Abstract group 11664.jb: $C_3^5:(S_3\times D_4)$

• Label: 11664.jb
• Order: \( 2^{4} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^5:(S_3\times D_4)$
Order:	\(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3\times C_3^4.C_{12}.C_2^4$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 6
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	9	12	18	36
Elements	1	219	404	180	6180	324	2088	1620	648	11664
Conjugacy classes	1	7	32	2	120	6	18	10	2	198
Divisions	1	7	20	2	68	3	10	5	1	117
Autjugacy classes	1	5	9	2	25	2	6	3	1	54

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	24	18	48	27	16	12	49	0	4	198
Irr. rational chars.	8	14	0	15	28	13	7	4	28	117

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$9434880$

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:	${\langle a, b, c, d, e, f \mid b^{6}=c^{12}=d^{3}=e^{3}=f^{3}=[a,e]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,4,7,3,6,8)(10,11,13,15)(12,14), (1,2,4)(3,5,7)(6,8,9)(10,11)(12,15,13,14), (1,3,7,5,8,6,9,4,2)(10,12)(11,14,15)\rangle$
Transitive group:	36T9358				more information
Direct product:	$\SOPlus(4,2)$ $\, \times\, $ $(C_3\wr S_3)$
Semidirect product:	$(C_3^5:D_4)$ $\,\rtimes\,$ $S_3$	$(C_3^5:S_3)$ $\,\rtimes\,$ $D_4$	$(C_3^5:S_3)$ $\,\rtimes\,$ $D_4$	$C_3^5$ $\,\rtimes\,$ $(S_3\times D_4)$	all 36
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $(C_{12}:D_6)$	$C_3^2$ . $(S_3^3:C_6)$	$(C_3^5:D_6)$ . $C_2^2$	$C_3^3$ . $(C_6^2:D_6)$	all 18

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:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 61736 subgroups in 2148 conjugacy classes, 70 normal (46 characteristic).

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

Special subgroups

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

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 11664: $C_3^5:(S_3\times D_4)$

Order 5832: $(C_3^3\times S_3^2):S_3$ x 2, $C_3^3:S_3^3$ x 2, $(C_3^3\times S_3^2):C_6$, $C_3^5:D_{12}$, $C_3^5:(C_4\times S_3)$

Order 3888: $C_3\times S_3^3:C_6$, $C_3^4:(S_3\times D_4)$

Order 2916: $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3^5:D_6$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:C_{12}$

Order 1944: $C_3^2\times S_3^3$ x 2, $C_3^5:D_4$ x 2, $C_3^2:S_3^3$ x 2, $\He_3:\SOPlus(4,2)$ x 2, $C_3\wr S_3\times D_6$ x 2, $C_3^4:D_{12}$, $C_3^5:D_4$, $S_3\times C_3^3:C_{12}$, $C_3^4:D_{12}$, $(C_9\times S_3^2):C_6$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$

Order 1458: $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1296: $S_3^3:C_6$ x 5, $S_3^2:C_6^2$, $C_3\wr S_3\times D_4$

Order 972: $C_3^3:S_3^2$ x 12, $C_3^3\times S_3^2$ x 4, $C_3^3:C_6^2$ x 3, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3.C_6^2$ x 2, $\He_3\times S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3.C_6^2$ x 2, $C_3^4:D_6$ x 2, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $(C_3^2\times \He_3):C_4$, $C_3^4.C_{12}$, $C_3^4:C_{12}$

Order 729: $C_3^5:C_3$

Order 648: $C_3\times S_3^3$ x 10, $C_3\wr D_4$ x 10, $C_3^4:D_4$ x 9, $C_3^3:D_{12}$ x 5, $S_3\times C_3^2:C_{12}$ x 5, $C_3^4:C_2^3$ x 4, $S_3\times C_3^2:D_6$ x 2, $C_2^2\times C_3\wr S_3$ x 2, $(C_3\times C_6^2):S_3$ x 2, $C_6\times C_3^2:C_{12}$, $S_3^2:C_{18}$, $C_3\wr C_3\times D_4$, $C_3^3:D_{12}$, $C_4\times C_3\wr S_3$

Order 486: $C_3^4:C_6$ x 10, $C_3^4:S_3$ x 8, $C_3^4:S_3$ x 8, $C_3^4:C_6$ x 3, $S_3\times C_3^4$ x 3, $C_3^4:S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $C_6\times \SOPlus(4,2)$ x 4, $S_3^3:C_2$ x 2, $C_{12}:C_6^2$, $C_6^2:D_6$

Order 324: $C_3^2\times S_3^2$ x 44, $C_3^2:S_3^2$ x 15, $C_3^2:S_3^2$ x 12, $C_3^3:D_6$ x 11, $C_3\wr C_2^2$ x 10, $C_3^3:C_{12}$ x 7, $D_6\times C_3^3$ x 6, $C_3\wr C_4$ x 5, $C_3^2:C_6^2$ x 3, $C_3^2.C_6^2$ x 2, $C_3^3:D_6$ x 2, $C_3^3:D_6$ x 2, $C_3^2.C_6^2$ x 2, $D_6\times \He_3$ x 2, $C_9\times S_3^2$ x 2, $C_3^3:C_{12}$, $\He_3:C_{12}$, $C_3^2:C_{36}$

Order 243: $C_3^4:C_3$ x 8, $C_3^5$, $C_9:C_3^3$, $C_3^2\times \He_3$

Order 216: $C_6\times S_3^2$ x 18, $S_3^2:C_6$ x 18, $S_3\times C_6^2$ x 4, $C_2\times C_3^2:C_{12}$ x 4, $S_3^3$ x 4, $S_3^2:S_3$ x 4, $C_6^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_6^2:C_6$ x 2, $C_6^2:S_3$ x 2, $C_{36}:C_6$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $D_4\times C_3^3$, $C_3^2\times D_{12}$, $C_6.C_6^2$

Order 162: $S_3\times C_3^3$ x 47, $C_3^2\wr C_2$ x 31, $C_3\wr S_3$ x 14, $C_3^3:C_6$ x 9, $C_3^3:S_3$ x 8, $C_3^3:S_3$ x 8, $C_3^3.C_6$ x 8, $S_3\times \He_3$ x 8, $C_3^3:C_6$ x 5, $C_3^3\times C_6$ x 3, $C_{18}:C_3^2$ x 2, $C_6\times \He_3$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$

Order 144: $C_{12}:D_6$ x 5, $S_3^2:C_2^2$, $C_4:C_6^2$

Order 108: $C_3\times S_3^2$ x 92, $C_3^2\times D_6$ x 65, $C_3^2:D_6$ x 14, $C_3^2:D_6$ x 11, $C_3^2:C_{12}$ x 10, $C_3:S_3^2$ x 6, $C_3\times C_6^2$ x 4, $C_3:S_3^2$ x 4, $C_3^3:C_4$ x 2, $C_{18}:C_6$ x 2, $C_2^2\times \He_3$ x 2, $S_3\times C_{18}$ x 2, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 81: $C_3^4$ x 18, $C_3\wr C_3$ x 11, $C_9:C_3^2$ x 6, $C_3\times \He_3$ x 6, $C_3^2\times C_9$

Order 72: $C_6\times D_6$ x 18, $C_6\wr C_2$ x 10, $D_4\times C_3^2$ x 9, $S_3\times D_6$ x 6, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $\SOPlus(4,2)$ x 4, $C_2\times C_6^2$ x 2, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $D_4\times C_9$

Order 54: $S_3\times C_3^2$ x 170, $C_3^2:C_6$ x 54, $C_3^2\times C_6$ x 41, $C_3^2:S_3$ x 14, $C_9:C_6$ x 7, $C_2\times \He_3$ x 7, $S_3\times C_9$ x 6, $C_3\times C_{18}$ x 2, $C_3^2:S_3$ x 2

Order 48: $C_6\times D_4$ x 4, $S_3\times D_4$ x 2

Order 36: $C_6\times S_3$ x 121, $C_6^2$ x 35, $S_3^2$ x 26, $C_3\times C_{12}$ x 7, $C_3:C_{12}$ x 5, $C_6:S_3$ x 5, $C_3^2:C_4$ x 2, $C_2\times C_{18}$ x 2, $C_{36}$

Order 27: $C_3^3$ x 83, $C_9:C_3$ x 7, $\He_3$ x 7, $C_3\times C_9$ x 4

Order 24: $C_3\times D_4$ x 18, $C_2^2\times C_6$ x 8, $C_2\times D_6$ x 6, $C_2\times C_{12}$ x 4, $C_3:D_4$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 18: $C_3\times S_3$ x 146, $C_3\times C_6$ x 115, $C_3:S_3$ x 14, $C_{18}$ x 5

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 50, $D_6$ x 30, $C_{12}$ x 10, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 87, $C_9$ x 3

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

Order 6: $C_6$ x 68, $S_3$ x 24

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 20

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11664: $C_3^5:(S_3\times D_4)$

Order 5832: $(C_3^3\times S_3^2):C_6$, $(C_3^3\times S_3^2):S_3$, $C_3^5:D_{12}$, $C_3^5:(C_4\times S_3)$, $C_3^3:S_3^3$

Order 3888: $C_3\times S_3^3:C_6$, $C_3^4:(S_3\times D_4)$

Order 2916: $C_3^5:D_6$, $C_3^5:D_6$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^5:C_{12}$

Order 1944: $C_3^2\times S_3^3$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^5:D_4$, $S_3\times C_3^3:C_{12}$, $C_3^2:S_3^3$, $C_3^4:D_{12}$, $\He_3:\SOPlus(4,2)$, $(C_9\times S_3^2):C_6$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$, $C_3\wr S_3\times D_6$

Order 1458: $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $S_3^3:C_6$ x 3, $S_3^2:C_6^2$, $C_3\wr S_3\times D_4$

Order 972: $C_3^3:S_3^2$ x 6, $C_3^3:C_6^2$ x 2, $C_3^3\times S_3^2$ x 2, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.C_6^2$, $\He_3\times S_3^2$, $(C_3^2\times \He_3):C_4$, $C_3^4.C_{12}$, $C_3^4:C_{12}$, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^4:D_6$

Order 729: $C_3^5:C_3$

Order 648: $C_3^4:D_4$ x 6, $C_3\times S_3^3$ x 3, $C_3^3:D_{12}$ x 3, $C_3\wr D_4$ x 3, $S_3\times C_3^2:C_{12}$ x 3, $C_3^4:C_2^3$ x 2, $C_6\times C_3^2:C_{12}$, $S_3\times C_3^2:D_6$, $S_3^2:C_{18}$, $C_2^2\times C_3\wr S_3$, $C_3\wr C_3\times D_4$, $(C_3\times C_6^2):S_3$, $C_3^3:D_{12}$, $C_4\times C_3\wr S_3$

Order 486: $C_3^4:C_6$ x 5, $C_3^4:S_3$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 2, $S_3\times C_3^4$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $C_6\times \SOPlus(4,2)$ x 2, $S_3^3:C_2$ x 2, $C_{12}:C_6^2$, $C_6^2:D_6$

Order 324: $C_3^2\times S_3^2$ x 16, $C_3^3:D_6$ x 6, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^3:C_{12}$ x 5, $D_6\times C_3^3$ x 3, $C_3\wr C_2^2$ x 3, $C_3\wr C_4$ x 3, $C_3^2:C_6^2$ x 2, $C_3^2.C_6^2$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $\He_3:C_{12}$, $C_9\times S_3^2$, $C_3^2:C_{36}$

Order 243: $C_3^4:C_3$ x 4, $C_3^5$, $C_9:C_3^3$, $C_3^2\times \He_3$

Order 216: $S_3^2:C_6$ x 8, $C_6\times S_3^2$ x 5, $S_3\times C_6^2$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $S_3^3$ x 2, $C_3^2:D_{12}$ x 2, $S_3^2:S_3$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_{36}:C_6$, $D_4\times \He_3$, $C_6^2:S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $D_4\times C_3^3$, $C_6^2:C_6$, $C_3^2\times D_{12}$, $C_6.C_6^2$, $C_6^2:S_3$

Order 162: $S_3\times C_3^3$ x 18, $C_3^2\wr C_2$ x 13, $C_3\wr S_3$ x 8, $C_3^3:C_6$ x 5, $C_3^3:S_3$ x 4, $C_3^3:S_3$ x 4, $C_3^3.C_6$ x 4, $S_3\times \He_3$ x 4, $C_3^3:C_6$ x 3, $C_3^3\times C_6$ x 2, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^2:C_{18}$, $C_3^2:C_{18}$

Order 144: $C_{12}:D_6$ x 3, $S_3^2:C_2^2$, $C_4:C_6^2$

Order 108: $C_3\times S_3^2$ x 28, $C_3^2\times D_6$ x 23, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 5, $C_3:S_3^2$ x 4, $C_3\times C_6^2$ x 2, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $S_3\times C_{18}$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 81: $C_3^4$ x 8, $C_3\wr C_3$ x 6, $C_9:C_3^2$ x 3, $C_3\times \He_3$ x 3, $C_3^2\times C_9$

Order 72: $D_4\times C_3^2$ x 6, $C_6\times D_6$ x 5, $S_3\times D_6$ x 3, $\SOPlus(4,2)$ x 3, $C_6\wr C_2$ x 3, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_2\times C_6^2$, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $D_4\times C_9$

Order 54: $S_3\times C_3^2$ x 53, $C_3^2:C_6$ x 20, $C_3^2\times C_6$ x 15, $C_3^2:S_3$ x 8, $C_9:C_6$ x 4, $C_2\times \He_3$ x 4, $S_3\times C_9$ x 3, $C_3^2:S_3$ x 2, $C_3\times C_{18}$

Order 48: $C_6\times D_4$ x 2, $S_3\times D_4$ x 2

Order 36: $C_6\times S_3$ x 37, $S_3^2$ x 13, $C_6^2$ x 12, $C_3\times C_{12}$ x 5, $C_3:C_{12}$ x 3, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $C_2\times C_{18}$, $C_{36}$

Order 27: $C_3^3$ x 25, $C_9:C_3$ x 4, $\He_3$ x 4, $C_3\times C_9$ x 2

Order 24: $C_3\times D_4$ x 8, $C_2\times D_6$ x 3, $C_2\times C_{12}$ x 2, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2

Order 18: $C_3\times S_3$ x 50, $C_3\times C_6$ x 36, $C_3:S_3$ x 8, $C_{18}$ x 3

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 16, $C_2\times C_6$ x 15, $C_{12}$ x 6, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 27, $C_9$ x 2

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

Order 6: $C_6$ x 25, $S_3$ x 14

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11664: $C_3^5:(S_3\times D_4)$ ($C_1$)

Order 5832: $(C_3^3\times S_3^2):S_3$ ($C_2$) x 2, $C_3^3:S_3^3$ ($C_2$) x 2, $(C_3^3\times S_3^2):C_6$ ($C_2$), $C_3^5:D_{12}$ ($C_2$), $C_3^5:(C_4\times S_3)$ ($C_2$)

Order 3888: $C_3^4:(S_3\times D_4)$ ($C_3$)

Order 2916: $C_3^5:D_6$ ($C_2^2$) x 2, $C_3^4.C_6^2$ ($C_2^2$) x 2, $C_3^5:D_6$ ($C_2^2$), $(C_3^2\times \He_3):C_{12}$ ($C_2^2$), $C_3^5:C_{12}$ ($C_2^2$)

Order 1944: $C_3^2:S_3^3$ ($C_6$) x 2, $\He_3:\SOPlus(4,2)$ ($C_6$) x 2, $C_3^5:D_4$ ($S_3$), $C_3^4:D_{12}$ ($C_6$), $\He_3\times \SOPlus(4,2)$ ($C_6$), $C_3^4:(C_4\times S_3)$ ($C_6$)

Order 1458: $C_3^5:S_3$ ($D_4$), $C_3^5:S_3$ ($D_4$), $C_3^5:C_6$ ($C_2^3$)

Order 972: $C_3^3\times S_3^2$ ($D_6$) x 2, $C_3^3:S_3^2$ ($C_2\times C_6$) x 2, $\He_3\times S_3^2$ ($C_2\times C_6$) x 2, $C_3^4:C_{12}$ ($D_6$), $C_3^3:S_3^2$ ($C_2\times C_6$), $(C_3^2\times \He_3):C_4$ ($C_2\times C_6$), $C_3^4:C_{12}$ ($C_2\times C_6$)

Order 729: $C_3^5:C_3$ ($C_2\times D_4$)

Order 648: $C_3^4:D_4$ ($C_3\times S_3$)

Order 486: $C_3^4:C_6$ ($C_2\times D_6$), $C_3^4:S_3$ ($C_3\times D_4$), $C_3^4:C_6$ ($C_2^2\times C_6$), $C_3^4:S_3$ ($C_3\times D_4$)

Order 324: $C_3^2\times S_3^2$ ($C_6\times S_3$) x 2, $C_3^3:C_{12}$ ($C_6\times S_3$)

Order 243: $C_3^5$ ($S_3\times D_4$), $C_3^2\times \He_3$ ($C_6\times D_4$)

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

Order 162: $C_3^2\wr C_2$ ($C_6\times D_6$), $C_3\wr S_3$ ($\SOPlus(4,2)$)

Order 108: $C_3\times S_3^2$ ($C_3^2:D_6$) x 2, $C_3^2:C_{12}$ ($C_3^2:D_6$)

Order 81: $C_3\wr C_3$ ($S_3^2:C_2^2$), $C_3^4$ ($C_{12}:D_6$)

Order 72: $\SOPlus(4,2)$ ($C_3\wr S_3$)

Order 54: $C_3^2:S_3$ ($S_3^2:C_6$), $C_3^2:C_6$ ($C_6^2:C_6$)

Order 36: $S_3^2$ ($C_3^3:D_6$) x 2, $C_3^2:C_4$ ($C_3^3:D_6$)

Order 27: $C_3^3$ ($S_3^3:C_2$), $C_3^3$ ($C_6^2:D_6$), $\He_3$ ($C_6\times \SOPlus(4,2)$)

Order 18: $C_3:S_3$ ($C_2^2\times C_3\wr S_3$)

Order 9: $C_3^2$ ($S_3^3:C_6$), $C_3^2$ ($C_3\wr S_3\times D_4$)

Order 3: $C_3$ ($C_3^4:(C_6\times D_4)$)

Order 1: $C_1$ ($C_3^5:(S_3\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11664: $C_3^5:(S_3\times D_4)$ ($C_1$)

Order 5832: $(C_3^3\times S_3^2):C_6$ ($C_2$), $(C_3^3\times S_3^2):S_3$ ($C_2$), $C_3^5:D_{12}$ ($C_2$), $C_3^5:(C_4\times S_3)$ ($C_2$), $C_3^3:S_3^3$ ($C_2$)

Order 3888: $C_3^4:(S_3\times D_4)$ ($C_3$)

Order 2916: $C_3^5:D_6$ ($C_2^2$), $(C_3^2\times \He_3):C_{12}$ ($C_2^2$), $C_3^5:D_6$ ($C_2^2$), $C_3^4.C_6^2$ ($C_2^2$), $C_3^5:C_{12}$ ($C_2^2$)

Order 1944: $C_3^5:D_4$ ($S_3$), $C_3^2:S_3^3$ ($C_6$), $C_3^4:D_{12}$ ($C_6$), $\He_3:\SOPlus(4,2)$ ($C_6$), $\He_3\times \SOPlus(4,2)$ ($C_6$), $C_3^4:(C_4\times S_3)$ ($C_6$)

Order 1458: $C_3^5:S_3$ ($D_4$), $C_3^5:S_3$ ($D_4$), $C_3^5:C_6$ ($C_2^3$)

Order 972: $C_3^3\times S_3^2$ ($D_6$), $C_3^4:C_{12}$ ($D_6$), $C_3^3:S_3^2$ ($C_2\times C_6$), $C_3^3:S_3^2$ ($C_2\times C_6$), $\He_3\times S_3^2$ ($C_2\times C_6$), $(C_3^2\times \He_3):C_4$ ($C_2\times C_6$), $C_3^4:C_{12}$ ($C_2\times C_6$)

Order 729: $C_3^5:C_3$ ($C_2\times D_4$)

Order 648: $C_3^4:D_4$ ($C_3\times S_3$)

Order 486: $C_3^4:C_6$ ($C_2\times D_6$), $C_3^4:S_3$ ($C_3\times D_4$), $C_3^4:C_6$ ($C_2^2\times C_6$), $C_3^4:S_3$ ($C_3\times D_4$)

Order 324: $C_3^2\times S_3^2$ ($C_6\times S_3$), $C_3^3:C_{12}$ ($C_6\times S_3$)

Order 243: $C_3^5$ ($S_3\times D_4$), $C_3^2\times \He_3$ ($C_6\times D_4$)

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

Order 162: $C_3^2\wr C_2$ ($C_6\times D_6$), $C_3\wr S_3$ ($\SOPlus(4,2)$)

Order 108: $C_3\times S_3^2$ ($C_3^2:D_6$), $C_3^2:C_{12}$ ($C_3^2:D_6$)

Order 81: $C_3\wr C_3$ ($S_3^2:C_2^2$), $C_3^4$ ($C_{12}:D_6$)

Order 72: $\SOPlus(4,2)$ ($C_3\wr S_3$)

Order 54: $C_3^2:S_3$ ($S_3^2:C_6$), $C_3^2:C_6$ ($C_6^2:C_6$)

Order 36: $C_3^2:C_4$ ($C_3^3:D_6$), $S_3^2$ ($C_3^3:D_6$)

Order 27: $C_3^3$ ($S_3^3:C_2$), $C_3^3$ ($C_6^2:D_6$), $\He_3$ ($C_6\times \SOPlus(4,2)$)

Order 18: $C_3:S_3$ ($C_2^2\times C_3\wr S_3$)

Order 9: $C_3^2$ ($S_3^3:C_6$), $C_3^2$ ($C_3\wr S_3\times D_4$)

Order 3: $C_3$ ($C_3^4:(C_6\times D_4)$)

Order 1: $C_1$ ($C_3^5:(S_3\times D_4)$)

Series

Derived series

$C_3^5:(S_3\times D_4)$

$\rhd$

$C_3^5:(S_3\times D_4)$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^5:(S_3\times D_4)$

$\rhd$

$C_3^5:(S_3\times D_4)$

$\rhd$

$(C_3^3\times S_3^2):C_6$

$\rhd$

$(C_3^3\times S_3^2):C_6$

$\rhd$

$C_3^4.C_6^2$

$\rhd$

$C_3^4.C_6^2$

$\rhd$

$C_3^5:C_6$

$\rhd$

$C_3^5:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^5:(S_3\times D_4)$

$\rhd$

$C_3^5:(S_3\times D_4)$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^2\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $198 \times 198$ 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 $117 \times 117$ rational character table.