Abstract group 15552.fa: $C_6^4:D_6$

• Label: 15552.fa
• Order: \( 2^{6} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^4:D_6$
Order:	\(15552\)\(\medspace = 2^{6} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_2\times C_6^3).C_3^4.C_2^4$, of order \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \)
Composition factors:	$C_2$ x 6, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12
Elements	1	555	944	1620	5520	1728	5184	15552
Conjugacy classes	1	6	14	5	117	3	16	162
Divisions	1	6	14	5	79	3	11	119
Autjugacy classes	1	6	8	5	38	1	7	66

Dimension	1	2	3	4	6	12	24
Irr. complex chars.	4	10	4	4	44	96	0	162
Irr. rational chars.	4	10	4	4	24	40	33	119

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$2177280$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{3}=d^{6}=e^{6}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(2,3,5)(4,7,6)(9,11,10)(12,14,15,19,20,18,17,13,16), (3,5)(4,6)(10,11)(13,15,14,18,20,16), (1,2)(3,4)(5,7)(6,8)(9,10)(12,13)(14,17)(15,16)(19,20)\rangle$
Transitive group:	36T9961	36T9962			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_6^4$ $\,\rtimes\,$ $D_6$	$(C_6^4:S_3)$ $\,\rtimes\,$ $C_2$	$(C_6^4:C_6)$ $\,\rtimes\,$ $C_2$	$(C_6^2:A_4)$ $\,\rtimes\,$ $S_3^2$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $(A_4:S_3^2)$	$(C_2\times C_6^3)$ . $S_3^2$	$C_6^2$ . $(C_6^2:D_6)$	$C_3^3$ . $(C_2^4:S_3^2)$	all 16
Aut. group:	$\Aut(C_6^3.C_2^3)$	$\Aut(C_6^3.C_2^3)$

Elements of the group are displayed as permutations of degree 20.

Homology

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

Subgroups

There are 168672 subgroups in 4249 conjugacy classes, 44 normal (35 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_6^4:D_6$
Commutator:	$G' \simeq$ $C_3\times C_3.C_2^4:\He_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_6^2$	$G/\Phi \simeq$ $A_4:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^4$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^4:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_6^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 15552: $C_6^4:D_6$

Order 7776: $C_6^4.S_3$, $C_6^4:S_3$, $C_6^4:C_6$

Order 5184: $(C_2\times C_6^3):D_6$ x 3, $C_3^4.C_2\wr C_2^2$, $(C_2\times C_6^3):D_6$

Order 3888: $C_3\times C_3.C_2^4:\He_3$, $C_3^4:(C_2\times S_4)$

Order 2592: $(C_3\times C_6^2):S_4$ x 6, $(C_3\times C_6^2):S_4$ x 3, $(C_2\times C_6^3):C_6$ x 3, $(C_3\times C_6^3).C_2^2$ x 2, $C_6^2.C_6^2.C_2$, $C_6^4.C_2$, $C_3\times C_6^3.C_2^2$, $C_3^4.Q_8:C_2^2$, $C_2^4.C_3^3:S_3$, $(C_3\times C_2^4:C_9).C_6$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^2):S_4$, $(C_2\times C_6^3):C_6$

Order 1944: $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3^3:(C_6\times A_4)$

Order 1728: $C_6^3.C_2^3$ x 4, $(C_2^2\times C_6^2):D_6$ x 3, $C_6^3.C_2^3$ x 2, $(C_6\times D_6):S_4$

Order 1296: $(C_2\times C_6^3):C_3$ x 6, $C_6^3:C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^2.S_3^2$ x 3, $C_6^3:C_6$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.C_6$ x 2, $C_6^4$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_3\times C_6^2:A_4$, $C_3\times C_6^2.A_4$, $C_6^2.S_3^2$

Order 972: $C_3^4:A_4$ x 3, $C_3^4:D_6$

Order 864: $C_6^3.C_2^2$ x 18, $C_6^2.D_{12}$ x 12, $C_6^3:C_2^2$ x 4, $C_6^2:S_4$ x 4, $(C_2^2\times C_6^2):C_6$ x 3, $C_6^2:S_4$ x 3, $C_6^2.S_4$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3:C_2^2$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:S_4$, $C_6^2:S_4$, $C_6^2.S_4$, $(C_6\times D_6):A_4$

Order 648: $C_3^3:S_4$ x 15, $C_3^4:D_4$ x 6, $C_3^3:S_4$ x 6, $C_3^4:D_4$ x 4, $C_3\times C_6^3$ x 3, $C_2\times C_3^3:C_{12}$ x 3, $C_3^3.S_4$ x 3, $(C_3\times C_6^2):C_6$ x 3, $C_3^4:C_2^3$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $C_3\times C_6^2:C_6$

Order 576: $C_2^4:S_3^2$ x 9, $C_2^4:S_3^2$ x 3, $(C_2^2\times C_6):S_4$, $(C_6\times C_{12}):D_4$

Order 486: $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $C_6^3:C_2$ x 39, $C_6^2:C_{12}$ x 39, $C_6^2:D_6$ x 12, $C_6^2.D_6$ x 12, $C_2\times C_6^3$ x 11, $D_6:C_6^2$ x 7, $C_6^2:C_{12}$ x 7, $C_6^2:A_4$ x 4, $C_6^2.A_4$ x 4, $C_6^3:C_2$ x 3, $C_6^2:D_6$ x 3, $C_6^3.C_2$ x 3, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2:A_4$, $A_4:S_3^2$, $C_6^2.A_4$

Order 324: $C_3^3:A_4$ x 21, $C_3^2\times C_6^2$ x 8, $C_3^3.A_4$ x 4, $C_3^3:D_6$ x 3, $C_3^3:D_6$ x 3, $C_3^3:C_{12}$ x 3, $C_3^3:A_4$ x 3, $C_3^2:C_6^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 288: $C_6^2:D_4$ x 26, $C_6^2.D_4$ x 18, $C_6^2:D_4$ x 15, $C_2^3:S_3^2$ x 9, $C_6^2.D_4$ x 9, $(C_2\times C_6):S_4$ x 6, $C_2^4:D_9$ x 3, $(C_2\times D_6):A_4$ x 3, $C_3\times C_2^2:S_4$ x 3, $C_6^2.D_4$ x 2, $C_6^2:D_4$, $C_3\times C_2^3:A_4$, $C_6^2.D_4$, $C_6^2.C_2^3$

Order 243: $C_3^4:C_3$

Order 216: $C_6^2:C_6$ x 78, $C_6^3$ x 70, $C_6^2.C_6$ x 39, $C_6^2:C_6$ x 18, $C_6^2:C_6$ x 14, $C_3^2:D_{12}$ x 12, $C_6.S_3^2$ x 12, $C_3^2:S_4$ x 9, $C_3^2.S_4$ x 9, $C_6.C_6^2$ x 7, $C_3^2:S_4$ x 6, $C_6^2:S_3$ x 6, $C_6:S_3^2$ x 4, $C_3^3:Q_8$ x 4, $C_6^2:C_6$ x 3, $C_3^2.S_4$ x 3, $S_3\times C_6^2$ x 3, $C_6^2.S_3$ x 3, $C_6^2:S_3$ x 2, $C_6:S_3^2$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$

Order 192: $C_2^4:D_6$ x 6, $C_2^3:S_4$ x 3

Order 162: $C_3^3:S_3$ x 6, $C_3^3\times C_6$ x 3, $C_3^3:C_6$ x 3, $C_3\wr S_3$ x 3, $C_3^3:C_6$ x 2, $C_3^2\wr C_2$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^3:C_6$, $C_3^3:C_6$

Order 144: $C_6^2:C_4$ x 59, $C_6^2:C_2^2$ x 59, $C_6^2:C_2^2$ x 34, $C_6^2:C_4$ x 33, $C_2^2\times C_6^2$ x 25, $D_6:D_6$ x 18, $D_6.D_6$ x 18, $D_6:D_6$ x 9, $D_6.D_6$ x 9, $C_2^4:C_3^2$ x 5, $S_3\times S_4$ x 3, $C_2^4:C_9$ x 3, $C_4:C_6^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $C_6:S_4$, $C_{12}.D_6$

Order 108: $C_3\times C_6^2$ x 167, $C_3^2:D_6$ x 39, $C_3^2:C_{12}$ x 39, $C_3^2.A_4$ x 17, $C_3^2:A_4$ x 13, $C_3^2\times D_6$ x 7, $C_3^2:C_{12}$ x 7, $C_3:S_3^2$ x 4, $C_3^2.A_4$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times A_4$ x 3, $C_3:S_3^2$ x 3, $C_3^3:C_4$ x 3, $C_3^2:D_6$ x 3

Order 96: $C_2^4:S_3$ x 17, $C_2^2.D_{12}$ x 12, $C_2^3.D_6$ x 6, $C_2^2:S_4$ x 6, $D_4:D_6$ x 6, $C_2^4:C_6$ x 6, $C_2^3:A_4$ x 3

Order 81: $C_3\wr C_3$ x 6, $C_3^4$, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $C_2\times C_6^2$ x 203, $C_6\wr C_2$ x 118, $C_6^2:C_2$ x 68, $C_6:C_{12}$ x 59, $C_6.D_6$ x 34, $C_6\times D_6$ x 26, $C_3:D_{12}$ x 18, $C_6.D_6$ x 18, $C_6:D_6$ x 16, $C_3:S_4$ x 13, $S_3\times D_6$ x 9, $C_3^2:Q_8$ x 9, $D_6:S_3$ x 9, $C_6.D_6$ x 9, $C_2^2:D_9$ x 9, $C_3\times S_4$ x 6, $D_4\times C_3^2$ x 4, $S_3\times A_4$ x 3, $C_6\times C_{12}$ x 2, $C_{12}:S_3$ x 2, $C_6\times A_4$, $C_3:D_{12}$, $C_3^2:Q_8$

Order 64: $C_2\wr C_2^2$

Order 54: $C_3^2\times C_6$ x 70, $C_3^2:C_6$ x 19, $C_3^2:C_6$ x 6, $C_3^2:S_3$ x 4, $S_3\times C_3^2$ x 4, $C_9:C_6$ x 3, $C_3^2:S_3$ x 3, $C_3:D_9$

Order 48: $C_6:D_4$ x 44, $C_6.D_4$ x 38, $D_4:S_3$ x 12, $S_3\times D_4$ x 12, $C_2^3\times C_6$ x 11, $C_6\times D_4$ x 11, $C_2^2:C_{12}$ x 11, $D_{12}:C_2$ x 6, $C_2^2:A_4$ x 3, $C_2\times S_4$ x 3

Order 36: $C_6^2$ x 477, $C_3:C_{12}$ x 59, $C_6\times S_3$ x 59, $C_6:S_3$ x 35, $C_3^2:C_4$ x 33, $C_3\times A_4$ x 16, $C_2^2:C_9$ x 12, $S_3^2$ x 12, $C_3\times C_{12}$ x 2

Order 32: $C_2^3:C_4$ x 3, $C_2^2\wr C_2$ x 3, $D_4:C_2^2$

Order 27: $C_3^3$ x 12, $C_9:C_3$ x 4, $\He_3$ x 4, $C_3\times C_9$

Order 24: $C_3:D_4$ x 88, $C_2^2\times C_6$ x 76, $C_6:C_4$ x 44, $C_2\times D_6$ x 23, $C_3\times D_4$ x 22, $C_4\times S_3$ x 12, $S_4$ x 12, $C_2\times C_{12}$ x 11, $D_{12}$ x 6, $C_3:Q_8$ x 6, $C_2\times A_4$ x 3

Order 18: $C_3\times C_6$ x 204, $C_3\times S_3$ x 32, $C_3:S_3$ x 21, $D_9$ x 3

Order 16: $C_2\times D_4$ x 8, $C_2^2:C_4$ x 5, $D_4:C_2$ x 3, $C_2^4$

Order 12: $C_2\times C_6$ x 178, $D_6$ x 47, $C_3:C_4$ x 38, $C_{12}$ x 11, $A_4$ x 9

Order 9: $C_3^2$ x 30, $C_9$ x 3

Order 8: $D_4$ x 13, $C_2\times C_4$ x 8, $C_2^3$ x 7, $Q_8$

Order 6: $C_6$ x 79, $S_3$ x 23

Order 4: $C_2^2$ x 14, $C_4$ x 5

Order 3: $C_3$ x 14

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 15552: $C_6^4:D_6$

Order 7776: $C_6^4.S_3$, $C_6^4:S_3$, $C_6^4:C_6$

Order 5184: $C_3^4.C_2\wr C_2^2$, $(C_2\times C_6^3):D_6$, $(C_2\times C_6^3):D_6$

Order 3888: $C_3\times C_3.C_2^4:\He_3$, $C_3^4:(C_2\times S_4)$

Order 2592: $(C_3\times C_6^3).C_2^2$ x 2, $(C_3\times C_6^2):S_4$ x 2, $C_6^2.C_6^2.C_2$, $C_6^4.C_2$, $C_3\times C_6^3.C_2^2$, $C_3^4.Q_8:C_2^2$, $C_2^4.C_3^3:S_3$, $(C_3\times C_2^4:C_9).C_6$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^2):S_4$, $(C_3\times C_6^2):S_4$, $(C_2\times C_6^3):C_6$, $(C_2\times C_6^3):C_6$

Order 1944: $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3^3:(C_6\times A_4)$

Order 1728: $C_6^3.C_2^3$ x 2, $C_6^3.C_2^3$ x 2, $(C_6\times D_6):S_4$, $(C_2^2\times C_6^2):D_6$

Order 1296: $C_6^3:C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^3:C_6$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.C_6$ x 2, $(C_2\times C_6^3):C_3$ x 2, $C_6^4$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_3\times C_6^2:A_4$, $C_3\times C_6^2.A_4$, $C_6^2.S_3^2$, $C_6^2.S_3^2$

Order 972: $C_3^4:A_4$ x 3, $C_3^4:D_6$

Order 864: $C_6^3.C_2^2$ x 10, $C_6^2.D_{12}$ x 6, $C_6^3.C_2^2$ x 3, $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2:S_4$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $(C_2^2\times C_6^2):C_6$, $C_6^2:S_4$, $C_6^2:S_4$, $C_6^2.S_4$, $C_6^2:S_4$, $C_6^2.S_4$, $(C_6\times D_6):A_4$

Order 648: $C_3^4:D_4$ x 6, $C_3^4:D_4$ x 4, $C_3^3:S_4$ x 4, $C_3\times C_6^3$ x 3, $C_2\times C_3^3:C_{12}$ x 3, $C_3^4:C_2^3$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $C_3^3.S_4$ x 2, $C_3^3:S_4$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $C_3\times C_6^2:C_6$, $(C_3\times C_6^2):C_6$

Order 576: $C_2^4:S_3^2$ x 5, $(C_2^2\times C_6):S_4$, $C_2^4:S_3^2$, $(C_6\times C_{12}):D_4$

Order 486: $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $C_6^3:C_2$ x 19, $C_6^2:C_{12}$ x 19, $C_2\times C_6^3$ x 7, $D_6:C_6^2$ x 6, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 6, $C_6^2:C_{12}$ x 6, $C_6^3:C_2$ x 3, $C_6^3.C_2$ x 3, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2:A_4$ x 2, $C_6^2.A_4$ x 2, $C_6^2:A_4$, $A_4:S_3^2$, $C_6^2.A_4$, $C_6^2:D_6$

Order 324: $C_3^2\times C_6^2$ x 8, $C_3^3:A_4$ x 6, $C_3^3:D_6$ x 3, $C_3^3:C_{12}$ x 3, $C_3^3:A_4$ x 3, $C_3^3.A_4$ x 3, $C_3^2:C_6^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^3:D_6$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 288: $C_6^2:D_4$ x 14, $C_6^2.D_4$ x 10, $C_6^2:D_4$ x 9, $C_2^3:S_3^2$ x 5, $C_6^2.D_4$ x 5, $(C_2\times C_6):S_4$ x 4, $C_6^2.D_4$ x 2, $C_2^4:D_9$, $C_6^2:D_4$, $C_3\times C_2^3:A_4$, $(C_2\times D_6):A_4$, $C_6^2.D_4$, $C_3\times C_2^2:S_4$, $C_6^2.C_2^3$

Order 243: $C_3^4:C_3$

Order 216: $C_6^2:C_6$ x 38, $C_6^3$ x 33, $C_6^2.C_6$ x 19, $C_6^2:C_6$ x 12, $C_6^2:C_6$ x 10, $C_6^2:S_3$ x 6, $C_6.C_6^2$ x 6, $C_3^2:D_{12}$ x 6, $C_6.S_3^2$ x 6, $C_3^2:S_4$ x 4, $S_3\times C_6^2$ x 3, $C_6^2.S_3$ x 3, $C_3^2.S_4$ x 2, $C_3^2:S_4$ x 2, $C_3^2.S_4$ x 2, $C_6^2:S_3$ x 2, $C_6:S_3^2$ x 2, $C_6:S_3^2$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$, $C_6^2:C_6$

Order 192: $C_2^4:D_6$ x 4, $C_2^3:S_4$

Order 162: $C_3^3\times C_6$ x 3, $C_3^3:C_6$ x 2, $C_3^3:S_3$ x 2, $C_3^2\wr C_2$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^3:C_6$, $C_3^3:C_6$, $C_3^3:C_6$, $C_3\wr S_3$

Order 144: $C_6^2:C_4$ x 27, $C_6^2:C_2^2$ x 27, $C_6^2:C_2^2$ x 18, $C_6^2:C_4$ x 17, $C_2^2\times C_6^2$ x 13, $D_6:D_6$ x 10, $D_6.D_6$ x 10, $D_6:D_6$ x 5, $D_6.D_6$ x 5, $C_2^4:C_3^2$ x 3, $C_4:C_6^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $C_6:S_4$, $S_3\times S_4$, $C_{12}.D_6$, $C_2^4:C_9$

Order 108: $C_3\times C_6^2$ x 80, $C_3^2:D_6$ x 19, $C_3^2:C_{12}$ x 19, $C_3^2\times D_6$ x 6, $C_3^2:C_{12}$ x 6, $C_3^2:A_4$ x 6, $C_3^2.A_4$ x 6, $C_3^2:D_6$ x 3, $C_3^2\times A_4$ x 3, $C_3:S_3^2$ x 3, $C_3^3:C_4$ x 3, $C_3^2.A_4$ x 3, $C_3:S_3^2$ x 2, $C_3^2:D_6$

Order 96: $C_2^4:S_3$ x 11, $C_2^2.D_{12}$ x 8, $C_2^3.D_6$ x 4, $D_4:D_6$ x 4, $C_2^4:C_6$ x 4, $C_2^2:S_4$ x 2, $C_2^3:A_4$

Order 81: $C_3\wr C_3$ x 2, $C_3^4$, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $C_2\times C_6^2$ x 73, $C_6\wr C_2$ x 54, $C_6^2:C_2$ x 36, $C_6:C_{12}$ x 27, $C_6.D_6$ x 18, $C_6\times D_6$ x 14, $C_6:D_6$ x 10, $C_3:D_{12}$ x 10, $C_6.D_6$ x 10, $C_3:S_4$ x 8, $S_3\times D_6$ x 5, $C_3^2:Q_8$ x 5, $D_6:S_3$ x 5, $C_6.D_6$ x 5, $D_4\times C_3^2$ x 4, $C_3\times S_4$ x 2, $C_6\times C_{12}$ x 2, $C_{12}:S_3$ x 2, $C_2^2:D_9$ x 2, $C_6\times A_4$, $S_3\times A_4$, $C_3:D_{12}$, $C_3^2:Q_8$

Order 64: $C_2\wr C_2^2$

Order 54: $C_3^2\times C_6$ x 33, $C_3^2:C_6$ x 11, $S_3\times C_3^2$ x 4, $C_3^2:S_3$ x 3, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $C_3:D_9$, $C_9:C_6$

Order 48: $C_6:D_4$ x 25, $C_6.D_4$ x 21, $D_4:S_3$ x 8, $S_3\times D_4$ x 8, $C_2^3\times C_6$ x 7, $C_6\times D_4$ x 7, $C_2^2:C_{12}$ x 7, $D_{12}:C_2$ x 4, $C_2^2:A_4$, $C_2\times S_4$

Order 36: $C_6^2$ x 173, $C_3:C_{12}$ x 27, $C_6\times S_3$ x 27, $C_6:S_3$ x 19, $C_3^2:C_4$ x 17, $C_3\times A_4$ x 9, $S_3^2$ x 6, $C_2^2:C_9$ x 3, $C_3\times C_{12}$ x 2

Order 32: $C_2^3:C_4$ x 3, $C_2^2\wr C_2$ x 3, $D_4:C_2^2$

Order 27: $C_3^3$ x 8, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_3:D_4$ x 50, $C_2^2\times C_6$ x 37, $C_6:C_4$ x 25, $C_2\times D_6$ x 15, $C_3\times D_4$ x 14, $C_4\times S_3$ x 8, $C_2\times C_{12}$ x 7, $D_{12}$ x 4, $C_3:Q_8$ x 4, $S_4$ x 4, $C_2\times A_4$

Order 18: $C_3\times C_6$ x 74, $C_3\times S_3$ x 16, $C_3:S_3$ x 13, $D_9$

Order 16: $C_2\times D_4$ x 8, $C_2^2:C_4$ x 5, $D_4:C_2$ x 3, $C_2^4$

Order 12: $C_2\times C_6$ x 87, $D_6$ x 26, $C_3:C_4$ x 21, $C_{12}$ x 7, $A_4$ x 3

Order 9: $C_3^2$ x 16, $C_9$

Order 8: $D_4$ x 13, $C_2\times C_4$ x 8, $C_2^3$ x 7, $Q_8$

Order 6: $C_6$ x 38, $S_3$ x 13

Order 4: $C_2^2$ x 14, $C_4$ x 5

Order 3: $C_3$ x 8

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 15552: $C_6^4:D_6$ ($C_1$)

Order 7776: $C_6^4.S_3$ ($C_2$), $C_6^4:S_3$ ($C_2$), $C_6^4:C_6$ ($C_2$)

Order 3888: $C_3\times C_3.C_2^4:\He_3$ ($C_2^2$)

Order 2592: $(C_2\times C_6^3):C_6$ ($S_3$) x 3, $C_3\times C_6^3.C_2^2$ ($S_3$), $(C_3\times C_6^2):S_4$ ($S_3$)

Order 1296: $(C_2\times C_6^3):C_3$ ($D_6$) x 3, $C_6^4$ ($D_6$), $C_3\times C_6^2:A_4$ ($D_6$)

Order 864: $C_6^3.C_2^2$ ($C_3:S_3$)

Order 648: $C_3^4:C_2^3$ ($S_4$)

Order 432: $C_6^2:A_4$ ($S_3^2$) x 3, $C_2\times C_6^3$ ($C_6:S_3$), $C_2\times C_6^3$ ($S_3^2$)

Order 324: $C_3^2\times C_6^2$ ($C_2\times S_4$)

Order 216: $C_6^2:S_3$ ($C_3:S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_3:S_3^2$), $C_2^2\times C_6^2$ ($C_3^2:D_6$)

Order 108: $C_3\times C_6^2$ ($C_6:S_4$), $C_3\times C_6^2$ ($S_3\times S_4$)

Order 81: $C_3^4$ ($C_2^3:S_4$)

Order 48: $C_2^3\times C_6$ ($C_3^3:D_6$), $C_2^3\times C_6$ ($C_3^2:S_3^2$)

Order 36: $C_6^2$ ($A_4:S_3^2$), $C_6^2$ ($C_6^2:D_6$)

Order 27: $C_3^3$ ($(C_2^2\times C_6):S_4$), $C_3^3$ ($C_2^4:S_3^2$)

Order 16: $C_2^4$ ($C_3^4:D_6$)

Order 12: $C_2\times C_6$ ($C_6^2.S_3^2$), $C_2\times C_6$ ($C_6^2.S_3^2$)

Order 9: $C_3^2$ ($(C_6\times D_6):S_4$), $C_3^2$ ($(C_2^2\times C_6^2):D_6$)

Order 4: $C_2^2$ ($C_3^4:(C_2\times S_4)$)

Order 3: $C_3$ ($(C_2\times C_6^3):D_6$), $C_3$ ($(C_2\times C_6^3):D_6$)

Order 1: $C_1$ ($C_6^4:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 15552: $C_6^4:D_6$ ($C_1$)

Order 7776: $C_6^4.S_3$ ($C_2$), $C_6^4:S_3$ ($C_2$), $C_6^4:C_6$ ($C_2$)

Order 3888: $C_3\times C_3.C_2^4:\He_3$ ($C_2^2$)

Order 2592: $C_3\times C_6^3.C_2^2$ ($S_3$), $(C_3\times C_6^2):S_4$ ($S_3$), $(C_2\times C_6^3):C_6$ ($S_3$)

Order 1296: $C_6^4$ ($D_6$), $C_3\times C_6^2:A_4$ ($D_6$), $(C_2\times C_6^3):C_3$ ($D_6$)

Order 864: $C_6^3.C_2^2$ ($C_3:S_3$)

Order 648: $C_3^4:C_2^3$ ($S_4$)

Order 432: $C_2\times C_6^3$ ($C_6:S_3$), $C_2\times C_6^3$ ($S_3^2$), $C_6^2:A_4$ ($S_3^2$)

Order 324: $C_3^2\times C_6^2$ ($C_2\times S_4$)

Order 216: $C_6^2:S_3$ ($C_3:S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_3:S_3^2$), $C_2^2\times C_6^2$ ($C_3^2:D_6$)

Order 108: $C_3\times C_6^2$ ($C_6:S_4$), $C_3\times C_6^2$ ($S_3\times S_4$)

Order 81: $C_3^4$ ($C_2^3:S_4$)

Order 48: $C_2^3\times C_6$ ($C_3^3:D_6$), $C_2^3\times C_6$ ($C_3^2:S_3^2$)

Order 36: $C_6^2$ ($A_4:S_3^2$), $C_6^2$ ($C_6^2:D_6$)

Order 27: $C_3^3$ ($(C_2^2\times C_6):S_4$), $C_3^3$ ($C_2^4:S_3^2$)

Order 16: $C_2^4$ ($C_3^4:D_6$)

Order 12: $C_2\times C_6$ ($C_6^2.S_3^2$), $C_2\times C_6$ ($C_6^2.S_3^2$)

Order 9: $C_3^2$ ($(C_6\times D_6):S_4$), $C_3^2$ ($(C_2^2\times C_6^2):D_6$)

Order 4: $C_2^2$ ($C_3^4:(C_2\times S_4)$)

Order 3: $C_3$ ($(C_2\times C_6^3):D_6$), $C_3$ ($(C_2\times C_6^3):D_6$)

Order 1: $C_1$ ($C_6^4:D_6$)

Series

Derived series

$C_6^4:D_6$

$\rhd$

$C_6^4:D_6$

$\rhd$

$C_3\times C_3.C_2^4:\He_3$

$\rhd$

$C_3\times C_3.C_2^4:\He_3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_6^4:D_6$

$\rhd$

$C_6^4:D_6$

$\rhd$

$C_6^4:S_3$

$\rhd$

$C_6^4:S_3$

$\rhd$

$C_3\times C_3.C_2^4:\He_3$

$\rhd$

$C_3\times C_3.C_2^4:\He_3$

$\rhd$

$C_6^4$

$\rhd$

$C_6^4$

$\rhd$

$C_3\times C_6^2:A_4$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_6^4:D_6$

$\rhd$

$C_6^4:D_6$

$\rhd$

$C_3\times C_3.C_2^4:\He_3$

$\rhd$

$C_3\times C_3.C_2^4:\He_3$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

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

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $162 \times 162$ 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 $119 \times 119$ rational character table.