Abstract group 139968.j: $C_3^6:C_2^3:S_4$

• Label: 139968.j
• Order: \( 2^{6} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $18$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_3^6:C_2^3:S_4$
Order:	\(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_2^4:S_4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
Composition factors:	$C_2$ x 6, $C_3$ 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	2187	3320	14580	36936	20736	46656	15552	139968
Conjugacy classes	1	6	14	5	26	5	13	2	72
Divisions	1	6	14	5	26	5	9	2	68
Autjugacy classes	1	6	14	5	26	3	9	1	65

Dimension	1	2	3	6	12	16	24	32	48	64	96	192
Irr. complex chars.	4	2	4	4	16	3	12	8	11	1	6	1	72
Irr. rational chars.	4	2	4	4	12	3	12	8	10	1	7	1	68

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$13104$

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:	${\langle a, b, c, d, e, f, g, h, i \mid c^{2}=d^{6}=e^{6}=f^{6}=g^{3}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,12,15,3,11,13,2,10,14)(4,17,8,5,18,9)(6,16,7), (1,8,17,11,3,9,16,10,2,7,18,12)(4,13)(5,15)(6,14), (1,12,18,8,3,10,17,9,2,11,16,7)(4,15,5,13)(6,14)\rangle$
Transitive group:	18T817	18T832	24T17978	24T17979	all 29
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^6:C_2^3)$ $\,\rtimes\,$ $S_4$	$C_3^6$ $\,\rtimes\,$ $(C_2^3:S_4)$	$(C_3^5:D_6:S_4)$ $\,\rtimes\,$ $C_2$	$(C_3^4:D_6\wr C_2)$ $\,\rtimes\,$ $S_3$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^5:D_6)$ . $(C_2\times S_4)$				more information

Elements of the group are displayed as permutations of degree 18.

Homology

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

Subgroups

There are 1749094 subgroups in 3960 conjugacy classes, 11 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^6:C_2^3:S_4$
Commutator:	$G' \simeq$ $C_3^6:(C_2^2:A_4)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^6:C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $C_2^3:S_4$
Radical:	$R \simeq$ $C_3^6:C_2^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $C_2^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
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: 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 139968: $C_3^6:C_2^3:S_4$

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

Order 46656: $C_3^6:C_2\wr C_2^2$

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

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

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

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

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

Order 7776: $C_3^4:D_4:D_6$, $(C_3^2\times S_3^2):D_{12}$

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

Order 5184: $S_3\wr C_2^2$

Order 4374: $C_3^6:S_3$, $C_3\wr S_3$, $C_3\wr C_6$

Order 3888: $C_3^4:(S_3\times D_4)$ x 7, $C_3^4:(C_2\times D_{12})$ x 4, $C_3^4:D_{12}:C_2$ x 3, $C_3^4:D_{12}:C_2$ x 2, $C_3:S_3^4$, $C_3^4:D_4:C_6$, $C_3^2\wr C_2.D_{12}$, $C_3^2\wr C_2.D_{12}$, $(C_3^2\times S_3^2):C_{12}$, $C_3^3:(S_3\times D_{12})$, $S_3\times C_3^3:D_{12}$, $(C_3\times S_3^2):S_3^2$

Order 2916: $C_3^5:D_6$ x 4, $C_3^5:D_6$ x 4, $C_3^6.C_2^2$ x 3, $C_3^5:C_{12}$ x 3, $C_3^4:C_6^2$ x 2, $C_3^5.A_4$ x 2, $C_3^5.D_6$ x 2, $C_3^6.C_2^2$, $C_3^5:C_{12}$, $C_3^5:C_{12}$, $C_3^5:D_6$, $C_3^5:A_4$, $C_3^5:D_6$

Order 2592: $S_3^2\wr C_2$ x 4, $C_3:S_3^3:C_4$ x 3, $C_3^4.C_2^2\wr C_2$, $C_3:S_3^3:C_2^2$, $(C_3\times C_6^2):S_4$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 12, $C_3^4:D_{12}$ x 12, $C_3^5:D_4$ x 8, $C_3^4:(C_4\times S_3)$ x 7, $C_3^2:S_3^3$ x 4, $C_3^5:D_4$ x 4, $C_3^4:(C_2\times C_{12})$ x 4, $C_3^4:D_{12}$ x 3, $C_3^4:D_{12}$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4.S_4$ x 2, $C_3^2:S_3^3$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^4:(C_4\times S_3)$, $C_3^3:(S_3\times C_{12})$, $C_3\wr C_4\times S_3$, $C_3^4:S_4$

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

Order 1296: $C_3^4:(C_2\times D_4)$ x 7, $S_3^2:S_3^2$ x 6, $(C_3^2\times S_3^2):C_4$ x 5, $C_3^2\wr C_2.D_4$ x 4, $C_3^4:D_4:C_2$ x 3, $S_3\times C_3^2:D_{12}$ x 3, $S_3^4$ x 2, $C_6:S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^3:D_{12}$, $(C_3\times C_6^2):A_4$, $S_3^3:S_3$, $C_2\times C_3^3:S_4$, $S_3\wr S_3$, $C_6.S_3^3$, $(C_3^2\times C_6).D_{12}$, $C_3\wr C_2^2:C_4$, $(C_3^2\times D_6):C_{12}$

Order 972: $C_3^3:S_3^2$ x 32, $C_3^3:S_3^2$ x 17, $C_3^3:S_3^2$ x 8, $C_3^4:C_{12}$ x 6, $C_3^3:C_6^2$ x 5, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^5:C_4$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4.A_4$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3\times S_3^2$, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^4:D_6$

Order 864: $C_6^2:D_{12}$, $S_3^2:D_{12}$

Order 729: $C_3^5.C_3$ x 5, $C_3^5:C_3$ x 3, $C_3^6$

Order 648: $C_3:S_3^3$ x 21, $C_3^3:D_{12}$ x 20, $C_3^4:D_4$ x 11, $C_3\wr D_4$ x 10, $C_3:S_3^3$ x 8, $C_3:S_3^3$ x 8, $C_3^4:(C_2\times C_4)$ x 7, $C_3^3:(C_4\times S_3)$ x 6, $C_3^3:S_4$ x 6, $C_3\times S_3^3$ x 5, $C_3^3:D_{12}$ x 5, $C_3^4:C_2^3$ x 3, $C_3^3:D_{12}$ x 3, $S_3\times C_3^2:C_{12}$ x 3, $C_3^4:C_2^3$ x 2, $C_3^3:S_4$ x 2, $C_3^4:C_2^3$, $C_2\times C_3\wr C_4$, $C_6\times C_3^2:C_{12}$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:C_4\times S_3$, $C_2\times C_3^3:A_4$, $S_3\wr C_3$, $C_3^3:D_{12}$, $C_3\times C_6.S_3^2$, $C_3^3:D_{12}$, $C_3^3:S_4$

Order 576: $D_6^2:C_2^2$

Order 486: $C_3^4:C_6$ x 35, $C_3^4:C_6$ x 19, $C_3^4:C_6$ x 6, $C_3^3:D_9$ x 5, $C_3^4.S_3$ x 5, $C_3^3:D_9$ x 4, $C_3^4:S_3$ x 3, $S_3\times C_3^4$ x 2, $C_3^3:C_{18}$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^4.S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 9, $C_2\times C_3^2:D_{12}$ x 6, $C_2\times S_3^3$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_3:D_6^2$, $S_3^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $D_6:S_3^2$, $D_6:S_3^2$, $C_6.\SOPlus(4,2)$, $S_3^2:C_{12}$, $(C_3^2\times D_6):C_4$

Order 324: $C_3^2:S_3^2$ x 63, $C_3^2:S_3^2$ x 63, $C_3\wr C_2^2$ x 49, $C_3\wr C_4$ x 22, $C_3^2\times S_3^2$ x 15, $C_3^2:S_3^2$ x 12, $C_3^3:A_4$ x 7, $C_3^3:C_{12}$ x 5, $C_3^2:S_3^2$ x 4, $C_3^4:C_4$ x 4, $C_3^3:D_6$ x 3, $C_3^2:D_{18}$ x 2, $C_3^2:C_6^2$ x 2, $C_3^2.S_3^2$ x 2, $C_3^3:D_6$, $C_3^3:A_4$, $C_3^3:D_6$, $D_6\times C_3^3$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 288: $D_6\wr C_2$ x 4, $C_6^2.D_4$ x 2, $D_{12}:D_6$, $D_6:D_{12}$, $(C_6\times C_{12}):C_4$

Order 243: $C_3^5$ x 13, $C_3^3:C_9$ x 10, $C_3^4:C_3$ x 6, $C_3^4.C_3$ x 5, $C_3^4:C_3$

Order 216: $S_3^3$ x 35, $C_3^2:D_{12}$ x 31, $S_3^2:S_3$ x 23, $C_6:S_3^2$ x 14, $S_3^2:C_6$ x 10, $C_3^2:C_4\times S_3$ x 9, $C_6\times S_3^2$ x 7, $C_6:S_3^2$ x 6, $C_2\times C_3^2:C_{12}$ x 6, $C_3^2:D_{12}$ x 3, $C_3^2:D_{12}$ x 3, $C_3^2.S_4$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:S_4$, $C_6^2:C_6$, $C_2\times C_3^3:C_4$, $C_3^3:Q_8$, $C_3^3:Q_8$, $C_6^2:C_6$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$

Order 192: $C_2^3:S_4$

Order 162: $C_3^2\wr C_2$ x 145, $C_3^3:C_6$ x 84, $S_3\times C_3^3$ x 35, $C_3^2:D_9$ x 13, $C_3^3:S_3$ x 9, $C_3^3.S_3$ x 5, $C_3^3:S_3$ x 3, $C_3^3:C_6$ x 3, $C_3^2:C_{18}$ x 2, $C_3^3.C_6$ x 2, $D_9:C_3^2$ x 2, $C_3^3:C_6$ x 2, $C_3^2:D_9$ x 2, $C_3\wr S_3$ x 2, $C_3^3\times C_6$, $C_3^3:S_3$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^3:C_6$

Order 144: $S_3^2:C_2^2$ x 9, $S_3^2:C_4$ x 6, $D_6:D_6$ x 5, $C_6^2:C_4$ x 4, $D_6^2$ x 3, $S_3\times D_{12}$ x 3, $C_{12}.D_6$ x 2, $D_6:C_{12}$, $C_6.D_{12}$, $C_6:D_{12}$, $D_{12}:C_6$, $D_6:D_6$, $C_6:D_{12}$

Order 108: $C_3\times S_3^2$ x 104, $C_3:S_3^2$ x 90, $C_3:S_3^2$ x 72, $C_3^2:D_6$ x 30, $C_3^2:C_{12}$ x 29, $C_3^3:C_4$ x 16, $C_3^2\times D_6$ x 10, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_{18}:C_6$ x 2, $C_3^2.A_4$ x 2, $S_3\times D_9$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2:A_4$, $C_3^2:D_6$

Order 96: $C_2^2:S_4$ x 2, $C_2^2:D_{12}$, $C_2^3:A_4$, $D_4:D_6$

Order 81: $C_3^4$ x 123, $C_3\wr C_3$ x 14, $C_3^2:C_9$ x 13, $C_9:C_3^2$ x 5, $C_3\times \He_3$ x 3

Order 72: $S_3\times D_6$ x 40, $\SOPlus(4,2)$ x 26, $C_3:D_{12}$ x 14, $C_2\times C_3^2:C_4$ x 9, $C_6\wr C_2$ x 6, $C_6.D_6$ x 5, $C_6:D_6$ x 4, $C_6\times D_6$ x 4, $C_3:D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_3\times D_{12}$ x 2, $C_2^2:D_9$ x 2, $C_3:S_4$, $\PSU(3,2)$, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6:C_{12}$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$

Order 64: $C_2\wr C_2^2$

Order 54: $C_3^2:C_6$ x 272, $S_3\times C_3^2$ x 148, $C_3^2:S_3$ x 30, $C_3^2\times C_6$ x 13, $C_9:C_6$ x 9, $C_3^2:C_6$ x 6, $C_3:D_9$ x 5, $C_3^2:S_3$ x 2, $S_3\times C_9$ x 2, $C_3\times D_9$ x 2, $C_9:C_6$ x 2, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 8, $C_2\times D_{12}$ x 5, $D_{12}:C_2$ x 3, $D_{12}:C_2$ x 2, $D_6:C_4$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2\times S_4$, $D_4:C_6$, $C_6:D_4$, $C_2^2:C_{12}$

Order 36: $S_3^2$ x 116, $C_6\times S_3$ x 60, $C_6:S_3$ x 33, $C_3^2:C_4$ x 18, $C_3:C_{12}$ x 12, $C_6^2$ x 5, $C_3\times C_{12}$ x 2, $D_{18}$ x 2, $C_2^2:C_9$ x 2, $C_3^2:C_4$, $C_3\times A_4$

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 307, $C_9:C_3$ x 10, $\He_3$ x 6, $C_3\times C_9$ x 5

Order 24: $C_2\times D_6$ x 20, $D_{12}$ x 16, $C_3:D_4$ x 13, $C_4\times S_3$ x 8, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 5, $S_4$ x 4, $C_6:C_4$, $C_3:Q_8$, $C_2^2\times C_6$, $C_2\times A_4$, $C_3\times Q_8$

Order 18: $C_3\times S_3$ x 159, $C_3:S_3$ x 80, $C_3\times C_6$ x 36, $D_9$ x 7, $C_{18}$ x 2

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: $D_6$ x 60, $C_2\times C_6$ x 14, $C_{12}$ x 9, $C_3:C_4$ x 6, $A_4$ x 3

Order 9: $C_3^2$ x 126, $C_9$ x 5

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

Order 6: $S_3$ x 39, $C_6$ x 26

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 139968: $C_3^6:C_2^3:S_4$

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

Order 46656: $C_3^6:C_2\wr C_2^2$

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

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

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

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

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

Order 7776: $C_3^4:D_4:D_6$, $(C_3^2\times S_3^2):D_{12}$

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

Order 5184: $S_3\wr C_2^2$

Order 4374: $C_3^6:S_3$, $C_3\wr S_3$, $C_3\wr C_6$

Order 3888: $C_3^4:(S_3\times D_4)$ x 7, $C_3^4:(C_2\times D_{12})$ x 4, $C_3^4:D_{12}:C_2$ x 3, $C_3^4:D_{12}:C_2$ x 2, $C_3:S_3^4$, $C_3^4:D_4:C_6$, $C_3^2\wr C_2.D_{12}$, $C_3^2\wr C_2.D_{12}$, $(C_3^2\times S_3^2):C_{12}$, $C_3^3:(S_3\times D_{12})$, $S_3\times C_3^3:D_{12}$, $(C_3\times S_3^2):S_3^2$

Order 2916: $C_3^5:D_6$ x 4, $C_3^5:D_6$ x 4, $C_3^6.C_2^2$ x 3, $C_3^5:C_{12}$ x 3, $C_3^4:C_6^2$ x 2, $C_3^6.C_2^2$, $C_3^5:C_{12}$, $C_3^5:C_{12}$, $C_3^5:D_6$, $C_3^5:A_4$, $C_3^5.A_4$, $C_3^5:D_6$, $C_3^5.D_6$

Order 2592: $S_3^2\wr C_2$ x 4, $C_3:S_3^3:C_4$ x 3, $C_3^4.C_2^2\wr C_2$, $C_3:S_3^3:C_2^2$, $(C_3\times C_6^2):S_4$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 12, $C_3^4:D_{12}$ x 12, $C_3^5:D_4$ x 8, $C_3^4:(C_4\times S_3)$ x 7, $C_3^2:S_3^3$ x 4, $C_3^5:D_4$ x 4, $C_3^4:(C_2\times C_{12})$ x 4, $C_3^4:D_{12}$ x 3, $C_3^4:D_{12}$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^2:S_3^3$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^4:(C_4\times S_3)$, $C_3^3:(S_3\times C_{12})$, $C_3\wr C_4\times S_3$, $C_3^4:S_4$, $C_3^4.S_4$

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

Order 1296: $C_3^4:(C_2\times D_4)$ x 7, $S_3^2:S_3^2$ x 6, $(C_3^2\times S_3^2):C_4$ x 5, $C_3^2\wr C_2.D_4$ x 4, $C_3^4:D_4:C_2$ x 3, $S_3\times C_3^2:D_{12}$ x 3, $S_3^4$ x 2, $C_6:S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^3:D_{12}$, $(C_3\times C_6^2):A_4$, $S_3^3:S_3$, $C_2\times C_3^3:S_4$, $S_3\wr S_3$, $C_6.S_3^3$, $(C_3^2\times C_6).D_{12}$, $C_3\wr C_2^2:C_4$, $(C_3^2\times D_6):C_{12}$

Order 972: $C_3^3:S_3^2$ x 32, $C_3^3:S_3^2$ x 17, $C_3^3:S_3^2$ x 8, $C_3^4:C_{12}$ x 6, $C_3^3:C_6^2$ x 5, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^5:C_4$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3\times S_3^2$, $C_3\wr A_4$, $C_3^4.A_4$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3.S_3^2$, $C_3^4:D_6$, $C_3^3:D_{18}$

Order 864: $C_6^2:D_{12}$, $S_3^2:D_{12}$

Order 729: $C_3^5.C_3$ x 3, $C_3^5:C_3$ x 3, $C_3^6$

Order 648: $C_3:S_3^3$ x 21, $C_3^3:D_{12}$ x 20, $C_3^4:D_4$ x 11, $C_3\wr D_4$ x 10, $C_3:S_3^3$ x 8, $C_3:S_3^3$ x 8, $C_3^4:(C_2\times C_4)$ x 7, $C_3^3:(C_4\times S_3)$ x 6, $C_3^3:S_4$ x 6, $C_3\times S_3^3$ x 5, $C_3^3:D_{12}$ x 5, $C_3^4:C_2^3$ x 3, $C_3^3:D_{12}$ x 3, $S_3\times C_3^2:C_{12}$ x 3, $C_3^4:C_2^3$ x 2, $C_3^3:S_4$ x 2, $C_3^4:C_2^3$, $C_2\times C_3\wr C_4$, $C_6\times C_3^2:C_{12}$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:C_4\times S_3$, $C_2\times C_3^3:A_4$, $S_3\wr C_3$, $C_3^3:D_{12}$, $C_3\times C_6.S_3^2$, $C_3^3:D_{12}$, $C_3^3:S_4$

Order 576: $D_6^2:C_2^2$

Order 486: $C_3^4:C_6$ x 35, $C_3^4:C_6$ x 19, $C_3^4:C_6$ x 6, $C_3^3:D_9$ x 3, $C_3^4:S_3$ x 3, $C_3^4.S_3$ x 3, $S_3\times C_3^4$ x 2, $C_3^3:D_9$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^3:C_{18}$, $C_3^4.S_3$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 9, $C_2\times C_3^2:D_{12}$ x 6, $C_2\times S_3^3$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_3:D_6^2$, $S_3^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $D_6:S_3^2$, $D_6:S_3^2$, $C_6.\SOPlus(4,2)$, $S_3^2:C_{12}$, $(C_3^2\times D_6):C_4$

Order 324: $C_3^2:S_3^2$ x 63, $C_3^2:S_3^2$ x 63, $C_3\wr C_2^2$ x 49, $C_3\wr C_4$ x 22, $C_3^2\times S_3^2$ x 15, $C_3^2:S_3^2$ x 12, $C_3^3:A_4$ x 7, $C_3^3:C_{12}$ x 5, $C_3^2:S_3^2$ x 4, $C_3^4:C_4$ x 4, $C_3^3:D_6$ x 3, $C_3^2:C_6^2$ x 2, $C_3^3:D_6$, $C_3^3:A_4$, $C_3^3:D_6$, $C_3^2:D_{18}$, $D_6\times C_3^3$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^2:S_3^2$, $C_3^2.S_3^2$, $C_3^2:S_3^2$

Order 288: $D_6\wr C_2$ x 4, $C_6^2.D_4$ x 2, $D_{12}:D_6$, $D_6:D_{12}$, $(C_6\times C_{12}):C_4$

Order 243: $C_3^5$ x 13, $C_3^4:C_3$ x 6, $C_3^3:C_9$ x 6, $C_3^4.C_3$ x 3, $C_3^4:C_3$

Order 216: $S_3^3$ x 35, $C_3^2:D_{12}$ x 31, $S_3^2:S_3$ x 23, $C_6:S_3^2$ x 14, $S_3^2:C_6$ x 10, $C_3^2:C_4\times S_3$ x 9, $C_6\times S_3^2$ x 7, $C_6:S_3^2$ x 6, $C_2\times C_3^2:C_{12}$ x 6, $C_3^2:D_{12}$ x 3, $C_3^2:D_{12}$ x 3, $C_3^2:D_{12}$ x 2, $C_3^2:S_4$, $C_3^2.S_4$, $C_6^2:C_6$, $C_2\times C_3^3:C_4$, $C_3^3:Q_8$, $C_3^3:Q_8$, $C_6^2:C_6$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$

Order 192: $C_2^3:S_4$

Order 162: $C_3^2\wr C_2$ x 145, $C_3^3:C_6$ x 84, $S_3\times C_3^3$ x 35, $C_3^3:S_3$ x 9, $C_3^2:D_9$ x 7, $C_3^3:S_3$ x 3, $C_3^3.S_3$ x 3, $C_3^3:C_6$ x 3, $C_3^3:C_6$ x 2, $C_3\wr S_3$ x 2, $C_3^3\times C_6$, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $D_9:C_3^2$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^2:D_9$, $C_3^3:C_6$

Order 144: $S_3^2:C_2^2$ x 9, $S_3^2:C_4$ x 6, $D_6:D_6$ x 5, $C_6^2:C_4$ x 4, $D_6^2$ x 3, $S_3\times D_{12}$ x 3, $C_{12}.D_6$ x 2, $D_6:C_{12}$, $C_6.D_{12}$, $C_6:D_{12}$, $D_{12}:C_6$, $D_6:D_6$, $C_6:D_{12}$

Order 108: $C_3\times S_3^2$ x 104, $C_3:S_3^2$ x 90, $C_3:S_3^2$ x 72, $C_3^2:D_6$ x 30, $C_3^2:C_{12}$ x 29, $C_3^3:C_4$ x 16, $C_3^2\times D_6$ x 10, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3\times C_6^2$, $C_{18}:C_6$, $C_3^2:D_6$, $C_3^2:A_4$, $C_3^2.A_4$, $C_3^2:D_6$, $S_3\times D_9$

Order 96: $C_2^2:S_4$ x 2, $C_2^2:D_{12}$, $C_2^3:A_4$, $D_4:D_6$

Order 81: $C_3^4$ x 123, $C_3\wr C_3$ x 14, $C_3^2:C_9$ x 7, $C_9:C_3^2$ x 3, $C_3\times \He_3$ x 3

Order 72: $S_3\times D_6$ x 40, $\SOPlus(4,2)$ x 26, $C_3:D_{12}$ x 14, $C_2\times C_3^2:C_4$ x 9, $C_6\wr C_2$ x 6, $C_6.D_6$ x 5, $C_6:D_6$ x 4, $C_6\times D_6$ x 4, $C_3:D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_3\times D_{12}$ x 2, $C_3:S_4$, $\PSU(3,2)$, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6:C_{12}$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$, $C_2^2:D_9$

Order 64: $C_2\wr C_2^2$

Order 54: $C_3^2:C_6$ x 272, $S_3\times C_3^2$ x 148, $C_3^2:S_3$ x 30, $C_3^2\times C_6$ x 13, $C_3^2:C_6$ x 6, $C_9:C_6$ x 5, $C_3:D_9$ x 3, $C_3^2:S_3$ x 2, $S_3\times C_9$, $C_3\times D_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 8, $C_2\times D_{12}$ x 5, $D_{12}:C_2$ x 3, $D_{12}:C_2$ x 2, $D_6:C_4$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2\times S_4$, $D_4:C_6$, $C_6:D_4$, $C_2^2:C_{12}$

Order 36: $S_3^2$ x 116, $C_6\times S_3$ x 60, $C_6:S_3$ x 33, $C_3^2:C_4$ x 18, $C_3:C_{12}$ x 12, $C_6^2$ x 5, $C_3\times C_{12}$ x 2, $C_3^2:C_4$, $D_{18}$, $C_2^2:C_9$, $C_3\times A_4$

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 307, $C_9:C_3$ x 6, $\He_3$ x 6, $C_3\times C_9$ x 3

Order 24: $C_2\times D_6$ x 20, $D_{12}$ x 16, $C_3:D_4$ x 13, $C_4\times S_3$ x 8, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 5, $S_4$ x 4, $C_6:C_4$, $C_3:Q_8$, $C_2^2\times C_6$, $C_2\times A_4$, $C_3\times Q_8$

Order 18: $C_3\times S_3$ x 159, $C_3:S_3$ x 80, $C_3\times C_6$ x 36, $D_9$ x 4, $C_{18}$

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: $D_6$ x 60, $C_2\times C_6$ x 14, $C_{12}$ x 9, $C_3:C_4$ x 6, $A_4$ x 3

Order 9: $C_3^2$ x 126, $C_9$ x 3

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

Order 6: $S_3$ x 39, $C_6$ x 26

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: $C_1$

Normal subgroups (quotient in parentheses)

Order 139968: $C_3^6:C_2^3:S_4$ ($C_1$)

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

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

Order 23328: $C_3^4:D_6\wr C_2$ ($S_3$)

Order 11664: $C_3^6.C_2^4$ ($D_6$)

Order 5832: $C_3^6:C_2^3$ ($S_4$)

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

Order 729: $C_3^6$ ($C_2^3:S_4$)

Order 1: $C_1$ ($C_3^6:C_2^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 139968: $C_3^6:C_2^3:S_4$ ($C_1$)

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

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

Order 23328: $C_3^4:D_6\wr C_2$ ($S_3$)

Order 11664: $C_3^6.C_2^4$ ($D_6$)

Order 5832: $C_3^6:C_2^3$ ($S_4$)

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

Order 729: $C_3^6$ ($C_2^3:S_4$)

Order 1: $C_1$ ($C_3^6:C_2^3:S_4$)

Series

Derived series

$C_3^6:C_2^3:S_4$

$\rhd$

$C_3^6:C_2^3:S_4$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^6:C_2^3:S_4$

$\rhd$

$C_3^6:C_2^3:S_4$

$\rhd$

$C_3^6:C_2^3:A_4$

$\rhd$

$C_3^6:C_2^3:A_4$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^5:D_6$

$\rhd$

$C_3^5:D_6$

$\rhd$

$C_3^6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^6:C_2^3:S_4$

$\rhd$

$C_3^6:C_2^3:S_4$

$\rhd$

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

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 17 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.