6\times A_4$ x 2, $C_6\times D_6$, $D_4\times C_3^2$, $C_3^2:Q_8$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_6.D_6$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_2^2:\SD_{16}$ x 8, $C_2^2.D_8$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_2^3:C_8$, $D_4\times C_2^3$, $C_2^3:Q_8$

Order 48: $S_3\times D_4$ x 15, $C_2^2\times D_6$ x 10, $C_3:\SD_{16}$ x 9, $C_2^2\times A_4$ x 6, $C_4\times D_6$ x 6, $C_{24}:C_2$ x 4, $C_6\times D_4$ x 4, $C_6:D_4$ x 4, $S_3\times Q_8$ x 4, $S_3\times C_8$ x 4, $C_3\times \SD_{16}$ x 4, $Q_8:S_3$ x 4, $C_6:C_8$ x 3, $C_2\times D_{12}$ x 3, $C_6:Q_8$ x 3, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_6.D_4$ x 3, $C_2\times C_{24}$ x 2, $C_4:C_{12}$ x 2, $D_6:C_4$ x 2, $C_{12}:C_4$ x 2, $C_2^3\times C_6$, $C_6\times Q_8$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_2^2:C_{12}$, $C_6.D_4$

Order 36: $C_6\times S_3$ x 4, $C_3:C_{12}$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 16, $C_2^2\times D_4$ x 9, $D_4:C_4$ x 8, $C_2^2:Q_8$ x 6, $C_2^2:C_8$ x 4, $C_2^2\times C_8$ x 2, $C_2^2.D_4$ x 2, $C_2^5$, $C_2^2\times Q_8$, $C_2^3\times C_4$, $C_2^3:C_4$

Order 24: $C_2\times D_6$ x 32, $C_4\times S_3$ x 9, $C_3\times D_4$ x 9, $C_3:Q_8$ x 8, $C_2\times A_4$ x 7, $C_3:D_4$ x 6, $C_2^2\times C_6$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 5, $D_{12}$ x 5, $C_3:C_8$ x 5, $C_{24}$ x 2, $C_3\times Q_8$ x 2

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

Order 16: $C_2\times D_4$ x 25, $\SD_{16}$ x 16, $C_2^4$ x 11, $C_2\times C_8$ x 8, $C_2^2\times C_4$ x 8, $C_4:C_4$ x 7, $C_2\times Q_8$ x 7, $C_2^2:C_4$ x 4

Order 12: $D_6$ x 27, $C_2\times C_6$ x 15, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 37, $C_2\times C_4$ x 17, $D_4$ x 16, $Q_8$ x 6, $C_8$ x 4

Order 6: $C_6$ x 12, $S_3$ x 6

Order 4: $C_2^2$ x 36, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $D_6.\GL(2,\mathbb{Z}/4)$

Order 576: $S_3\times D_4\times A_4$, $S_3\times A_4:Q_8$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_8$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$, $S_3\times C_2^2:\SD_{16}$

Order 288: $C_2^3.C_6^2$, $A_4\times C_3:D_4$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $C_{12}.S_4$, $C_{12}.S_4$, $D_6.S_4$, $C_2^3.S_3^2$, $D_{12}.D_6$, $C_{12}.S_4$, $A_4:C_{24}$, $C_2\times A_4\times D_6$

Order 192: $A_4:\SD_{16}$ x 5, $D_6:\SD_{16}$ x 2, $D_6:\SD_{16}$ x 2, $D_6.D_8$ x 2, $D_6\times \SD_{16}$ x 2, $C_2\times A_4:C_8$, $C_3\times C_2^2:\SD_{16}$, $(C_2\times C_6):\SD_{16}$, $D_{12}.D_4$, $D_{12}.D_4$, $D_6.\OD_{16}$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4.D_6$, $S_3\times C_2^2:Q_8$

Order 144: $A_4\times D_6$ x 3, $C_{12}.D_6$, $C_{12}.D_6$, $C_6.D_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_2^2:C_6^2$, $C_{12}:D_6$, $C_{12}\times A_4$, $C_{12}.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$

Order 128: $C_2^3:\SD_{16}$

Order 96: $D_4\times D_6$ x 6, $S_3\times \SD_{16}$ x 6, $D_4\times A_4$ x 4, $A_4:Q_8$ x 4, $A_4:C_8$ x 3, $C_6:\SD_{16}$ x 3, $D_6.D_4$ x 2, $D_4:C_{12}$ x 2, $C_6.D_8$ x 2, $D_{12}:C_4$ x 2, $D_6:C_8$ x 2, $C_6\times \SD_{16}$ x 2, $D_{12}:C_4$ x 2, $Q_8:D_6$ x 2, $C_8:D_6$ x 2, $C_8\times D_6$ x 2, $D_6.D_4$, $C_2^3.D_6$, $C_2^2:C_{24}$, $C_{12}.D_4$, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^3:D_6$, $Q_8\times D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_{12}.D_4$, $D_6:Q_8$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:Q_8$

Order 72: $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_6\times D_6$, $D_4\times C_3^2$, $C_3^2:Q_8$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_6.D_6$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_2^2:\SD_{16}$ x 8, $C_2^2.D_8$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_2^3:C_8$, $D_4\times C_2^3$, $C_2^3:Q_8$

Order 48: $S_3\times D_4$ x 11, $C_2^2\times D_6$ x 8, $C_3:\SD_{16}$ x 8, $C_4\times D_6$ x 6, $C_2^2\times A_4$ x 5, $C_6\times D_4$ x 4, $C_6:D_4$ x 4, $S_3\times Q_8$ x 4, $S_3\times C_8$ x 4, $C_6:C_8$ x 3, $C_{24}:C_2$ x 3, $C_2\times D_{12}$ x 3, $C_6:Q_8$ x 3, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_3\times \SD_{16}$ x 3, $Q_8:S_3$ x 3, $C_6.D_4$ x 3, $C_2\times C_{24}$ x 2, $C_4:C_{12}$ x 2, $D_6:C_4$ x 2, $C_{12}:C_4$ x 2, $C_2^3\times C_6$, $C_6\times Q_8$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_2^2:C_{12}$, $C_6.D_4$

Order 36: $C_6\times S_3$ x 3, $C_3:C_{12}$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 14, $C_2^2\times D_4$ x 9, $D_4:C_4$ x 8, $C_2^2:Q_8$ x 6, $C_2^2:C_8$ x 4, $C_2^2\times C_8$ x 2, $C_2^2.D_4$ x 2, $C_2^5$, $C_2^2\times Q_8$, $C_2^3\times C_4$, $C_2^3:C_4$

Order 24: $C_2\times D_6$ x 21, $C_4\times S_3$ x 8, $C_3:Q_8$ x 8, $C_3\times D_4$ x 8, $C_2^2\times C_6$ x 6, $C_2\times A_4$ x 6, $C_2\times C_{12}$ x 5, $C_3:D_4$ x 5, $C_6:C_4$ x 5, $C_3:C_8$ x 5, $D_{12}$ x 4, $C_{24}$ x 2, $C_3\times Q_8$ x 2

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

Order 16: $C_2\times D_4$ x 21, $\SD_{16}$ x 12, $C_2^4$ x 9, $C_2\times C_8$ x 8, $C_2^2\times C_4$ x 8, $C_4:C_4$ x 7, $C_2\times Q_8$ x 7, $C_2^2:C_4$ x 4

Order 12: $D_6$ x 17, $C_2\times C_6$ x 14, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 26, $C_2\times C_4$ x 16, $D_4$ x 13, $Q_8$ x 6, $C_8$ x 4

Order 6: $C_6$ x 11, $S_3$ x 4

Order 4: $C_2^2$ x 26, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $D_6.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 576: $S_3\times D_4\times A_4$ ($C_2$), $S_3\times A_4:Q_8$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $S_3\times A_4:C_8$ ($C_2$)

Order 288: $C_2^3.C_6^2$ ($C_2^2$), $A_4\times D_{12}$ ($C_2^2$), $C_4\times S_3\times A_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $A_4:C_{24}$ ($C_2^2$)

Order 192: $A_4:\SD_{16}$ ($S_3$), $C_{12}:C_2^4$ ($S_3$)

Order 144: $A_4\times D_6$ ($D_4$), $C_{12}\times A_4$ ($C_2^3$), $A_4\times C_3:C_4$ ($D_4$)

Order 96: $A_4:C_8$ ($D_6$), $C_{12}:C_2^3$ ($D_6$), $C_2^2\times D_{12}$ ($D_6$), $C_{12}:C_2^3$ ($D_6$), $D_4\times A_4$ ($D_6$), $A_4:Q_8$ ($D_6$)

Order 72: $S_3\times A_4$ ($\SD_{16}$) x 2, $C_6\times A_4$ ($C_2\times D_4$)

Order 48: $C_2^2\times D_6$ ($C_3:D_4$), $C_2^2\times C_{12}$ ($C_2\times D_6$), $C_6.C_2^3$ ($C_3:D_4$), $S_3\times D_4$ ($S_4$), $C_4\times A_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($C_2\times \SD_{16}$)

Order 32: $C_2^2\times D_4$ ($S_3^2$)

Order 24: $C_2\times D_6$ ($C_3:\SD_{16}$) x 2, $D_{12}$ ($C_2\times S_4$), $C_4\times S_3$ ($C_2\times S_4$), $C_2^2\times C_6$ ($C_6:D_4$), $C_2\times A_4$ ($S_3\times D_4$), $C_3\times D_4$ ($C_2\times S_4$)

Order 16: $C_2^2\times C_4$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_6:\SD_{16}$), $D_6$ ($\GL(2,\mathbb{Z}/4)$), $A_4$ ($S_3\times \SD_{16}$), $C_{12}$ ($C_2^2\times S_4$), $C_3:C_4$ ($\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($D_6:D_6$), $D_4$ ($S_3\times S_4$)

Order 6: $S_3$ ($A_4:\SD_{16}$) x 2, $C_6$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($D_{12}.D_6$), $C_4$ ($D_6\times S_4$)

Order 3: $C_3$ ($C_4.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($S_3\times \GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($D_6.\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $D_6.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 576: $S_3\times D_4\times A_4$ ($C_2$), $S_3\times A_4:Q_8$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $S_3\times A_4:C_8$ ($C_2$)

Order 288: $C_2^3.C_6^2$ ($C_2^2$), $A_4\times D_{12}$ ($C_2^2$), $C_4\times S_3\times A_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$), $A_4:C_{24}$ ($C_2^2$)

Order 192: $A_4:\SD_{16}$ ($S_3$), $C_{12}:C_2^4$ ($S_3$)

Order 144: $A_4\times D_6$ ($D_4$), $C_{12}\times A_4$ ($C_2^3$), $A_4\times C_3:C_4$ ($D_4$)

Order 96: $A_4:C_8$ ($D_6$), $C_{12}:C_2^3$ ($D_6$), $C_2^2\times D_{12}$ ($D_6$), $C_{12}:C_2^3$ ($D_6$), $D_4\times A_4$ ($D_6$), $A_4:Q_8$ ($D_6$)

Order 72: $C_6\times A_4$ ($C_2\times D_4$), $S_3\times A_4$ ($\SD_{16}$)

Order 48: $C_2^2\times D_6$ ($C_3:D_4$), $C_2^2\times C_{12}$ ($C_2\times D_6$), $C_6.C_2^3$ ($C_3:D_4$), $S_3\times D_4$ ($S_4$), $C_4\times A_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($C_2\times \SD_{16}$)

Order 32: $C_2^2\times D_4$ ($S_3^2$)

Order 24: $D_{12}$ ($C_2\times S_4$), $C_4\times S_3$ ($C_2\times S_4$), $C_2^2\times C_6$ ($C_6:D_4$), $C_2\times D_6$ ($C_3:\SD_{16}$), $C_2\times A_4$ ($S_3\times D_4$), $C_3\times D_4$ ($C_2\times S_4$)

Order 16: $C_2^2\times C_4$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_6:\SD_{16}$), $D_6$ ($\GL(2,\mathbb{Z}/4)$), $A_4$ ($S_3\times \SD_{16}$), $C_{12}$ ($C_2^2\times S_4$), $C_3:C_4$ ($\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($D_6:D_6$), $D_4$ ($S_3\times S_4$)

Order 6: $C_6$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $S_3$ ($A_4:\SD_{16}$)

Order 4: $C_2^2$ ($D_{12}.D_6$), $C_4$ ($D_6\times S_4$)

Order 3: $C_3$ ($C_4.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($S_3\times \GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($D_6.\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$D_6.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$D_6.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_6.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_{12}.S_4$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_6.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

See the $57 \times 57$ 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.