Abstract group 7776.is: $C_3^4:(D_4\times D_6)$

• Label: 7776.is
• Order: \( 2^{5} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_3^4:(D_4\times D_6)$
Order:	\(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_3:S_3\times \He_3).D_4^2$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	615	242	504	4974	1440	7776
Conjugacy classes	1	15	14	4	58	7	99
Divisions	1	15	14	4	58	7	99
Autjugacy classes	1	8	7	3	24	5	48

Dimension	1	2	4	6	8	12	16	24
Irr. complex chars.	16	20	24	8	17	2	4	8	99
Irr. rational chars.	16	20	24	8	17	2	4	8	99

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	$33484389120$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{6}=d^{12}=e^{3}=f^{3}=[a,e]=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3)(2,6)(4,5)(7,9)(10,12,11,15,14,13), (1,4,6)(2,7,3,8,5,9)(10,13,14,15)(11,12), (2,5,3)(7,8,9)(10,11,14)(13,15), (1,2,4,5,6,3)(7,8,9)(10,11,14)\rangle$
Transitive group:	36T7342				more information
Direct product:	$\SOPlus(4,2)$ $\, \times\, $ $(C_3^2:D_6)$
Semidirect product:	$(C_3:S_3^3)$ $\,\rtimes\,$ $D_6$	$(C_3^4:D_6)$ $\,\rtimes\,$ $D_4$	$(C_3^4:D_4)$ $\,\rtimes\,$ $D_6$	$(C_3^4:D_4)$ $\,\rtimes\,$ $D_6$	all 58
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $(S_3^4:C_2)$	$(S_3^2:C_6)$ . $S_3^2$	$C_3^3$ . $(D_4\times S_3^2)$	$(C_3^2:C_6)$ . $D_6^2$	all 13
Aut. group:	$\Aut(C_3^3:S_3^2)$	$\Aut(\He_3:\SOPlus(4,2))$	$\Aut(C_3^4:D_{12})$	$\Aut(C_3^2:S_3^3)$	all 6

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 136108 subgroups in 3456 conjugacy classes, 139 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^4:(D_4\times D_6)$
Commutator:	$G' \simeq$ $C_3^4:C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3^4:C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2^2\times D_4$
Radical:	$R \simeq$ $C_3^4:(D_4\times D_6)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3$	$G/\operatorname{soc} \simeq$ $D_4\times S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times \He_3$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 7776: $C_3^4:(D_4\times D_6)$

Order 3888: $C_3^4:(S_3\times D_4)$ x 4, $C_3^4:(S_3\times D_4)$ x 2, $C_3^4:(C_6\times D_4)$ x 2, $C_3^4:(C_2\times D_{12})$ x 2, $C_3^3:D_6^2$ x 2, $C_3^4:(S_3\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3^4:(C_4\times D_6)$

Order 2592: $S_3^4:C_2$ x 2

Order 1944: $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 4, $C_3^4:D_{12}$ x 4, $\He_3:\SOPlus(4,2)$ x 4, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2.S_3^3$ x 2, $\He_3:\SOPlus(4,2)$ x 2, $\He_3:\SOPlus(4,2)$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:(C_2\times C_{12})$ x 2, $C_3^2.S_3^3$, $C_3^4:D_{12}$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$

Order 1296: $S_3^3:S_3$ x 8, $S_3^3:C_6$ x 5, $S_3^4$ x 4, $S_3\times C_3^2:D_{12}$ x 4, $S_3^2:S_3^2$ x 4, $C_3^3:(S_3\times D_4)$ x 2, $(C_3\times S_3^2):D_6$ x 2, $C_3^2:C_4\times S_3^2$ x 2, $C_3^2:D_6^2$ x 2

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

Order 864: $S_3^3:C_2^2$ x 3, $C_6.D_6^2$

Order 648: $C_3.S_3^3$ x 20, $C_3:S_3^3$ x 14, $C_3\times S_3^3$ x 14, $C_3:S_3^3$ x 12, $C_3\wr D_4$ x 10, $C_3^3:D_{12}$ x 8, $C_3^3:D_{12}$ x 5, $S_3\times C_3^2:C_{12}$ x 5, $C_3:S_3^3$ x 4, $C_3^4:D_4$ x 4, $C_3^4:D_4$ x 4, $C_3^3:C_4\times S_3$ x 4, $C_2\times C_3^2:S_3^2$ x 4, $S_3\times C_3^2:D_6$ x 4, $C_3^4:D_4$ x 3, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_2\times C_3^2:S_3^2$ x 2, $S_3\times C_3^2:D_6$ x 2, $C_6\times C_3^2:D_6$ x 2

Order 486: $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 25, $C_2\times S_3^3$ x 10, $S_3^2:D_6$ x 6, $C_6^2:D_6$ x 4, $C_2\times C_3^2:D_{12}$ x 3, $C_6\times \SOPlus(4,2)$ x 3, $C_3^2:C_4\times D_6$ x 3, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_6^2:D_6$, $C_{12}.S_3^2$, $C_{12}.S_3^2$

Order 324: $C_3^2:S_3^2$ x 27, $C_3^2:S_3^2$ x 24, $C_3^2:S_3^2$ x 24, $C_3^2\times S_3^2$ x 18, $C_3^2:S_3^2$ x 14, $C_3\wr C_2^2$ x 14, $C_3^2:S_3^2$ x 14, $C_3^2:S_3^2$ x 12, $C_3^3:D_6$ x 12, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 6, $C_3\wr C_4$ x 5, $C_3^3:D_6$ x 4, $C_3^2:C_6^2$ x 4, $C_3^3:C_{12}$ x 3, $C_3^4:C_4$ x 2, $C_3^3:D_6$ x 2, $C_3^3:D_6$ x 2, $D_6\times \He_3$ x 2

Order 288: $D_4\times S_3^2$ x 2, $C_6^2:D_4$

Order 243: $C_3^2\times \He_3$

Order 216: $S_3^3$ x 78, $C_6\times S_3^2$ x 26, $S_3^2:S_3$ x 26, $C_6:S_3^2$ x 21, $C_6.S_3^2$ x 19, $C_3^2:D_{12}$ x 13, $S_3^2:C_6$ x 13, $C_3^2:C_4\times S_3$ x 13, $C_6:S_3^2$ x 10, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_6^2:C_6$ x 4, $C_2\times C_3^3:C_4$ x 3, $C_2\times C_3^2:C_{12}$ x 3, $C_6^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_4\times C_3^2:C_6$ x 2, $\He_3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6^2:S_3$ x 2, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_6.S_3^2$

Order 162: $C_3^3:C_6$ x 16, $C_3^3:C_6$ x 16, $C_3^2\wr C_2$ x 15, $S_3\times C_3^3$ x 11, $C_3^3:S_3$ x 10, $C_3^3:S_3$ x 10, $S_3\times \He_3$ x 10, $C_3^3:C_6$ x 9, $C_3^3:S_3$ x 2, $C_6\times \He_3$ x 2

Order 144: $D_6^2$ x 12, $S_3^2:C_2^2$ x 12, $D_6:D_6$ x 8, $C_{12}:D_6$ x 5, $D_6:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $C_4\times S_3^2$ x 2, $C_6^2:C_4$

Order 108: $C_3\times S_3^2$ x 130, $C_3:S_3^2$ x 111, $C_3:S_3^2$ x 44, $C_3^2:D_6$ x 34, $C_3^2\times D_6$ x 25, $C_3^2:D_6$ x 23, $C_3^2:D_6$ x 22, $C_3^2:D_6$ x 13, $C_3^2:D_6$ x 7, $C_3^3:C_4$ x 7, $C_3^2:C_{12}$ x 7, $C_3^2:C_{12}$ x 2, $C_2^2\times \He_3$ x 2, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $D_4\times D_6$ x 3

Order 81: $C_3\times \He_3$ x 8, $C_3^4$ x 3

Order 72: $S_3\times D_6$ x 136, $C_6\times D_6$ x 24, $\SOPlus(4,2)$ x 16, $C_6:D_6$ x 11, $C_6\wr C_2$ x 10, $C_3:D_{12}$ x 8, $C_2\times C_3^2:C_4$ x 6, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $C_6^2:C_2$ x 4, $D_6:S_3$ x 4, $C_6.D_6$ x 4, $D_4\times C_3^2$ x 3, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_6.D_6$ x 2

Order 54: $S_3\times C_3^2$ x 85, $C_3^2:C_6$ x 77, $C_3^2:C_6$ x 28, $C_3^2:S_3$ x 23, $C_3^2:S_3$ x 20, $C_3^2\times C_6$ x 9, $C_2\times \He_3$ x 9

Order 48: $S_3\times D_4$ x 25, $C_2^2\times D_6$ x 8, $C_6:D_4$ x 6, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3

Order 36: $S_3^2$ x 218, $C_6\times S_3$ x 155, $C_6:S_3$ x 72, $C_6^2$ x 13, $C_3:C_{12}$ x 5, $C_3^2:C_4$ x 4, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$ x 22, $\He_3$ x 11

Order 24: $C_2\times D_6$ x 93, $C_3:D_4$ x 26, $D_{12}$ x 13, $C_4\times S_3$ x 13, $C_3\times D_4$ x 13, $C_2^2\times C_6$ x 8, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3

Order 18: $C_3\times S_3$ x 162, $C_3:S_3$ x 74, $C_3\times C_6$ x 50

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 191, $C_2\times C_6$ x 49, $C_{12}$ x 7, $C_3:C_4$ x 7

Order 9: $C_3^2$ x 38

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $S_3$ x 72, $C_6$ x 58

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$ x 14

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 7776: $C_3^4:(D_4\times D_6)$

Order 3888: $C_3^4:(S_3\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3^4:(C_6\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3^4:(C_2\times D_{12})$, $C_3^4:(C_4\times D_6)$, $C_3^3:D_6^2$

Order 2592: $S_3^4:C_2$

Order 1944: $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2.S_3^3$, $C_3^2:S_3^3$, $C_3^2.S_3^3$, $C_3^2:S_3^3$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $\He_3:\SOPlus(4,2)$, $\He_3:\SOPlus(4,2)$, $C_3^4:D_{12}$, $\He_3:\SOPlus(4,2)$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_2\times C_{12})$

Order 1296: $S_3^3:C_6$ x 3, $S_3\times C_3^2:D_{12}$ x 2, $S_3^3:S_3$ x 2, $S_3^4$, $C_3^3:(S_3\times D_4)$, $S_3^2:S_3^2$, $(C_3\times S_3^2):D_6$, $C_3^2:C_4\times S_3^2$, $C_3^2:D_6^2$

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

Order 864: $S_3^3:C_2^2$ x 2, $C_6.D_6^2$

Order 648: $C_3.S_3^3$ x 7, $C_3:S_3^3$ x 4, $C_3\times S_3^3$ x 4, $C_3:S_3^3$ 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, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^3:C_4\times S_3$ x 2, $C_3:S_3^3$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^2:S_3^2$, $S_3\times C_3^2:D_6$, $C_2\times C_3^2:S_3^2$, $C_6\times C_3^2:D_6$, $S_3\times C_3^2:D_6$

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

Order 432: $S_3^3:C_2$ x 10, $C_2\times S_3^3$ x 3, $C_2\times C_3^2:D_{12}$ x 2, $S_3^2:D_6$ x 2, $C_6\times \SOPlus(4,2)$ x 2, $C_3^2:C_4\times D_6$ x 2, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$

Order 324: $C_3^2:S_3^2$ x 9, $C_3^2\times S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 4, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 3, $C_3\wr C_4$ x 3, $C_3^2:S_3^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^4:C_4$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^3:D_6$, $D_6\times \He_3$, $C_3^2:C_6^2$

Order 288: $D_4\times S_3^2$, $C_6^2:D_4$

Order 243: $C_3^2\times \He_3$

Order 216: $S_3^3$ x 20, $C_6\times S_3^2$ x 8, $C_6:S_3^2$ x 7, $S_3^2:S_3$ x 7, $S_3^2:C_6$ x 7, $C_3^2:C_4\times S_3$ x 7, $C_6.S_3^2$ x 7, $C_3^2:D_{12}$ x 6, $C_6:S_3^2$ x 3, $C_2\times C_3^3:C_4$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $D_4\times \He_3$, $C_6^2:S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_4\times C_3^2:C_6$, $C_3^2:D_{12}$, $C_6.S_3^2$, $\He_3:D_4$, $C_6.S_3^2$, $C_6^2:S_3$, $C_6^2:C_6$

Order 162: $C_3^2\wr C_2$ x 6, $S_3\times C_3^3$ x 5, $C_3^3:C_6$ x 4, $C_3^3:S_3$ 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 4, $C_3^3:S_3$, $C_6\times \He_3$

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

Order 108: $C_3\times S_3^2$ x 38, $C_3:S_3^2$ x 32, $C_3:S_3^2$ x 12, $C_3^2:D_6$ x 12, $C_3^2\times D_6$ x 9, $C_3^2:D_6$ x 8, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 6, $C_3^2:C_{12}$ x 5, $C_3^3:C_4$ x 4, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$, $C_2^2\times \He_3$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $D_4\times D_6$ x 2

Order 81: $C_3\times \He_3$ x 3, $C_3^4$ x 2

Order 72: $S_3\times D_6$ x 39, $C_6\times D_6$ x 8, $\SOPlus(4,2)$ x 7, $C_6:D_6$ x 4, $C_2\times C_3^2:C_4$ x 4, $C_6\wr C_2$ x 3, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_6.D_6$ x 2, $C_6^2:C_2$, $C_3:D_{12}$, $C_{12}:S_3$, $D_6:S_3$, $C_6.D_6$

Order 54: $S_3\times C_3^2$ x 29, $C_3^2:C_6$ x 25, $C_3^2:S_3$ x 9, $C_3^2:C_6$ x 8, $C_3^2:S_3$ x 8, $C_3^2\times C_6$ x 4, $C_2\times \He_3$ x 4

Order 48: $S_3\times D_4$ x 10, $C_2^2\times D_6$ x 3, $C_6\times D_4$ x 2, $C_6:D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

Order 36: $S_3^2$ x 61, $C_6\times S_3$ x 50, $C_6:S_3$ x 23, $C_6^2$ x 5, $C_3^2:C_4$ x 3, $C_3:C_{12}$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$ x 9, $\He_3$ x 5

Order 24: $C_2\times D_6$ x 30, $C_3:D_4$ x 7, $C_4\times S_3$ x 7, $C_3\times D_4$ x 7, $D_{12}$ x 6, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 18: $C_3\times S_3$ x 54, $C_3:S_3$ x 23, $C_3\times C_6$ x 19

Order 16: $C_2\times D_4$ x 6, $C_2^4$, $C_2^2\times C_4$

Order 12: $D_6$ x 62, $C_2\times C_6$ x 18, $C_{12}$ x 5, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 15

Order 8: $C_2^3$ x 8, $D_4$ x 7, $C_2\times C_4$ x 4

Order 6: $S_3$ x 26, $C_6$ x 24

Order 4: $C_2^2$ x 16, $C_4$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7776: $C_3^4:(D_4\times D_6)$ ($C_1$)

Order 3888: $C_3^4:(S_3\times D_4)$ ($C_2$) x 4, $C_3^4:(S_3\times D_4)$ ($C_2$) x 2, $C_3^4:(C_6\times D_4)$ ($C_2$) x 2, $C_3^4:(C_2\times D_{12})$ ($C_2$) x 2, $C_3^3:D_6^2$ ($C_2$) x 2, $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3^4:(C_4\times D_6)$ ($C_2$)

Order 1944: $C_3^2:S_3^3$ ($C_2^2$) x 4, $C_3^2:S_3^3$ ($C_2^2$) x 4, $C_3^4:D_{12}$ ($C_2^2$) x 4, $\He_3:\SOPlus(4,2)$ ($C_2^2$) x 4, $C_3^2:S_3^3$ ($C_2^2$) x 2, $C_3^2:S_3^3$ ($C_2^2$) x 2, $\He_3:\SOPlus(4,2)$ ($C_2^2$) x 2, $\He_3:\SOPlus(4,2)$ ($C_2^2$) x 2, $C_3^4:D_{12}$ ($C_2^2$) x 2, $C_3^4:(C_4\times S_3)$ ($C_2^2$) x 2, $C_3^4:(C_2\times C_{12})$ ($C_2^2$) x 2, $C_3^2.S_3^3$ ($C_2^2$), $C_3^4:D_{12}$ ($C_2^2$), $\He_3\times \SOPlus(4,2)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$)

Order 1296: $(C_3\times S_3^2):D_6$ ($S_3$) x 2

Order 972: $C_3^3:S_3^2$ ($C_2^3$) x 4, $C_3^3:S_3^2$ ($C_2^3$) x 2, $C_3^3:S_3^2$ ($D_4$) x 2, $C_3^3:S_3^2$ ($C_2^3$) x 2, $\He_3\times S_3^2$ ($C_2^3$) x 2, $C_3^4:C_{12}$ ($C_2^3$) x 2, $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($C_2^3$), $C_3^4:D_6$ ($D_4$), $(C_3^2\times \He_3):C_4$ ($C_2^3$), $C_3^4:C_{12}$ ($C_2^3$)

Order 648: $C_3:S_3^3$ ($D_6$) x 4, $C_3^4:D_4$ ($D_6$) x 4, $C_3^3:D_{12}$ ($D_6$) x 2, $C_3^4:D_4$ ($D_6$) x 2, $C_3^3:(C_4\times S_3)$ ($D_6$) x 2

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

Order 324: $C_3^2:S_3^2$ ($C_2\times D_6$) x 4, $C_3^2\times S_3^2$ ($C_2\times D_6$) x 4, $C_3^2:S_3^2$ ($C_2\times D_6$) x 2, $C_3^4:C_4$ ($C_2\times D_6$) x 2, $C_3^3:C_{12}$ ($C_2\times D_6$) x 2

Order 243: $C_3^2\times \He_3$ ($C_2^2\times D_4$)

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

Order 162: $C_3^3:S_3$ ($S_3\times D_4$) x 2, $C_3^2\wr C_2$ ($C_2^2\times D_6$) x 2, $C_3^2\wr C_2$ ($S_3\times D_4$) x 2

Order 108: $C_3\times S_3^2$ ($S_3\times D_6$) x 2, $C_3^2:C_{12}$ ($S_3\times D_6$), $C_3^2:D_6$ ($\SOPlus(4,2)$)

Order 81: $C_3^4$ ($D_4\times D_6$) x 2

Order 72: $\SOPlus(4,2)$ ($C_3^2:D_6$)

Order 54: $C_3^2:C_6$ ($S_3^2:C_2^2$) x 2, $C_3^2:S_3$ ($S_3^2:C_2^2$), $C_3^2:C_6$ ($D_6^2$)

Order 36: $S_3^2$ ($C_6.S_3^2$) x 2, $C_3^2:C_4$ ($C_6.S_3^2$)

Order 27: $C_3^3$ ($D_4\times S_3^2$), $\He_3$ ($C_6^2:D_4$)

Order 18: $C_3:S_3$ ($S_3^3:C_2$) x 2, $C_3:S_3$ ($C_6^2:D_6$)

Order 9: $C_3^2$ ($S_3^3:C_2^2$) x 2, $C_3^2$ ($C_6.D_6^2$)

Order 3: $C_3$ ($S_3^4:C_2$)

Order 1: $C_1$ ($C_3^4:(D_4\times D_6)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7776: $C_3^4:(D_4\times D_6)$ ($C_1$)

Order 3888: $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3^4:(C_6\times D_4)$ ($C_2$), $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3^4:(C_2\times D_{12})$ ($C_2$), $C_3^4:(C_4\times D_6)$ ($C_2$), $C_3^3:D_6^2$ ($C_2$)

Order 1944: $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^2.S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^4:D_{12}$ ($C_2^2$), $C_3^4:D_{12}$ ($C_2^2$), $\He_3:\SOPlus(4,2)$ ($C_2^2$), $\He_3:\SOPlus(4,2)$ ($C_2^2$), $C_3^4:D_{12}$ ($C_2^2$), $\He_3:\SOPlus(4,2)$ ($C_2^2$), $\He_3\times \SOPlus(4,2)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$), $C_3^4:(C_2\times C_{12})$ ($C_2^2$)

Order 1296: $(C_3\times S_3^2):D_6$ ($S_3$)

Order 972: $C_3^3:S_3^2$ ($C_2^3$), $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($C_2^3$), $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($C_2^3$), $C_3^3:S_3^2$ ($C_2^3$), $\He_3\times S_3^2$ ($C_2^3$), $C_3^4:D_6$ ($D_4$), $(C_3^2\times \He_3):C_4$ ($C_2^3$), $C_3^4:C_{12}$ ($C_2^3$), $C_3^4:C_{12}$ ($C_2^3$)

Order 648: $C_3:S_3^3$ ($D_6$), $C_3^3:D_{12}$ ($D_6$), $C_3^4:D_4$ ($D_6$), $C_3^4:D_4$ ($D_6$), $C_3^3:(C_4\times S_3)$ ($D_6$)

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

Order 324: $C_3^2:S_3^2$ ($C_2\times D_6$), $C_3^2:S_3^2$ ($C_2\times D_6$), $C_3^2\times S_3^2$ ($C_2\times D_6$), $C_3^4:C_4$ ($C_2\times D_6$), $C_3^3:C_{12}$ ($C_2\times D_6$)

Order 243: $C_3^2\times \He_3$ ($C_2^2\times D_4$)

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

Order 162: $C_3^3:S_3$ ($S_3\times D_4$), $C_3^2\wr C_2$ ($C_2^2\times D_6$), $C_3^2\wr C_2$ ($S_3\times D_4$)

Order 108: $C_3\times S_3^2$ ($S_3\times D_6$), $C_3^2:C_{12}$ ($S_3\times D_6$), $C_3^2:D_6$ ($\SOPlus(4,2)$)

Order 81: $C_3^4$ ($D_4\times D_6$)

Order 72: $\SOPlus(4,2)$ ($C_3^2:D_6$)

Order 54: $C_3^2:S_3$ ($S_3^2:C_2^2$), $C_3^2:C_6$ ($S_3^2:C_2^2$), $C_3^2:C_6$ ($D_6^2$)

Order 36: $C_3^2:C_4$ ($C_6.S_3^2$), $S_3^2$ ($C_6.S_3^2$)

Order 27: $C_3^3$ ($D_4\times S_3^2$), $\He_3$ ($C_6^2:D_4$)

Order 18: $C_3:S_3$ ($S_3^3:C_2$), $C_3:S_3$ ($C_6^2:D_6$)

Order 9: $C_3^2$ ($S_3^3:C_2^2$), $C_3^2$ ($C_6.D_6^2$)

Order 3: $C_3$ ($S_3^4:C_2$)

Order 1: $C_1$ ($C_3^4:(D_4\times D_6)$)

Series

Derived series

$C_3^4:(D_4\times D_6)$

$\rhd$

$C_3^4:(D_4\times D_6)$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^4:(D_4\times D_6)$

$\rhd$

$C_3^4:(D_4\times D_6)$

$\rhd$

$C_3^3:D_6^2$

$\rhd$

$C_3^3:D_6^2$

$\rhd$

$C_3^2:S_3^3$

$\rhd$

$C_3^2:S_3^3$

$\rhd$

$\He_3\times S_3^2$

$\rhd$

$\He_3\times S_3^2$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^4:(D_4\times D_6)$

$\rhd$

$C_3^4:(D_4\times D_6)$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^2\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_1$

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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $99 \times 99$ rational character table. Alternatively, you may search for characters of this group with desired properties.