Abstract group 4608.ss: $A_4\times D_4^2:S_3$

• Label: 4608.ss
• Order: \( 2^{9} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$A_4\times D_4^2:S_3$
Order:	\(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_5^4:(C_2^2\times C_4)$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$ 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	8	12	24
Elements	1	239	26	592	1318	192	1856	384	4608
Conjugacy classes	1	21	5	16	81	2	52	2	180
Divisions	1	21	3	16	44	2	27	1	115
Autjugacy classes	1	13	3	10	22	2	14	1	66

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	24	66	8	42	22	3	14	0	1	180
Irr. rational chars.	8	18	8	34	10	19	12	1	5	115

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{6}=b^{2}=c^{4}=d^{4}=e^{2}=f^{12}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3,2)(4,7,6)(8,15)(12,14), (4,5,7), (1,2,3)(4,7,5)(11,13)(12,14), (4,5) \!\cdots\! \rangle$
Direct product:	$A_4$ $\, \times\, $ $(D_4^2:S_3)$
Semidirect product:	$(A_4\times D_4^2)$ $\,\rtimes\,$ $S_3$	$D_4^2$ $\,\rtimes\,$ $(S_3\times A_4)$	$(C_2^6:D_6)$ $\,\rtimes\,$ $C_6$	$C_2^6$ $\,\rtimes\,$ $(C_6\wr C_2)$	all 55
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^6:C_6)$ . $D_6$	$(A_4\times C_4:D_4)$ . $D_6$	$C_2^4$ . $(C_6^2:D_4)$	$(C_4:D_4)$ . $(A_4\times D_6)$	all 42

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 80650 subgroups in 6436 conjugacy classes, 135 normal (69 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5.(C_6\times D_6)$
Commutator:	$G' \simeq$ $C_{12}:C_2^4$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2\times D_4$	$G/\Phi \simeq$ $C_2\times A_4\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6\times D_4^2$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $A_4\times D_4^2:S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3\times C_2\wr C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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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 4608: $A_4\times D_4^2:S_3$

Order 2304: $A_4\times C_2^4:D_6$ x 2, $(C_2^5\times C_6):C_{12}$ x 2, $C_6.(C_2\times D_4\times A_4).C_2$, $A_4\times D_{12}:D_4$, $C_2^6:C_6^2$

Order 1536: $A_4\times C_2\wr D_4$, $C_2\times D_4^2:D_6$

Order 1152: $C_2^5:C_6^2$ x 3, $A_4\times C_2^4:S_3$ x 2, $(C_2^3\times C_{12}):C_{12}$ x 2, $A_4\times C_2^2.D_{12}$ x 2, $C_2^5:C_6^2$ x 2, $C_2^5.C_6^2$ x 2, $C_{12}^2:D_4$, $A_4\times D_4:D_6$, $A_4\times D_{12}:C_2^2$, $A_4\times C_{12}.D_4$, $A_4\times D_{12}:C_4$, $C_2^5.C_6^2$, $C_2^5.C_6^2$

Order 768: $D_4^2:D_6$ x 8, $(C_2^5\times C_6):C_4$ x 2, $A_4\times D_4^2$ x 2, $A_4\times C_2\wr C_2^2$ x 2, $C_2^6:C_{12}$ x 2, $C_2^6:D_6$ x 2, $C_2\times C_6\times D_4^2$, $(C_6\times D_4).C_2^4$, $A_4\times D_4:D_4$, $A_4\times C_4^2:C_4$, $C_2^2\times D_{12}:D_4$

Order 576: $C_2^4:C_6^2$ x 17, $C_2^4.C_6^2$ x 5, $C_6:D_4\times A_4$ x 4, $A_4\times C_6.D_4$ x 4, $C_2^4:C_6^2$ x 3, $C_2^4.C_6^2$ x 2, $C_3\times C_2^4:D_6$ x 2, $(C_2^3\times C_6):C_{12}$ x 2, $A_4\times D_4:S_3$, $S_3\times D_4\times A_4$, $A_4\times D_{12}:C_2$, $C_3^2\times D_4^2$, $C_2^4.C_6^2$, $C_2^2:C_{12}^2$, $A_4\times C_3:\SD_{16}$, $A_4\times C_3:D_8$, $A_4\times C_3:\OD_{16}$, $C_{12}^2:C_2^2$, $C_{12}^2:C_4$

Order 512: $D_4^2:C_2^3$

Order 384: $D_4^2:S_3$ x 22, $C_2^5:D_6$ x 12, $C_6\times D_4^2$ x 10, $C_2^6:C_6$ x 9, $C_6:C_2\wr C_4$ x 8, $C_2^6:C_6$ x 6, $A_4\times C_4:D_4$ x 6, $C_2\times D_{12}:D_4$ x 6, $C_2^5:C_{12}$ x 4, $(C_6\times D_4).D_4$ x 4, $C_2^6:C_6$ x 3, $C_4\times D_4\times A_4$ x 3, $C_{12}:C_2^5$ x 2, $C_2^2\times C_{12}:D_4$ x 2, $C_2^5:D_6$ x 2, $A_4\times C_4:D_4$ x 2, $C_2^5.D_6$ x 2, $C_2^4.D_{12}$ x 2, $A_4\times C_4\wr C_2$, $C_2^5.C_{12}$, $D_4^2:C_6$, $D_{12}:C_2^4$, $A_4\times D_4:C_2^2$, $C_2^2\times C_{12}:D_4$, $C_2^2\times D_4\times C_{12}$, $D_{12}:C_2^4$, $A_4\times D_8:C_2$, $C_2^2\times C_{12}.D_4$, $C_2\times C_4^2:D_6$

Order 288: $C_2^3.C_6^2$ x 18, $C_2^3:C_6^2$ x 16, $C_2^3.C_6^2$ x 8, $A_4\times C_3:D_4$ x 7, $C_6:C_4\times A_4$ x 4, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_2\times A_4\times D_6$ x 2, $D_4\times C_6^2$ x 2, $C_6^2.C_2^3$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $A_4\times C_3:Q_8$, $C_{12}^2:C_2$, $C_{12}^2:C_2$, $C_6^2.D_4$, $A_4\times C_3:C_8$, $C_3\times C_{12}.D_4$, $C_{12}\wr C_2$

Order 256: $D_4^2:C_2^2$ x 8, $C_2^5:D_4$ x 2, $C_2^6:C_4$ x 2, $C_2^2\times D_4^2$, $C_4^2:C_2^4$, $C_2^5.D_4$

Order 192: $C_2^5:C_6$ x 54, $C_3\times D_4^2$ x 48, $C_{12}:C_2^4$ x 41, $C_2^4:D_6$ x 28, $C_2^5:C_6$ x 20, $C_2^4:C_{12}$ x 17, $C_2\times C_{12}:D_4$ x 16, $D_{12}:D_4$ x 14, $(C_2^3\times C_6):C_4$ x 12, $C_2^4:D_6$ x 12, $C_2^3.D_{12}$ x 8, $C_2\times D_4\times C_{12}$ x 8, $A_4\times C_2^4$ x 7, $D_{12}:C_2^3$ x 7, $C_2^4.D_6$ x 6, $C_2^4:C_{12}$ x 6, $C_2\times C_{12}:D_4$ x 6, $D_{12}:C_2^3$ x 6, $(C_4\times C_{12}):C_4$ x 6, $C_2^4:C_{12}$ x 5, $C_4^2:D_6$ x 4, $D_6:C_2^4$ x 4, $C_2^4.D_6$ x 4, $A_4\times C_4:C_4$ x 3, $C_2\times C_{12}.D_4$ x 3, $C_2^5\times C_6$ x 3, $A_4\times C_4^2$ x 2, $C_3\times C_2\wr C_2^2$ x 2, $C_2^4:C_{12}$ x 2, $C_2^4\times C_{12}$ x 2, $A_4\times D_4:C_2$ x 2, $C_3\times D_4:D_4$, $C_4^2:C_{12}$, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $D_{12}:C_2^3$, $C_2^2\times C_4:C_{12}$, $C_2^2\times C_4\times C_{12}$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $C_{12}.C_2^4$, $A_4\times \SD_{16}$, $A_4\times D_8$, $A_4\times \OD_{16}$

Order 144: $C_2^2:C_6^2$ x 26, $C_4:C_6^2$ x 17, $C_{12}\times A_4$ x 5, $C_6^2:C_4$ x 5, $C_6^2:C_4$ x 4, $C_6^2:C_2^2$ x 4, $C_2^2\times C_6^2$ x 3, $A_4\times D_6$ x 3, $A_4\times C_3:C_4$ x 3, $C_2\times C_6\times C_{12}$ x 2, $C_{12}.D_6$, $D_{12}:C_6$, $C_{12}.C_{12}$, $C_{12}.D_6$, $C_{12}:D_6$, $D_{12}:C_6$, $C_{12}:C_{12}$, $C_{12}^2$

Order 128: $C_2\wr D_4$ x 22, $C_2^4:D_4$ x 12, $C_2\times D_4^2$ x 10, $C_2^5:C_4$ x 8, $C_4^2:C_2^3$ x 6, $C_2^4:D_4$ x 5, $C_2^4.D_4$ x 4, $C_2^5:C_4$ x 4, $D_4\times C_2^4$ x 2, $C_2^4:D_4$ x 2, $D_4:C_2^4$, $C_8:C_2^4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $\OD_{16}:C_2^3$

Order 96: $C_{12}:C_2^3$ x 249, $D_4\times A_4$ x 60, $C_2^4:C_6$ x 59, $C_{12}:D_4$ x 50, $C_2^3\times A_4$ x 46, $C_2^4\times C_6$ x 37, $C_2^3:D_6$ x 31, $C_2^3:C_{12}$ x 28, $C_2^4:S_3$ x 28, $D_4:D_6$ x 26, $D_4\times C_{12}$ x 25, $C_2^3\times C_{12}$ x 20, $C_2^3:C_{12}$ x 18, $C_{12}:D_4$ x 16, $C_2^3.D_6$ x 14, $D_{12}:C_2^2$ x 14, $C_2^2.D_{12}$ x 12, $C_2^3.D_6$ x 8, $D_4:D_6$ x 8, $D_4\times D_6$ x 8, $D_{12}:C_2^2$ x 6, $D_{12}:C_4$ x 6, $C_2^3:C_{12}$ x 4, $C_{12}.D_4$ x 4, $C_6.C_2^4$ x 4, $C_2\times C_4:C_{12}$ x 4, $C_6:\SD_{16}$ x 4, $C_6:D_8$ x 4, $C_2\times C_4\times C_{12}$ x 3, $C_6:\OD_{16}$ x 3, $C_2^3\times D_6$ x 2, $C_8\times A_4$, $D_4:C_{12}$, $\OD_{16}:C_6$, $C_6.C_2^4$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_6.C_2^4$, $Q_8\times A_4$, $D_8:C_6$, $C_{12}.C_2^3$

Order 72: $D_4\times C_3^2$ x 18, $C_2\times C_6^2$ x 16, $C_6\times A_4$ x 9, $C_6\times C_{12}$ x 8, $C_6\wr C_2$ x 7, $C_6:C_{12}$ x 4, $C_6\times D_6$ x 2, $S_3\times A_4$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3:C_{24}$

Order 64: $D_4\times C_2^3$ x 46, $D_4^2$ x 46, $C_2^3:D_4$ x 32, $C_2\wr C_2^2$ x 28, $C_2^3:D_4$ x 16, $C_2^4:C_4$ x 14, $D_4:D_4$ x 14, $C_2\wr C_4$ x 12, $C_2^4:C_4$ x 9, $C_4^2:C_2^2$ x 8, $D_4:C_2^3$ x 7, $C_4^2:C_4$ x 6, $C_8:C_2^3$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 4, $\OD_{16}:C_2^2$ x 3, $C_2^6$ x 3, $D_4:C_2^3$ x 2, $C_2^4\times C_4$ x 2, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 48: $C_6\times D_4$ x 424, $C_2^3\times C_6$ x 198, $C_6:D_4$ x 84, $C_2^2\times A_4$ x 74, $C_2^2\times C_{12}$ x 68, $C_2^2:C_{12}$ x 43, $D_4:S_3$ x 22, $S_3\times D_4$ x 22, $C_6.D_4$ x 20, $C_4\times A_4$ x 17, $C_6.C_2^3$ x 15, $D_{12}:C_2$ x 14, $C_2^2\times D_6$ x 11, $C_4:C_{12}$ x 9, $C_4\times C_{12}$ x 6, $C_3:\SD_{16}$ x 6, $C_3:D_8$ x 6, $C_4\times D_6$ x 4, $C_3:\OD_{16}$ x 4, $C_2\times D_{12}$ x 3, $C_6:Q_8$ x 3, $C_6:C_8$ x 2, $D_4:C_6$ x 2, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_3\times \OD_{16}$

Order 36: $C_6^2$ x 26, $C_3\times C_{12}$ x 5, $C_3:C_{12}$ x 3, $C_6\times S_3$ x 3, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 283, $C_2^2\wr C_2$ x 78, $C_4:D_4$ x 44, $C_2^5$ x 39, $C_2^3:C_4$ x 32, $D_4:C_2^2$ x 26, $C_2^3\times C_4$ x 25, $C_4\times D_4$ x 22, $C_2^3:C_4$ x 20, $D_4:C_2^2$ x 14, $D_8:C_2$ x 14, $C_4:D_4$ x 14, $C_4\wr C_2$ x 6, $\OD_{16}:C_2$ x 4, $C_2\times \SD_{16}$ x 4, $C_2\times D_8$ x 4, $C_2^2.D_4$ x 4, $C_2\times \OD_{16}$ x 3, $C_2\times C_4^2$ x 3, $C_2^2\times Q_8$, $C_2^2\times C_8$

Order 24: $C_2^2\times C_6$ x 357, $C_3\times D_4$ x 166, $C_2\times C_{12}$ x 80, $C_3:D_4$ x 42, $C_2\times A_4$ x 25, $C_2\times D_6$ x 24, $C_6:C_4$ x 22, $C_4\times S_3$ x 6, $D_{12}$ x 4, $C_3:Q_8$ x 4, $C_3:C_8$ x 2, $C_{24}$, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 9, $C_3\times S_3$

Order 16: $C_2\times D_4$ x 479, $C_2^4$ x 202, $C_2^2\times C_4$ x 81, $C_2^2:C_4$ x 46, $D_4:C_2$ x 36, $\SD_{16}$ x 6, $D_8$ x 6, $C_4:C_4$ x 6, $\OD_{16}$ x 4, $C_4^2$ x 4, $C_2\times Q_8$ x 3, $C_2\times C_8$ x 2

Order 12: $C_2\times C_6$ x 217, $C_{12}$ x 27, $D_6$ x 14, $C_3:C_4$ x 6, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 335, $D_4$ x 152, $C_2\times C_4$ x 80, $Q_8$ x 4, $C_8$ x 2

Order 6: $C_6$ x 44, $S_3$ x 2

Order 4: $C_2^2$ x 157, $C_4$ x 16

Order 3: $C_3$ x 3

Order 2: $C_2$ x 21

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4608: $A_4\times D_4^2:S_3$

Order 2304: $C_6.(C_2\times D_4\times A_4).C_2$, $A_4\times C_2^4:D_6$, $A_4\times D_{12}:D_4$, $(C_2^5\times C_6):C_{12}$, $C_2^6:C_6^2$

Order 1536: $A_4\times C_2\wr D_4$, $C_2\times D_4^2:D_6$

Order 1152: $C_2^5:C_6^2$ x 2, $C_{12}^2:D_4$, $A_4\times D_4:D_6$, $A_4\times C_2^4:S_3$, $A_4\times D_{12}:C_2^2$, $A_4\times C_{12}.D_4$, $(C_2^3\times C_{12}):C_{12}$, $A_4\times C_2^2.D_{12}$, $A_4\times D_{12}:C_4$, $C_2^5:C_6^2$, $C_2^5.C_6^2$, $C_2^5.C_6^2$, $C_2^5.C_6^2$

Order 768: $D_4^2:D_6$ x 6, $A_4\times D_4^2$ x 2, $C_2\times C_6\times D_4^2$, $(C_6\times D_4).C_2^4$, $(C_2^5\times C_6):C_4$, $A_4\times C_2\wr C_2^2$, $A_4\times D_4:D_4$, $A_4\times C_4^2:C_4$, $C_2^6:C_{12}$, $C_2^6:D_6$, $C_2^2\times D_{12}:D_4$

Order 576: $C_2^4:C_6^2$ x 7, $C_2^4.C_6^2$ x 3, $C_2^4:C_6^2$ x 2, $C_6:D_4\times A_4$ x 2, $A_4\times C_6.D_4$ x 2, $C_2^4.C_6^2$, $A_4\times D_4:S_3$, $S_3\times D_4\times A_4$, $A_4\times D_{12}:C_2$, $C_3^2\times D_4^2$, $C_2^4.C_6^2$, $C_2^2:C_{12}^2$, $A_4\times C_3:\SD_{16}$, $A_4\times C_3:D_8$, $A_4\times C_3:\OD_{16}$, $C_{12}^2:C_2^2$, $C_3\times C_2^4:D_6$, $C_{12}^2:C_4$, $(C_2^3\times C_6):C_{12}$

Order 512: $D_4^2:C_2^3$

Order 384: $D_4^2:S_3$ x 11, $C_6\times D_4^2$ x 7, $C_2\times D_{12}:D_4$ x 6, $C_2^5:D_6$ x 6, $C_2^6:C_6$ x 5, $C_6:C_2\wr C_4$ x 4, $(C_6\times D_4).D_4$ x 3, $C_2^5:C_{12}$ x 2, $C_2^6:C_6$ x 2, $C_2^6:C_6$ x 2, $A_4\times C_4:D_4$ x 2, $A_4\times C_4:D_4$ x 2, $C_4\times D_4\times A_4$ x 2, $A_4\times C_4\wr C_2$, $C_2^5.C_{12}$, $D_4^2:C_6$, $C_{12}:C_2^5$, $D_{12}:C_2^4$, $A_4\times D_4:C_2^2$, $C_2^2\times C_{12}:D_4$, $C_2^2\times C_{12}:D_4$, $C_2^2\times D_4\times C_{12}$, $C_2^5:D_6$, $D_{12}:C_2^4$, $A_4\times D_8:C_2$, $C_2^5.D_6$, $C_2^2\times C_{12}.D_4$, $C_2^4.D_{12}$, $C_2\times C_4^2:D_6$

Order 288: $C_2^3:C_6^2$ x 7, $C_2^3.C_6^2$ x 6, $C_2^3.C_6^2$ x 4, $A_4\times C_3:D_4$ x 3, $C_6:C_4\times A_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $A_4\times C_3:Q_8$, $C_{12}^2:C_2$, $C_6^2:D_4$, $C_{12}^2:C_2$, $C_6^2:D_4$, $C_6^2.D_4$, $A_4\times C_3:C_8$, $C_6^2.D_4$, $C_3\times C_{12}.D_4$, $C_6^2.D_4$, $C_{12}\wr C_2$, $C_2\times A_4\times D_6$, $D_4\times C_6^2$

Order 256: $D_4^2:C_2^2$ x 6, $C_2^2\times D_4^2$, $C_2^5:D_4$, $C_4^2:C_2^4$, $C_2^5.D_4$, $C_2^6:C_4$

Order 192: $C_3\times D_4^2$ x 23, $C_2^5:C_6$ x 17, $C_{12}:C_2^4$ x 16, $C_2^5:C_6$ x 12, $D_{12}:D_4$ x 11, $C_2^4:D_6$ x 11, $C_2^4:C_{12}$ x 8, $C_2\times C_{12}:D_4$ x 8, $D_{12}:C_2^3$ x 6, $C_2\times D_4\times C_{12}$ x 6, $C_2^4:D_6$ x 6, $D_{12}:C_2^3$ x 6, $(C_2^3\times C_6):C_4$ x 5, $C_2\times C_{12}:D_4$ x 5, $C_2^3.D_{12}$ x 4, $C_4^2:D_6$ x 4, $A_4\times C_2^4$ x 4, $(C_4\times C_{12}):C_4$ x 4, $C_2^4.D_6$ x 3, $C_2\times C_{12}.D_4$ x 3, $C_2^4:C_{12}$ x 3, $A_4\times C_4:C_4$ x 2, $A_4\times C_4^2$ x 2, $C_2^5\times C_6$ x 2, $D_6:C_2^4$ x 2, $A_4\times D_4:C_2$ x 2, $C_2^4:C_{12}$ x 2, $C_2^4.D_6$ x 2, $C_3\times C_2\wr C_2^2$, $C_3\times D_4:D_4$, $C_4^2:C_{12}$, $C_2^4:C_{12}$, $C_2^4\times C_{12}$, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $D_{12}:C_2^3$, $C_2^2\times C_4:C_{12}$, $C_2^2\times C_4\times C_{12}$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $C_{12}.C_2^4$, $A_4\times \SD_{16}$, $A_4\times D_8$, $A_4\times \OD_{16}$

Order 144: $C_2^2:C_6^2$ x 11, $C_4:C_6^2$ x 7, $C_{12}\times A_4$ x 3, $C_6^2:C_4$ x 3, $C_6^2:C_4$ x 2, $C_2^2\times C_6^2$ x 2, $A_4\times D_6$ x 2, $C_6^2:C_2^2$ x 2, $A_4\times C_3:C_4$ x 2, $C_{12}.D_6$, $D_{12}:C_6$, $C_{12}.C_{12}$, $C_2\times C_6\times C_{12}$, $C_{12}.D_6$, $C_{12}:D_6$, $D_{12}:C_6$, $C_{12}:C_{12}$, $C_{12}^2$

Order 128: $C_2\wr D_4$ x 11, $C_2\times D_4^2$ x 7, $C_2^4:D_4$ x 6, $C_4^2:C_2^3$ x 6, $C_2^5:C_4$ x 4, $C_2^4.D_4$ x 3, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 2, $D_4:C_2^4$, $D_4\times C_2^4$, $C_8:C_2^4$, $C_4^2:C_2^3$, $C_2^4:D_4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $\OD_{16}:C_2^3$

Order 96: $C_{12}:C_2^3$ x 66, $C_2^4:C_6$ x 27, $C_2^4\times C_6$ x 17, $C_{12}:D_4$ x 17, $D_4:D_6$ x 16, $D_4\times A_4$ x 16, $C_2^3\times A_4$ x 15, $C_2^3:D_6$ x 14, $D_4\times C_{12}$ x 13, $C_2^3:C_{12}$ x 11, $C_{12}:D_4$ x 11, $C_2^4:S_3$ x 11, $D_{12}:C_2^2$ x 11, $C_2^3:C_{12}$ x 10, $C_2^3\times C_{12}$ x 9, $C_2^3.D_6$ x 7, $D_4:D_6$ x 6, $D_4\times D_6$ x 6, $D_{12}:C_2^2$ x 6, $C_2^2.D_{12}$ x 5, $D_{12}:C_4$ x 5, $C_2^3.D_6$ x 4, $C_{12}.D_4$ x 4, $C_6:\SD_{16}$ x 4, $C_6:D_8$ x 4, $C_2\times C_4:C_{12}$ x 3, $C_2\times C_4\times C_{12}$ x 3, $C_6:\OD_{16}$ x 3, $C_2^3:C_{12}$ x 2, $C_6.C_2^4$ x 2, $C_8\times A_4$, $D_4:C_{12}$, $\OD_{16}:C_6$, $C_2^3\times D_6$, $C_6.C_2^4$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_6.C_2^4$, $Q_8\times A_4$, $D_8:C_6$, $C_{12}.C_2^3$

Order 72: $C_2\times C_6^2$ x 7, $D_4\times C_3^2$ x 6, $C_6\times A_4$ x 5, $C_6\times C_{12}$ x 4, $C_6\wr C_2$ x 3, $C_6:C_{12}$ x 2, $C_6\times D_6$, $S_3\times A_4$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3:C_{24}$

Order 64: $D_4^2$ x 21, $D_4\times C_2^3$ x 19, $C_2^3:D_4$ x 18, $C_2\wr C_2^2$ x 11, $D_4:D_4$ x 11, $C_2^3:D_4$ x 8, $C_2^4:C_4$ x 7, $D_4:C_2^3$ x 6, $C_8:C_2^3$ x 6, $C_4^2:C_2^2$ x 6, $C_2\wr C_4$ x 5, $C_4^2:C_2^2$ x 5, $C_2^4:C_4$ x 5, $C_4^2:C_4$ x 4, $C_4^2:C_2^2$ x 4, $\OD_{16}:C_2^2$ x 3, $C_2^6$ x 2, $D_4:C_2^3$ x 2, $C_2^4\times C_4$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 48: $C_6\times D_4$ x 101, $C_2^3\times C_6$ x 73, $C_6:D_4$ x 32, $C_2^2\times C_{12}$ x 27, $C_2^2\times A_4$ x 24, $C_2^2:C_{12}$ x 21, $D_4:S_3$ x 11, $S_3\times D_4$ x 11, $D_{12}:C_2$ x 11, $C_6.D_4$ x 9, $C_6.C_2^3$ x 8, $C_4\times A_4$ x 8, $C_2^2\times D_6$ x 6, $C_4:C_{12}$ x 6, $C_4\times C_{12}$ x 6, $C_3:\SD_{16}$ x 5, $C_3:D_8$ x 5, $C_4\times D_6$ x 4, $C_3:\OD_{16}$ x 4, $C_2\times D_{12}$ x 3, $C_6:Q_8$ x 3, $C_6:C_8$ x 2, $D_4:C_6$ x 2, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_3\times \OD_{16}$

Order 36: $C_6^2$ x 11, $C_3\times C_{12}$ x 3, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 85, $C_2^2\wr C_2$ x 33, $C_2^5$ x 18, $C_2^3:C_4$ x 17, $D_4:C_2^2$ x 16, $C_4:D_4$ x 15, $D_4:C_2^2$ x 12, $C_2^3\times C_4$ x 12, $D_8:C_2$ x 11, $C_4\times D_4$ x 11, $C_2^3:C_4$ x 9, $C_4:D_4$ x 9, $C_4\wr C_2$ x 5, $\OD_{16}:C_2$ x 4, $C_2\times \SD_{16}$ x 4, $C_2\times D_8$ x 4, $C_2\times \OD_{16}$ x 3, $C_2^2.D_4$ x 3, $C_2\times C_4^2$ x 3, $C_2^2\times Q_8$, $C_2^2\times C_8$

Order 24: $C_2^2\times C_6$ x 112, $C_3\times D_4$ x 43, $C_2\times C_{12}$ x 34, $C_3:D_4$ x 15, $C_2\times D_6$ x 14, $C_6:C_4$ x 12, $C_2\times A_4$ x 11, $C_4\times S_3$ x 5, $D_{12}$ x 4, $C_3:Q_8$ x 4, $C_3:C_8$ x 2, $C_{24}$, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 5, $C_3\times S_3$

Order 16: $C_2\times D_4$ x 130, $C_2^4$ x 75, $C_2^2\times C_4$ x 37, $C_2^2:C_4$ x 22, $D_4:C_2$ x 22, $\SD_{16}$ x 5, $D_8$ x 5, $\OD_{16}$ x 4, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2\times Q_8$ x 3, $C_2\times C_8$ x 2

Order 12: $C_2\times C_6$ x 79, $C_{12}$ x 14, $D_6$ x 10, $C_3:C_4$ x 4, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 111, $D_4$ x 46, $C_2\times C_4$ x 40, $Q_8$ x 4, $C_8$ x 2

Order 6: $C_6$ x 22, $S_3$ x 2

Order 4: $C_2^2$ x 65, $C_4$ x 10

Order 3: $C_3$ x 3

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4608: $A_4\times D_4^2:S_3$ ($C_1$)

Order 2304: $A_4\times C_2^4:D_6$ ($C_2$) x 2, $(C_2^5\times C_6):C_{12}$ ($C_2$) x 2, $C_6.(C_2\times D_4\times A_4).C_2$ ($C_2$), $A_4\times D_{12}:D_4$ ($C_2$), $C_2^6:C_6^2$ ($C_2$)

Order 1536: $C_2\times D_4^2:D_6$ ($C_3$)

Order 1152: $(C_2^3\times C_{12}):C_{12}$ ($C_2^2$) x 2, $C_2^5:C_6^2$ ($C_2^2$) x 2, $A_4\times D_4:D_6$ ($C_2^2$), $A_4\times C_{12}.D_4$ ($C_2^2$), $C_2^5.C_6^2$ ($C_2^2$)

Order 768: $(C_2^5\times C_6):C_4$ ($C_6$) x 2, $C_2^6:D_6$ ($C_6$) x 2, $C_2\times C_6\times D_4^2$ ($C_6$), $(C_6\times D_4).C_2^4$ ($C_6$), $A_4\times D_4^2$ ($S_3$), $C_2^2\times D_{12}:D_4$ ($C_6$)

Order 576: $C_2^4:C_6^2$ ($D_4$) x 2, $C_2^4.C_6^2$ ($D_4$) x 2, $C_2^4:C_6^2$ ($C_2^3$), $C_2^4:C_6^2$ ($D_4$), $C_2^2:C_{12}^2$ ($D_4$)

Order 384: $C_2^6:C_6$ ($C_2\times C_6$) x 2, $C_2^6:C_6$ ($D_6$) x 2, $C_2^5.D_6$ ($C_2\times C_6$) x 2, $D_4^2:S_3$ ($A_4$), $D_{12}:C_2^4$ ($C_2\times C_6$), $C_2^2\times C_{12}:D_4$ ($C_2\times C_6$), $A_4\times C_4:D_4$ ($D_6$), $C_2^2\times C_{12}.D_4$ ($C_2\times C_6$)

Order 288: $C_2^3:C_6^2$ ($C_2\times D_4$) x 2, $C_2^3.C_6^2$ ($C_2\times D_4$)

Order 256: $C_2^2\times D_4^2$ ($C_3\times S_3$)

Order 192: $C_2^4:C_{12}$ ($C_3:D_4$) x 2, $(C_2^3\times C_6):C_4$ ($C_2\times A_4$) x 2, $C_2^4:D_6$ ($C_2\times A_4$) x 2, $C_2^5\times C_6$ ($C_3\times D_4$) x 2, $A_4\times C_2^4$ ($C_3:D_4$) x 2, $C_2^4:C_{12}$ ($C_3\times D_4$) x 2, $A_4\times C_4^2$ ($C_3:D_4$), $D_{12}:D_4$ ($C_2\times A_4$), $C_{12}:C_2^4$ ($C_2^2\times C_6$), $C_{12}:C_2^4$ ($C_3\times D_4$), $C_2^5:C_6$ ($C_3:D_4$), $C_2^5:C_6$ ($C_2\times D_6$), $C_3\times D_4^2$ ($C_2\times A_4$), $C_2^2\times C_4\times C_{12}$ ($C_3\times D_4$), $(C_4\times C_{12}):C_4$ ($C_2\times A_4$)

Order 144: $C_2^2:C_6^2$ ($C_2^2\wr C_2$)

Order 128: $C_2^4:D_4$ ($C_6\times S_3$) x 2, $C_4^2:C_2^3$ ($C_6\times S_3$)

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

Order 72: $C_6\times A_4$ ($C_2\wr C_2^2$)

Order 64: $C_2^6$ ($C_6\wr C_2$) x 2, $C_2^4:C_4$ ($C_6\wr C_2$) x 2, $D_4\times C_2^3$ ($C_6\times D_6$), $D_4\times C_2^3$ ($C_6\wr C_2$), $D_4^2$ ($S_3\times A_4$), $C_2^2\times C_4^2$ ($C_6\wr C_2$)

Order 48: $C_2^3\times C_6$ ($D_4\times A_4$) x 2, $C_2^2:C_{12}$ ($D_4\times A_4$) x 2, $C_2^3\times C_6$ ($C_2^4:C_6$), $C_2^2\times A_4$ ($C_2^4:S_3$), $C_6\times D_4$ ($C_2^3\times A_4$), $C_6\times D_4$ ($D_4\times A_4$), $C_4\times C_{12}$ ($D_4\times A_4$)

Order 36: $C_3\times A_4$ ($C_2\wr D_4$)

Order 32: $C_2^5$ ($C_6^2:C_2^2$) x 2, $C_2^2\wr C_2$ ($A_4\times D_6$) x 2, $C_2^3\times C_4$ ($C_6^2:C_2^2$), $C_4:D_4$ ($A_4\times D_6$)

Order 24: $C_2^2\times C_6$ ($C_2^5:C_6$) x 2, $C_2\times C_{12}$ ($C_2^5:C_6$), $C_2^2\times C_6$ ($C_3\times C_2\wr C_2^2$), $C_2\times A_4$ ($(C_2\times C_{12}):D_4$)

Order 16: $C_2^2:C_4$ ($A_4\times C_3:D_4$) x 2, $C_2^4$ ($A_4\times C_3:D_4$) x 2, $C_4^2$ ($A_4\times C_3:D_4$), $C_2^4$ ($C_6^2:D_4$), $C_2\times D_4$ ($A_4\times C_3:D_4$), $C_2\times D_4$ ($C_2\times A_4\times D_6$)

Order 12: $C_2\times C_6$ ($D_4^2:C_6$), $C_2\times C_6$ ($C_2^6:C_6$), $A_4$ ($D_4^2:S_3$)

Order 8: $C_2^3$ ($C_6:D_4\times A_4$) x 2, $C_2^3$ ($(C_6\times C_{12}):D_4$), $C_2\times C_4$ ($C_6:D_4\times A_4$)

Order 6: $C_6$ ($A_4\times C_2\wr C_2^2$)

Order 4: $C_2^2$ ($C_{12}^2:D_4$), $C_2^2$ ($A_4\times C_2^4:S_3$)

Order 3: $C_3$ ($A_4\times C_2\wr D_4$)

Order 2: $C_2$ ($C_2^5.(C_6\times D_6)$)

Order 1: $C_1$ ($A_4\times D_4^2:S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4608: $A_4\times D_4^2:S_3$ ($C_1$)

Order 2304: $C_6.(C_2\times D_4\times A_4).C_2$ ($C_2$), $A_4\times C_2^4:D_6$ ($C_2$), $A_4\times D_{12}:D_4$ ($C_2$), $(C_2^5\times C_6):C_{12}$ ($C_2$), $C_2^6:C_6^2$ ($C_2$)

Order 1536: $C_2\times D_4^2:D_6$ ($C_3$)

Order 1152: $A_4\times D_4:D_6$ ($C_2^2$), $A_4\times C_{12}.D_4$ ($C_2^2$), $(C_2^3\times C_{12}):C_{12}$ ($C_2^2$), $C_2^5.C_6^2$ ($C_2^2$), $C_2^5:C_6^2$ ($C_2^2$)

Order 768: $C_2\times C_6\times D_4^2$ ($C_6$), $(C_6\times D_4).C_2^4$ ($C_6$), $(C_2^5\times C_6):C_4$ ($C_6$), $A_4\times D_4^2$ ($S_3$), $C_2^6:D_6$ ($C_6$), $C_2^2\times D_{12}:D_4$ ($C_6$)

Order 576: $C_2^4:C_6^2$ ($D_4$), $C_2^4:C_6^2$ ($C_2^3$), $C_2^4:C_6^2$ ($D_4$), $C_2^4.C_6^2$ ($D_4$), $C_2^2:C_{12}^2$ ($D_4$)

Order 384: $D_4^2:S_3$ ($A_4$), $D_{12}:C_2^4$ ($C_2\times C_6$), $C_2^2\times C_{12}:D_4$ ($C_2\times C_6$), $C_2^6:C_6$ ($C_2\times C_6$), $A_4\times C_4:D_4$ ($D_6$), $C_2^6:C_6$ ($D_6$), $C_2^5.D_6$ ($C_2\times C_6$), $C_2^2\times C_{12}.D_4$ ($C_2\times C_6$)

Order 288: $C_2^3.C_6^2$ ($C_2\times D_4$), $C_2^3:C_6^2$ ($C_2\times D_4$)

Order 256: $C_2^2\times D_4^2$ ($C_3\times S_3$)

Order 192: $C_2^4:C_{12}$ ($C_3:D_4$), $A_4\times C_4^2$ ($C_3:D_4$), $(C_2^3\times C_6):C_4$ ($C_2\times A_4$), $D_{12}:D_4$ ($C_2\times A_4$), $C_2^4:D_6$ ($C_2\times A_4$), $C_2^5\times C_6$ ($C_3\times D_4$), $A_4\times C_2^4$ ($C_3:D_4$), $C_{12}:C_2^4$ ($C_2^2\times C_6$), $C_{12}:C_2^4$ ($C_3\times D_4$), $C_2^5:C_6$ ($C_3:D_4$), $C_2^5:C_6$ ($C_2\times D_6$), $C_3\times D_4^2$ ($C_2\times A_4$), $C_2^4:C_{12}$ ($C_3\times D_4$), $C_2^2\times C_4\times C_{12}$ ($C_3\times D_4$), $(C_4\times C_{12}):C_4$ ($C_2\times A_4$)

Order 144: $C_2^2:C_6^2$ ($C_2^2\wr C_2$)

Order 128: $C_4^2:C_2^3$ ($C_6\times S_3$), $C_2^4:D_4$ ($C_6\times S_3$)

Order 96: $C_2^3.D_6$ ($C_2^2\times A_4$), $C_{12}.D_4$ ($C_2^2\times A_4$), $C_2^4\times C_6$ ($C_6\times D_4$), $C_2^3\times A_4$ ($C_6:D_4$), $C_2^3\times C_{12}$ ($C_6\times D_4$), $D_4:D_6$ ($C_2^2\times A_4$), $C_2^3:C_{12}$ ($C_6:D_4$), $C_{12}:D_4$ ($C_2^2\times A_4$), $C_2^4:C_6$ ($C_2^2\times A_4$)

Order 72: $C_6\times A_4$ ($C_2\wr C_2^2$)

Order 64: $C_2^6$ ($C_6\wr C_2$), $D_4\times C_2^3$ ($C_6\times D_6$), $D_4\times C_2^3$ ($C_6\wr C_2$), $D_4^2$ ($S_3\times A_4$), $C_2^4:C_4$ ($C_6\wr C_2$), $C_2^2\times C_4^2$ ($C_6\wr C_2$)

Order 48: $C_2^3\times C_6$ ($D_4\times A_4$), $C_2^3\times C_6$ ($C_2^4:C_6$), $C_2^2\times A_4$ ($C_2^4:S_3$), $C_6\times D_4$ ($C_2^3\times A_4$), $C_6\times D_4$ ($D_4\times A_4$), $C_2^2:C_{12}$ ($D_4\times A_4$), $C_4\times C_{12}$ ($D_4\times A_4$)

Order 36: $C_3\times A_4$ ($C_2\wr D_4$)

Order 32: $C_2^5$ ($C_6^2:C_2^2$), $C_2^3\times C_4$ ($C_6^2:C_2^2$), $C_4:D_4$ ($A_4\times D_6$), $C_2^2\wr C_2$ ($A_4\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_2^5:C_6$), $C_2^2\times C_6$ ($C_3\times C_2\wr C_2^2$), $C_2^2\times C_6$ ($C_2^5:C_6$), $C_2\times A_4$ ($(C_2\times C_{12}):D_4$)

Order 16: $C_2^2:C_4$ ($A_4\times C_3:D_4$), $C_4^2$ ($A_4\times C_3:D_4$), $C_2^4$ ($A_4\times C_3:D_4$), $C_2^4$ ($C_6^2:D_4$), $C_2\times D_4$ ($A_4\times C_3:D_4$), $C_2\times D_4$ ($C_2\times A_4\times D_6$)

Order 12: $C_2\times C_6$ ($D_4^2:C_6$), $C_2\times C_6$ ($C_2^6:C_6$), $A_4$ ($D_4^2:S_3$)

Order 8: $C_2^3$ ($C_6:D_4\times A_4$), $C_2^3$ ($(C_6\times C_{12}):D_4$), $C_2\times C_4$ ($C_6:D_4\times A_4$)

Order 6: $C_6$ ($A_4\times C_2\wr C_2^2$)

Order 4: $C_2^2$ ($C_{12}^2:D_4$), $C_2^2$ ($A_4\times C_2^4:S_3$)

Order 3: $C_3$ ($A_4\times C_2\wr D_4$)

Order 2: $C_2$ ($C_2^5.(C_6\times D_6)$)

Order 1: $C_1$ ($A_4\times D_4^2:S_3$)

Series

Derived series

$A_4\times D_4^2:S_3$

$\rhd$

$C_{12}:C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$A_4\times D_4^2:S_3$

$\rhd$

$C_2^6:C_6^2$

$\rhd$

$C_2^5:C_6^2$

$\rhd$

$C_2^4:C_6^2$

$\rhd$

$C_{12}:C_2^4$

$\rhd$

$C_2^4\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$A_4\times D_4^2:S_3$

$\rhd$

$C_{12}:C_2^4$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

$\lhd$

$D_4^2$

Supergroups

This group is a maximal subgroup of 4 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 $180 \times 180$ 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 $115 \times 115$ rational character table.