Abstract group 279936.mf: $C_3^6.C_2^4:S_4$

• Label: 279936.mf
• Order: \( 2^{7} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^6.C_2^4:S_4$
Order:	\(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_2^4.C_6.C_2^4$, of order \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \)
Composition factors:	$C_2$ x 7, $C_3$ x 7
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	4375	3320	29160	77192	20736	93312	51840	279936
Conjugacy classes	1	13	14	10	66	5	26	9	144
Divisions	1	13	14	10	66	5	18	9	136
Autjugacy classes	1	10	14	5	55	3	9	4	101

Dimension	1	2	3	6	12	16	24	32	48	64	96	192
Irr. complex chars.	8	4	8	8	32	6	24	16	22	2	12	2	144
Irr. rational chars.	8	4	8	8	24	6	24	16	20	2	14	2	136

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

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:	${\langle a, b, c, d, e, f, g, h, i, j \mid c^{2}=e^{6}=f^{6}=g^{3}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2,4,7,13,11,16,12,17,8,15,9,6,3,5,10,14,18), (1,3,4,8,13,11,6,12,17,10,15,9,16,2,5,7,14,18) \!\cdots\! \rangle$
Transitive group:	36T25809	36T25810	36T25811	36T25812	all 8
Direct product:	$C_2$ $\, \times\, $ $(C_3^6:C_2^3:S_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^6.C_2^4)$ . $S_4$	$C_3^6$ . $(C_2^4:S_4)$	$(C_3^6.C_2^5)$ . $D_6$	$(C_3^5:D_6:S_4)$ . $C_2^2$	all 20

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 30 normal subgroups (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^6.C_2^4.D_6$
Commutator:	$G' \simeq$ $C_3^6:(C_2^2:A_4)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^6.C_2^4:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2^3:S_4$
Radical:	$R \simeq$ $C_3^6.C_2^4:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^5\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 192 (order at least 1458) are shown, as well as all normal subgroups of any index.

Order 279936: $C_3^6.C_2^4:S_4$

Order 139968: $C_3^6:C_2^3:S_4$ x 4, $C_3^6.C_2^3:S_4$, $C_3^3.C_6^3:S_4$, $C_3^6.C_2^4:A_4$

Order 93312: $C_3^6.C_2^4.C_2^3$

Order 69984: $C_3^5:D_6:S_4$ x 2, $C_3^6.C_2^4.S_3$ x 2, $C_3^6:C_2^3:A_4$ x 2, $C_3^3.C_6^3:A_4$, $C_3^6.(C_2^2\times S_4)$

Order 46656: $C_3^6:C_2\wr C_2^2$ x 8, $C_3^4.D_6\wr C_2.C_2$ x 2, $C_3^6.C_2^3.C_2^3$ x 2, $C_3^6.C_2^4.C_2^2$, $C_3^5.D_6.C_2^4$, $C_3^6.C_2^3.C_2^3$

Order 34992: $C_3^6:(C_2\times S_4)$ x 4, $C_2\times C_3^6.S_4$ x 2, $C_3^6.S_4.C_2$, $C_2\times C_3^6:S_4$, $C_3^6.(C_2^2\times A_4)$, $C_3^6:(C_2^2:A_4)$

Order 23328: $C_3^4:D_{12}:D_6$ x 10, $C_3^4:D_6\wr C_2$ x 10, $C_3^6:C_2^3:C_4$ x 8, $C_3^6.C_2.D_4.C_2$ x 4, $C_3^5:D_6.D_4$ x 4, $C_3^6.C_2^3.C_2^2$ x 3, $C_3^6.C_2.C_2^4$ x 3, $C_3^6.D_4.C_2^2$ x 2, $C_3^6.D_4.C_2^2$ x 2, $C_3^6.C_2^4.C_2$ x 2, $C_3^5.C_6.C_2.C_2^3$, $C_3^5.D_6.C_2^3$, $C_3^6.C_2.C_2^4$, $C_3^5.C_6.C_2^4$, $C_3^6.C_2^5$, $C_3^5.D_6.C_2^3$, $C_3^5.D_6:C_4.C_2$

Order 17496: $C_3^6.S_4$ x 4, $C_2\times C_3^6.A_4$ x 2, $C_3^6:S_4$ x 2, $C_3^6:S_4$ x 2, $C_3^6:(C_2\times A_4)$ x 2, $C_2\times C_3^6:D_6$, $C_2\times C_3\wr A_4$

Order 15552: $C_3^5.C_2^4.C_2^2$, $C_3^4.D_6.C_2^4$

Order 11664: $C_3^5:(S_3\times D_4)$ x 26, $C_3^5:D_{12}:C_2$ x 16, $C_3^4:(S_3\times D_{12})$ x 16, $C_3^4:C_6^2:C_4$ x 10, $C_3^4:D_{12}:C_6$ x 8, $C_3^6.D_4.C_2$ x 5, $C_3^6.C_2.C_2^3$ x 4, $C_3^6.C_2^4$ x 4, $C_3^6.C_2^4$ x 3, $C_3^5.C_6.C_2^3$ x 3, $C_3^6.C_2^4$ x 3, $C_3^5:D_6:C_4$ x 3, $C_3^6.C_4.C_2^2$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^6.C_2^3.C_2$ x 2, $C_3^6.C_2.C_2^3$ x 2, $C_3^5.C_6.C_4.C_2$ x 2, $C_3^5.C_6.C_2^2.C_2$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^6.C_2^3.C_2$ x 2, $C_3^4.D_6:D_6$ x 2, $C_3^6.D_4.C_2$ x 2, $C_2\times C_3^5.S_4$ x 2, $C_3^5.C_6.C_2^3$, $C_3^6.D_4.C_2$, $C_3^5.C_6.C_2^3$, $C_3^6.C_2.C_2^3$, $C_3^5.C_6.C_2^3$, $C_3^6.C_2.C_2^3$, $C_3^6.C_2^4$, $C_2\times C_3^5:S_4$

Order 10368: $S_3^4:C_2^3$

Order 8748: $C_3^6:D_6$ x 4, $C_3^6.A_4$ x 2, $C_3^6.D_6$, $C_3^6.D_6$, $C_3^5.C_6^2$, $C_3\wr A_4$

Order 7776: $C_3^4:D_4:D_6$ x 16, $(C_3^2\times S_3^2):D_{12}$ x 8, $C_3^5.C_2^3.C_2^2$ x 5, $C_3^5.D_4.C_2^2$ x 2, $C_3^5.D_4.C_2^2$ x 2, $C_3^4.D_6.C_2^3$ x 2, $C_3^5.D_4.C_2^2$ x 2, $C_3^5.D_4.C_2^2$ x 2, $C_3^5.C_2^3.C_2^2$, $C_3^3.D_6:D_6.C_2$, $C_3^4.D_6.C_2^3$, $C_3^4.(C_6\times D_4:C_2)$, $C_3^5.C_2^5$, $C_3^5.C_2^3.C_2^2$, $C_3^4.(C_2\times C_6).C_2^3$, $C_3^4.D_6:C_4.C_2$, $C_3^5.D_4:C_2.C_2$

Order 5832: $C_3^5:D_{12}$ x 20, $C_3^6.C_2^3$ x 15, $C_3^6:(C_2\times C_4)$ x 13, $C_3^6:D_4$ x 12, $C_3^6:C_2^3$ x 12, $C_3^5:D_{12}$ x 8, $C_3^4:(S_3\times C_{12})$ x 8, $C_3^6.C_2^3$ x 6, $C_3^6.C_2^3$ x 4, $C_3^6.D_4$ x 4, $C_3^6.C_2^3$ x 4, $C_3^6:Q_8$ x 4, $C_3^5:D_{12}$ x 4, $C_3^5.S_4$ x 4, $C_3^6.C_2^3$ x 3, $C_3^6.C_4.C_2$ x 3, $C_3^4:(S_3\times C_{12})$ x 3, $C_3^6.C_2^3$ x 2, $C_3^5.C_{12}.C_2$ x 2, $C_3^5.C_6.C_2^2$ x 2, $C_3^6.C_2^2.C_2$ x 2, $C_2\times C_3^5.A_4$ x 2, $C_2\times C_3^5.D_6$ x 2, $C_3^5:S_4$ x 2, $C_3^6.C_2^3$, $C_3^6.C_4.C_2$, $C_3^6.C_4.C_2$, $C_3^6.C_4.C_2$, $C_2\times C_3^5:D_6$, $C_2\times C_3^5:A_4$, $C_2\times C_3^5:D_6$

Order 5184: $S_3\wr C_2^2$ x 8, $S_3^4:C_2^2$ x 4, $C_6:S_3^3:C_4$ x 3, $C_3^4.C_2^4.C_2^2$, $C_6:S_3^3:C_2^2$, $C_6^3:S_4$

Order 4374: $C_3^6:S_3$ x 2, $C_3\wr S_3$ x 2, $C_3\wr C_6$ x 2, $C_2:(C_3^6.C_3)$

Order 3888: $C_3^4:(S_3\times D_4)$ x 56, $C_3^4:(C_2\times D_{12})$ x 30, $C_3^4:D_{12}:C_2$ x 24, $C_3^4:D_{12}:C_2$ x 16, $C_3^5.C_2^4$ x 12, $C_3^5.D_4.C_2$ x 12, $C_3:S_3^4$ x 10, $C_3^5.D_4.C_2$ x 8, $C_3^4:D_4:C_6$ x 8, $S_3\times C_3^3:D_{12}$ x 8, $C_3^3:(S_3\times D_{12})$ x 6, $(C_3\times S_3^2):S_3^2$ x 6, $C_3^5.C_2^2.C_2^2$ x 5, $C_3^5.C_2^4$ x 4, $C_3^4.(C_6\times D_4)$ x 4, $C_3^2\wr C_2.D_{12}$ x 4, $C_3^2\wr C_2.D_{12}$ x 4, $(C_3^2\times S_3^2):C_{12}$ x 4, $C_3^5.D_4.C_2$ x 3, $C_3^4.(C_6\times D_4)$ x 3, $C_3^5.C_2^4$ x 2, $C_3^5.C_4.C_2^2$ x 2, $C_3^5.C_2^2.C_2^2$ x 2, $C_3^5.C_2^4$ x 2, $C_3^5.D_4.C_2$ x 2, $C_3^5.C_4.C_2^2$ x 2, $C_2\times C_3^4.S_4$ x 2, $C_3^4.(C_6\times D_4)$, $C_3^5.C_2^4$, $C_3^5.C_2^2.C_2^2$, $C_3^5.C_2^2.C_2^2$, $C_3^4.(C_6\times Q_8)$, $C_3^5.C_2^2.C_2^2$, $C_3^5.Q_8.C_2$, $C_3^5.D_4.C_2$, $C_2\times C_3^4:S_4$

Order 2916: $C_3^5:D_6$ x 13, $C_3^6.C_2^2$ x 10, $C_3^5:D_6$ x 10, $C_3^5.D_6$ x 8, $C_3^4:C_6^2$ x 6, $C_3^5:C_{12}$ x 6, $C_3^4.(C_6\times S_3)$ x 5, $C_3^5:D_6$ x 4, $C_3^5:D_6$ x 4, $C_3^4.(C_6\times S_3)$ x 3, $C_3^6.C_2^2$ x 3, $C_3^6.C_2^2$ x 2, $C_3^6.C_2^2$ x 2, $C_3^5:C_{12}$ x 2, $C_3^5:C_{12}$ x 2, $C_3^5.D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3^5.A_4$ x 2, $C_3^6.C_2^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^5:A_4$

Order 2592: $S_3^2\wr C_2$ x 26, $C_3:S_3^3:C_4$ x 12, $C_3:S_3^3:C_2^2$ x 10, $C_3^4:(C_2^2\times D_4)$ x 7, $C_3^4.C_2^2\wr C_2$ x 6, $C_2\times S_3^2:S_3^2$ x 6, $C_3:S_3^3:C_4$ x 5, $C_3:S_3^3:C_4$ x 4, $C_3^2:D_{12}\times D_6$ x 3, $C_3^4:D_4:C_2^2$ x 3, $C_3^4.C_2^4.C_2$ x 2, $C_2\times S_3^4$ x 2, $(C_3\times C_6^2):S_4$ x 2, $C_3^3.(C_6\times C_2^2:C_4)$, $C_3^4.C_2^5$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^3.C_2^2$, $C_3^2.(C_6\times D_6:C_4)$, $C_3^4.C_2^3.C_2^2$, $C_3^3.D_6:C_4.C_2$, $S_3^3:D_6$, $C_2^2\times C_3^3:S_4$, $S_3^3:D_6$, $C_6^3:A_4$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 69, $C_3^4:D_{12}$ x 48, $C_3^5:C_2^3$ x 32, $C_3^5:D_4$ x 32, $C_3^2:S_3^3$ x 30, $C_3^4:(C_4\times S_3)$ x 28, $C_3^5:C_2^3$ x 17, $C_3^5:D_4$ x 16, $C_3^4:(C_2\times C_{12})$ x 15, $C_3^2:S_3^3$ x 14, $C_3^2:S_3^3$ x 14, $C_3^4:D_{12}$ x 12, $C_3^4:D_{12}$ x 12, $C_3^5:C_2^3$ x 8, $C_3^4:D_{12}$ x 8, $C_2\times C_3^4:C_{12}$ x 6, $C_3^2:S_3^3$ x 6, $C_3^5:C_2^3$ x 5, $C_3^5:C_2^3$ x 4, $C_3^5:Q_8$ x 4, $C_3^5:Q_8$ x 4, $C_3^4:D_{12}$ x 4, $C_3^5:D_4$ x 4, $C_3\wr C_4\times S_3$ x 4, $C_3^4.S_4$ x 4, $C_3^5:C_2^3$ x 3, $C_3^5:C_2^3$ x 3, $C_2\times C_3^5:C_4$ x 3, $C_2\times C_3^4:C_{12}$ x 3, $C_3\wr C_4\times C_6$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3^3:(S_3\times C_{12})$ x 3, $C_2\times C_3^3:D_{18}$ x 2, $C_3^4:S_4$ x 2, $C_2\times C_3^4.A_4$ x 2, $C_2\times C_3^3.S_3^2$ x 2, $C_3^2:C_6^3$, $C_2\times C_3\wr A_4$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^4:D_6$, $C_2\times C_3^4:D_6$

Order 1728: $C_6^3:D_4$, $D_6^2.D_6$

Order 1458: $C_3^5.S_3$ x 10, $C_3^5:C_6$ x 6, $C_3^5:S_3$ x 6, $C_3^5.C_6$ x 5, $C_3^3\wr C_2$ x 4, $C_3^5.S_3$ x 4, $C_3^5.C_6$ x 4, $C_3^5:C_6$ x 3, $C_3^5:C_6$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^5\times C_6$

Order 729: $C_3^6$

Order 384: $C_2^4:S_4$

Order 128: $C_2^4:D_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 192 (order at least 1458) are shown, as well as all normal subgroups of any index.

Order 279936: $C_3^6.C_2^4:S_4$

Order 139968: $C_3^6.C_2^3:S_4$, $C_3^3.C_6^3:S_4$, $C_3^6.C_2^4:A_4$, $C_3^6:C_2^3:S_4$

Order 93312: $C_3^6.C_2^4.C_2^3$

Order 69984: $C_3^5:D_6:S_4$, $C_3^6.C_2^4.S_3$, $C_3^6:C_2^3:A_4$, $C_3^3.C_6^3:A_4$, $C_3^6.(C_2^2\times S_4)$

Order 46656: $C_3^4.D_6\wr C_2.C_2$ x 2, $C_3^6.C_2^3.C_2^3$ x 2, $C_3^6:C_2\wr C_2^2$ x 2, $C_3^6.C_2^4.C_2^2$, $C_3^5.D_6.C_2^4$, $C_3^6.C_2^3.C_2^3$

Order 34992: $C_2\times C_3^6.S_4$ x 2, $C_3^6.S_4.C_2$, $C_3^6:(C_2\times S_4)$, $C_2\times C_3^6:S_4$, $C_3^6.(C_2^2\times A_4)$, $C_3^6:(C_2^2:A_4)$

Order 23328: $C_3^4:D_6\wr C_2$ x 5, $C_3^4:D_{12}:D_6$ x 4, $C_3^6.C_2^3.C_2^2$ x 3, $C_3^6.C_2.C_2^4$ x 3, $C_3^6.D_4.C_2^2$ x 2, $C_3^6.D_4.C_2^2$ x 2, $C_3^6.C_2.D_4.C_2$ x 2, $C_3^6:C_2^3:C_4$ x 2, $C_3^5.C_6.C_2.C_2^3$, $C_3^6.C_2^4.C_2$, $C_3^5.D_6.C_2^3$, $C_3^6.C_2.C_2^4$, $C_3^5.C_6.C_2^4$, $C_3^6.C_2^5$, $C_3^5.D_6.C_2^3$, $C_3^5.D_6:C_4.C_2$, $C_3^5:D_6.D_4$

Order 17496: $C_2\times C_3^6.A_4$ x 2, $C_3^6.S_4$ x 2, $C_2\times C_3^6:D_6$, $C_2\times C_3\wr A_4$, $C_3^6:S_4$, $C_3^6:S_4$, $C_3^6:(C_2\times A_4)$

Order 15552: $C_3^5.C_2^4.C_2^2$, $C_3^4.D_6.C_2^4$

Order 11664: $C_3^5:(S_3\times D_4)$ x 14, $C_3^4:C_6^2:C_4$ x 6, $C_3^6.D_4.C_2$ x 5, $C_3^6.C_2.C_2^3$ x 4, $C_3^6.C_2^4$ x 4, $C_3^5:D_{12}:C_2$ x 4, $C_3^4:(S_3\times D_{12})$ x 4, $C_3^6.C_2^4$ x 3, $C_3^5.C_6.C_2^3$ x 3, $C_3^6.C_2^4$ x 3, $C_3^6.C_4.C_2^2$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^4:D_{12}:C_6$ x 2, $C_3^5:D_6:C_4$ x 2, $C_3^5.C_6.C_2^3$, $C_3^6.D_4.C_2$, $C_3^5.C_6.C_2^3$, $C_3^6.C_2^3.C_2$, $C_3^6.C_2.C_2^3$, $C_3^6.C_2.C_2^3$, $C_3^5.C_6.C_2^3$, $C_3^6.C_2.C_2^3$, $C_3^5.C_6.C_4.C_2$, $C_3^5.C_6.C_2^2.C_2$, $C_3^6.C_2^4$, $C_3^6.C_2^3.C_2$, $C_3^4.D_6:D_6$, $C_2\times C_3^5:S_4$, $C_2\times C_3^5.S_4$

Order 10368: $S_3^4:C_2^3$

Order 8748: $C_3^6.A_4$ x 2, $C_3^6.D_6$, $C_3^6.D_6$, $C_3^5.C_6^2$, $C_3^6:D_6$, $C_3\wr A_4$

Order 7776: $C_3^5.C_2^3.C_2^2$ x 5, $C_3^4:D_4:D_6$ x 4, $(C_3^2\times S_3^2):D_{12}$ x 4, $C_3^5.D_4.C_2^2$ x 2, $C_3^5.D_4.C_2^2$ x 2, $C_3^4.D_6.C_2^3$ x 2, $C_3^5.D_4.C_2^2$ x 2, $C_3^5.D_4.C_2^2$ x 2, $C_3^5.C_2^3.C_2^2$, $C_3^3.D_6:D_6.C_2$, $C_3^4.D_6.C_2^3$, $C_3^4.(C_6\times D_4:C_2)$, $C_3^5.C_2^5$, $C_3^5.C_2^3.C_2^2$, $C_3^4.(C_2\times C_6).C_2^3$, $C_3^4.D_6:C_4.C_2$, $C_3^5.D_4:C_2.C_2$

Order 5832: $C_3^6.C_2^3$ x 14, $C_3^5:D_{12}$ x 8, $C_3^6:C_2^3$ x 7, $C_3^6:(C_2\times C_4)$ x 7, $C_3^6.C_2^3$ x 6, $C_3^6:D_4$ x 6, $C_3^6.C_2^3$ x 4, $C_3^6.C_2^3$ x 4, $C_3^5:D_{12}$ x 4, $C_3^6.C_2^3$ x 3, $C_3^6.C_4.C_2$ x 3, $C_3^6.C_2^3$ x 2, $C_3^6.D_4$ x 2, $C_3^4:(S_3\times C_{12})$ x 2, $C_3^4:(S_3\times C_{12})$ x 2, $C_3^6.C_2^3$, $C_3^6.C_4.C_2$, $C_3^5.C_{12}.C_2$, $C_3^5.C_6.C_2^2$, $C_3^6.C_4.C_2$, $C_3^6.C_2^2.C_2$, $C_3^6.C_4.C_2$, $C_2\times C_3^5:D_6$, $C_2\times C_3^5:A_4$, $C_2\times C_3^5.A_4$, $C_2\times C_3^5:D_6$, $C_2\times C_3^5.D_6$, $C_3^6:Q_8$, $C_3^5:D_{12}$, $C_3^5:S_4$, $C_3^5.S_4$

Order 5184: $S_3^4:C_2^2$ x 4, $C_6:S_3^3:C_4$ x 3, $S_3\wr C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$, $C_6:S_3^3:C_2^2$, $C_6^3:S_4$

Order 4374: $C_2:(C_3^6.C_3)$, $C_3^6:S_3$, $C_3\wr S_3$, $C_3\wr C_6$

Order 3888: $C_3^4:(S_3\times D_4)$ x 20, $C_3^5.C_2^4$ x 12, $C_3^5.D_4.C_2$ x 12, $C_3^4:(C_2\times D_{12})$ x 12, $C_3:S_3^4$ x 10, $C_3^5.D_4.C_2$ x 8, $C_3^4:D_{12}:C_2$ x 6, $C_3^5.C_2^2.C_2^2$ x 5, $C_3^5.C_2^4$ x 4, $C_3^4.(C_6\times D_4)$ x 4, $C_3^4:D_{12}:C_2$ x 4, $C_3^3:(S_3\times D_{12})$ x 4, $S_3\times C_3^3:D_{12}$ x 4, $(C_3\times S_3^2):S_3^2$ x 4, $C_3^5.D_4.C_2$ x 3, $C_3^4.(C_6\times D_4)$ x 3, $C_3^5.C_2^4$ x 2, $C_3^5.C_4.C_2^2$ x 2, $C_3^5.C_2^2.C_2^2$ x 2, $C_3^5.C_2^4$ x 2, $C_3^5.D_4.C_2$ x 2, $C_3^5.C_4.C_2^2$ x 2, $C_3^4:D_4:C_6$ x 2, $C_3^2\wr C_2.D_{12}$ x 2, $C_3^2\wr C_2.D_{12}$ x 2, $(C_3^2\times S_3^2):C_{12}$ x 2, $C_3^4.(C_6\times D_4)$, $C_3^5.C_2^4$, $C_3^5.C_2^2.C_2^2$, $C_3^5.C_2^2.C_2^2$, $C_3^4.(C_6\times Q_8)$, $C_3^5.C_2^2.C_2^2$, $C_3^5.Q_8.C_2$, $C_3^5.D_4.C_2$, $C_2\times C_3^4.S_4$, $C_2\times C_3^4:S_4$

Order 2916: $C_3^6.C_2^2$ x 9, $C_3^5:D_6$ x 9, $C_3^5:D_6$ x 7, $C_3^4:C_6^2$ x 6, $C_3^4.(C_6\times S_3)$ x 3, $C_3^6.C_2^2$ x 3, $C_3^4.(C_6\times S_3)$ x 3, $C_3^5:C_{12}$ x 3, $C_3^6.C_2^2$ x 2, $C_3^6.C_2^2$ x 2, $C_3^6.C_2^2$, $C_3^5:C_{12}$, $C_3^5:C_{12}$, $C_3^5:D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^5:D_6$, $C_3^5:A_4$, $C_3^5.A_4$, $C_3^5:D_6$, $C_3^5.D_6$

Order 2592: $S_3^2\wr C_2$ x 13, $C_3^4:(C_2^2\times D_4)$ x 7, $C_2\times S_3^2:S_3^2$ x 6, $C_3:S_3^3:C_4$ x 5, $C_3:S_3^3:C_4$ x 4, $C_3:S_3^3:C_2^2$ x 4, $C_3^4.C_2^2\wr C_2$ x 3, $C_3^2:D_{12}\times D_6$ x 3, $C_3^4:D_4:C_2^2$ x 3, $C_3:S_3^3:C_4$ x 3, $C_2\times S_3^4$ x 2, $C_3^3.(C_6\times C_2^2:C_4)$, $C_3^4.C_2^5$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^3.C_2^2$, $C_3^2.(C_6\times D_6:C_4)$, $C_3^4.C_2^3.C_2^2$, $C_3^3.D_6:C_4.C_2$, $C_3^4.C_2^4.C_2$, $S_3^3:D_6$, $C_2^2\times C_3^3:S_4$, $S_3^3:D_6$, $(C_3\times C_6^2):S_4$, $C_6^3:A_4$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 48, $C_3^5:C_2^3$ x 32, $C_3^2:S_3^3$ x 28, $C_3^4:D_{12}$ x 21, $C_3^5:C_2^3$ x 17, $C_3^2:S_3^3$ x 14, $C_3^2:S_3^3$ x 14, $C_3^5:D_4$ x 14, $C_3^4:(C_4\times S_3)$ x 10, $C_3^5:C_2^3$ x 8, $C_3^5:D_4$ x 7, $C_2\times C_3^4:C_{12}$ x 6, $C_3^2:S_3^3$ x 6, $C_3^4:D_{12}$ x 6, $C_3^4:D_{12}$ x 6, $C_3^4:(C_2\times C_{12})$ x 6, $C_3^5:C_2^3$ x 5, $C_3^5:C_2^3$ x 4, $C_3^4:D_{12}$ x 4, $C_3^5:C_2^3$ x 3, $C_3^5:C_2^3$ x 3, $C_2\times C_3^5:C_4$ x 3, $C_2\times C_3^4:C_{12}$ x 3, $C_3\wr C_4\times C_6$ x 3, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $C_3\wr C_4\times S_3$ x 2, $C_2\times C_3^3:D_{18}$, $C_3^2:C_6^3$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_2\times C_3\wr A_4$, $C_3^4:S_4$, $C_2\times C_3^4.A_4$, $C_3^4.S_4$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^4:D_6$, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^4:D_6$

Order 1728: $C_6^3:D_4$, $D_6^2.D_6$

Order 1458: $C_3^5:C_6$ x 5, $C_3^5.C_6$ x 3, $C_3^5:C_6$ x 3, $C_3^5.S_3$ x 3, $C_3^5:S_3$ x 3, $C_3^3\wr C_2$ x 2, $C_3^5:C_6$ x 2, $C_3^5\times C_6$, $C_3^5.S_3$, $C_3^5.C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$

Order 729: $C_3^6$

Order 384: $C_2^4:S_4$

Order 128: $C_2^4:D_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 279936: $C_3^6.C_2^4:S_4$ ($C_1$)

Order 139968: $C_3^6:C_2^3:S_4$ ($C_2$) x 4, $C_3^6.C_2^3:S_4$ ($C_2$), $C_3^3.C_6^3:S_4$ ($C_2$), $C_3^6.C_2^4:A_4$ ($C_2$)

Order 69984: $C_3^5:D_6:S_4$ ($C_2^2$) x 2, $C_3^6.C_2^4.S_3$ ($C_2^2$) x 2, $C_3^6:C_2^3:A_4$ ($C_2^2$) x 2, $C_3^3.C_6^3:A_4$ ($C_2^2$)

Order 46656: $C_3^4.D_6\wr C_2.C_2$ ($S_3$)

Order 34992: $C_3^6:(C_2^2:A_4)$ ($C_2^3$)

Order 23328: $C_3^4:D_6\wr C_2$ ($D_6$) x 2, $C_3^6.C_2^5$ ($D_6$)

Order 11664: $C_3^6.C_2^4$ ($C_2\times D_6$), $C_3^6.C_2^4$ ($S_4$)

Order 5832: $C_3^6:C_2^3$ ($C_2\times S_4$) x 2, $C_3^6.C_2^3$ ($C_2\times S_4$)

Order 2916: $C_3^5:D_6$ ($C_2^2\times S_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^3:S_4$)

Order 729: $C_3^6$ ($C_2^4:S_4$)

Order 2: $C_2$ ($C_3^6.C_2^4.D_6$)

Order 1: $C_1$ ($C_3^6.C_2^4:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 279936: $C_3^6.C_2^4:S_4$ ($C_1$)

Order 139968: $C_3^6.C_2^3:S_4$ ($C_2$), $C_3^3.C_6^3:S_4$ ($C_2$), $C_3^6.C_2^4:A_4$ ($C_2$), $C_3^6:C_2^3:S_4$ ($C_2$)

Order 69984: $C_3^5:D_6:S_4$ ($C_2^2$), $C_3^6.C_2^4.S_3$ ($C_2^2$), $C_3^6:C_2^3:A_4$ ($C_2^2$), $C_3^3.C_6^3:A_4$ ($C_2^2$)

Order 46656: $C_3^4.D_6\wr C_2.C_2$ ($S_3$)

Order 34992: $C_3^6:(C_2^2:A_4)$ ($C_2^3$)

Order 23328: $C_3^6.C_2^5$ ($D_6$), $C_3^4:D_6\wr C_2$ ($D_6$)

Order 11664: $C_3^6.C_2^4$ ($C_2\times D_6$), $C_3^6.C_2^4$ ($S_4$)

Order 5832: $C_3^6.C_2^3$ ($C_2\times S_4$), $C_3^6:C_2^3$ ($C_2\times S_4$)

Order 2916: $C_3^5:D_6$ ($C_2^2\times S_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^3:S_4$)

Order 729: $C_3^6$ ($C_2^4:S_4$)

Order 2: $C_2$ ($C_3^6.C_2^4.D_6$)

Order 1: $C_1$ ($C_3^6.C_2^4:S_4$)

Series

Derived series

$C_3^6.C_2^4:S_4$

$\rhd$

$C_3^6:(C_2^2:A_4)$

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^6.C_2^4:S_4$

$\rhd$

$C_3^6:C_2^3:S_4$

$\rhd$

$C_3^6.C_2^4.S_3$

$\rhd$

$C_3^6:(C_2^2:A_4)$

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^5:D_6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$C_3^6.C_2^4:S_4$

$\rhd$

$C_3^6:(C_2^2:A_4)$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 11 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 $136 \times 136$ rational character table (warning: may be slow to load).