Abstract group 7776.cb: $(C_3^3\times S_3^2):D_4$

• Label: 7776.cb
• Order: \( 2^{5} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$(C_3^3\times S_3^2):D_4$
Order:	\(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_3^5:C_2^2$, 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 and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	483	242	1404	4134	1512	7776
Conjugacy classes	1	10	22	3	57	6	99
Divisions	1	10	19	3	50	4	87
Autjugacy classes	1	9	17	3	41	4	75

Dimension	1	2	4	8	16	32
Irr. complex chars.	8	14	34	24	18	1	99
Irr. rational chars.	8	10	28	24	12	5	87

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$4300800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{6}=d^{6}=e^{6}=f^{3}=[b,f]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2,5)(3,4)(7,8)(10,11,13,14,15,12), (1,3,2,4,5,6)(7,9,8)(10,12,13,11)(14,15), (1,4,2,3,5,6)(7,9,8)(11,12,14)(13,15)\rangle$
Transitive group:	24T9702	36T7344	36T7372	36T7375	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3:S_3^3)$ $\,\rtimes\,$ $D_6$	$(C_3:S_3^4)$ $\,\rtimes\,$ $C_2$	$C_3^4$ $\,\rtimes\,$ $(D_6:D_4)$	$(S_3^2:S_3^2)$ $\,\rtimes\,$ $S_3$	all 30
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3^3)$ . $D_6$	$(C_3^2:S_3^3)$ . $C_2^2$	$(C_3^2:S_3^3)$ . $C_2^2$	$(C_3^2:S_3^3)$ . $C_2^2$	all 14

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 106680 subgroups in 2706 conjugacy classes, 50 normal (48 characteristic).

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

Special subgroups

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

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 7776: $(C_3^3\times S_3^2):D_4$

Order 3888: $C_3:S_3^4$, $C_3^2\wr C_2.D_{12}$, $C_3^3:C_6^2:C_4$, $(C_3^3\times S_3^2):C_4$, $C_3^5:(C_2\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3\times S_3^2:S_3^2$

Order 2592: $S_3^2\wr C_2$

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

Order 1296: $S_3^4$ x 2, $S_3^2:S_3^2$ x 2, $(C_3^2\times S_3^2):C_4$ x 2, $C_6:S_3^3$, $C_3^4:(C_2\times D_4)$, $S_3^3:S_3$, $S_3^3:C_6$, $C_3^2\wr C_2.D_4$

Order 972: $C_3^3:S_3^2$ x 5, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3\times S_3^2$, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 864: $(S_3\times D_6):D_6$, $D_6:\SOPlus(4,2)$

Order 648: $C_3:S_3^3$ x 18, $C_3:S_3^3$ x 10, $C_3\times S_3^3$ x 10, $C_3\wr D_4$ x 6, $C_3^3:D_{12}$ x 4, $C_3^4:C_2^3$ x 3, $C_3^4:C_2^3$ x 3, $C_3:S_3^3$ x 3, $C_3^4:D_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^4:C_2^3$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:(C_2\times C_4)$, $C_3^3:C_4\times S_3$, $S_3\times C_3^2:C_{12}$

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

Order 432: $C_2\times S_3^3$ x 5, $S_3^2:D_6$ x 3, $S_3^3:C_2$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_3:D_6^2$, $C_6\times \SOPlus(4,2)$, $C_6^2:D_6$, $(C_3\times C_6^2):C_4$, $D_6:S_3^2$, $C_6.\SOPlus(4,2)$, $(C_3\times S_3^2):C_4$, $(C_3^2\times D_6):C_4$

Order 324: $C_3^2:S_3^2$ x 52, $C_3^2:S_3^2$ x 35, $C_3\wr C_2^2$ x 22, $C_3^2\times S_3^2$ x 20, $C_3^2:S_3^2$ x 7, $C_3^2:S_3^2$ x 6, $C_3\wr C_4$ x 6, $C_3^3:D_6$ x 3, $C_3^2:C_6^2$ x 3, $D_6\times C_3^3$, $C_3^4:C_4$, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 288: $D_6\wr C_2$ x 2

Order 243: $C_3^5$

Order 216: $S_3^3$ x 48, $C_6\times S_3^2$ x 20, $C_6:S_3^2$ x 15, $C_6:S_3^2$ x 14, $S_3^2:S_3$ x 13, $S_3^2:C_6$ x 5, $C_2\times C_3^3:C_4$ x 3, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_2\times C_3^2:C_{12}$, $C_6^2:C_6$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_6.S_3^2$

Order 162: $C_3^2\wr C_2$ x 45, $C_3^3:C_6$ x 33, $S_3\times C_3^3$ x 25, $C_3^3:S_3$ x 2, $C_3^3\times C_6$

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

Order 108: $C_3\times S_3^2$ x 123, $C_3:S_3^2$ x 91, $C_3:S_3^2$ x 59, $C_3^2:D_6$ x 40, $C_3^2\times D_6$ x 25, $C_3^3:C_4$ x 9, $C_3^2:D_6$ x 5, $C_3^2:C_{12}$ x 4, $C_3\times C_6^2$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $D_6:D_4$

Order 81: $C_3^4$ x 19

Order 72: $S_3\times D_6$ x 79, $C_6\times D_6$ x 14, $\SOPlus(4,2)$ x 10, $C_6:D_6$ x 6, $C_6\wr C_2$ x 6, $C_2\times C_3^2:C_4$ x 4, $C_3:D_{12}$ x 4, $C_6.D_6$ x 2, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $D_6:S_3$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 145, $S_3\times C_3^2$ x 125, $C_3^2:S_3$ x 27, $C_3^2\times C_6$ x 19

Order 48: $C_2^2\times D_6$ x 3, $C_6:D_4$ x 2, $S_3\times D_4$ x 2, $D_6:C_4$ x 2, $C_6\times D_4$, $C_6.D_4$

Order 36: $S_3^2$ x 161, $C_6\times S_3$ x 120, $C_6:S_3$ x 52, $C_6^2$ x 11, $C_3^2:C_4$ x 7, $C_3:C_{12}$ x 6, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 96

Order 24: $C_2\times D_6$ x 44, $C_3:D_4$ x 9, $C_3\times D_4$ x 5, $C_2^2\times C_6$ x 4, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 170, $C_3:S_3$ x 83, $C_3\times C_6$ x 63

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

Order 12: $D_6$ x 116, $C_2\times C_6$ x 33, $C_3:C_4$ x 5, $C_{12}$ x 4

Order 9: $C_3^2$ x 96

Order 8: $C_2^3$ x 10, $D_4$ x 6, $C_2\times C_4$ x 3

Order 6: $S_3$ x 56, $C_6$ x 50

Order 4: $C_2^2$ x 24, $C_4$ x 3

Order 3: $C_3$ x 19

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 7776: $(C_3^3\times S_3^2):D_4$

Order 3888: $C_3:S_3^4$, $C_3^2\wr C_2.D_{12}$, $C_3^3:C_6^2:C_4$, $(C_3^3\times S_3^2):C_4$, $C_3^5:(C_2\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3\times S_3^2:S_3^2$

Order 2592: $S_3^2\wr C_2$

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

Order 1296: $S_3^4$ x 2, $S_3^2:S_3^2$ x 2, $(C_3^2\times S_3^2):C_4$ x 2, $C_6:S_3^3$, $C_3^4:(C_2\times D_4)$, $S_3^3:S_3$, $S_3^3:C_6$, $C_3^2\wr C_2.D_4$

Order 972: $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^3:S_3^2$ 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, $C_3^3\times S_3^2$, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 864: $(S_3\times D_6):D_6$, $D_6:\SOPlus(4,2)$

Order 648: $C_3:S_3^3$ x 14, $C_3\times S_3^3$ x 9, $C_3:S_3^3$ x 7, $C_3\wr D_4$ x 6, $C_3^3:D_{12}$ x 4, $C_3:S_3^3$ x 3, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^4:C_2^3$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:(C_2\times C_4)$, $C_3^3:C_4\times S_3$, $S_3\times C_3^2:C_{12}$

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

Order 432: $C_2\times S_3^3$ x 4, $S_3^2:D_6$ x 3, $S_3^3:C_2$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_3:D_6^2$, $C_6\times \SOPlus(4,2)$, $C_6^2:D_6$, $(C_3\times C_6^2):C_4$, $D_6:S_3^2$, $C_6.\SOPlus(4,2)$, $(C_3\times S_3^2):C_4$, $(C_3^2\times D_6):C_4$

Order 324: $C_3^2:S_3^2$ x 35, $C_3^2:S_3^2$ x 23, $C_3\wr C_2^2$ x 19, $C_3^2\times S_3^2$ x 16, $C_3\wr C_4$ x 6, $C_3^2:S_3^2$ x 5, $C_3^2:S_3^2$ x 5, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$ x 2, $D_6\times C_3^3$, $C_3^4:C_4$, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 288: $D_6\wr C_2$ x 2

Order 243: $C_3^5$

Order 216: $S_3^3$ x 32, $C_6\times S_3^2$ x 14, $C_6:S_3^2$ x 12, $S_3^2:S_3$ x 10, $C_6:S_3^2$ x 9, $S_3^2:C_6$ x 5, $C_2\times C_3^3:C_4$ x 3, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_2\times C_3^2:C_{12}$, $C_6^2:C_6$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_6.S_3^2$

Order 162: $C_3^2\wr C_2$ x 34, $C_3^3:C_6$ x 25, $S_3\times C_3^3$ x 21, $C_3^3:S_3$ x 2, $C_3^3\times C_6$

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

Order 108: $C_3\times S_3^2$ x 82, $C_3:S_3^2$ x 58, $C_3:S_3^2$ x 39, $C_3^2:D_6$ x 27, $C_3^2\times D_6$ x 18, $C_3^3:C_4$ x 8, $C_3^2:D_6$ x 4, $C_3^2:C_{12}$ x 4, $C_3\times C_6^2$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $D_6:D_4$

Order 81: $C_3^4$ x 17

Order 72: $S_3\times D_6$ x 52, $C_6\times D_6$ x 12, $\SOPlus(4,2)$ x 7, $C_6\wr C_2$ x 6, $C_6:D_6$ x 5, $C_2\times C_3^2:C_4$ x 4, $C_3:D_{12}$ x 4, $C_6.D_6$ x 2, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $D_6:S_3$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 99, $S_3\times C_3^2$ x 92, $C_3^2:S_3$ x 21, $C_3^2\times C_6$ x 15

Order 48: $C_2^2\times D_6$ x 3, $C_6:D_4$ x 2, $S_3\times D_4$ x 2, $D_6:C_4$ x 2, $C_6\times D_4$, $C_6.D_4$

Order 36: $S_3^2$ x 99, $C_6\times S_3$ x 84, $C_6:S_3$ x 36, $C_6^2$ x 10, $C_3^2:C_4$ x 6, $C_3:C_{12}$ x 6, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 73

Order 24: $C_2\times D_6$ x 34, $C_3:D_4$ x 8, $C_3\times D_4$ x 5, $C_2^2\times C_6$ x 4, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 122, $C_3:S_3$ x 57, $C_3\times C_6$ x 48

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

Order 12: $D_6$ x 80, $C_2\times C_6$ x 28, $C_3:C_4$ x 5, $C_{12}$ x 4

Order 9: $C_3^2$ x 73

Order 8: $C_2^3$ x 9, $D_4$ x 5, $C_2\times C_4$ x 3

Order 6: $S_3$ x 42, $C_6$ x 41

Order 4: $C_2^2$ x 19, $C_4$ x 3

Order 3: $C_3$ x 17

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7776: $(C_3^3\times S_3^2):D_4$ ($C_1$)

Order 3888: $C_3:S_3^4$ ($C_2$), $C_3^2\wr C_2.D_{12}$ ($C_2$), $C_3^3:C_6^2:C_4$ ($C_2$), $(C_3^3\times S_3^2):C_4$ ($C_2$), $C_3^5:(C_2\times D_4)$ ($C_2$), $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3\times S_3^2:S_3^2$ ($C_2$)

Order 1944: $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^5:(C_2\times C_4)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$), $C_3^3:(S_3\times C_{12})$ ($C_2^2$)

Order 1296: $S_3^2:S_3^2$ ($S_3$)

Order 972: $C_3^3:S_3^2$ ($D_4$) x 2, $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($C_2^3$), $C_3^3:C_6^2$ ($D_4$), $C_3^3\times S_3^2$ ($D_4$)

Order 648: $C_3:S_3^3$ ($D_6$), $C_3:S_3^3$ ($D_6$), $C_3^3:(C_4\times S_3)$ ($D_6$)

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

Order 324: $C_3^2:S_3^2$ ($C_3:D_4$), $C_3^2:S_3^2$ ($C_2\times D_6$), $C_3^2\times S_3^2$ ($C_3:D_4$)

Order 243: $C_3^5$ ($C_2^2\wr C_2$)

Order 162: $C_3^3:S_3$ ($S_3\times D_4$), $C_3^2\wr C_2$ ($C_6:D_4$), $C_3^2\wr C_2$ ($S_3\times D_4$)

Order 108: $C_3:S_3^2$ ($\SOPlus(4,2)$), $C_3\times S_3^2$ ($\SOPlus(4,2)$)

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

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

Order 36: $S_3^2$ ($S_3^2:S_3$)

Order 27: $C_3^3$ ($D_6\wr C_2$) x 2

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

Order 9: $C_3^2$ ($(S_3\times D_6):D_6$), $C_3^2$ ($D_6:\SOPlus(4,2)$)

Order 3: $C_3$ ($S_3^2\wr C_2$)

Order 1: $C_1$ ($(C_3^3\times S_3^2):D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7776: $(C_3^3\times S_3^2):D_4$ ($C_1$)

Order 3888: $C_3:S_3^4$ ($C_2$), $C_3^2\wr C_2.D_{12}$ ($C_2$), $C_3^3:C_6^2:C_4$ ($C_2$), $(C_3^3\times S_3^2):C_4$ ($C_2$), $C_3^5:(C_2\times D_4)$ ($C_2$), $C_3^4:(S_3\times D_4)$ ($C_2$), $C_3\times S_3^2:S_3^2$ ($C_2$)

Order 1944: $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^2:S_3^3$ ($C_2^2$), $C_3^5:(C_2\times C_4)$ ($C_2^2$), $C_3^4:(C_4\times S_3)$ ($C_2^2$), $C_3^3:(S_3\times C_{12})$ ($C_2^2$)

Order 1296: $S_3^2:S_3^2$ ($S_3$)

Order 972: $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($D_4$), $C_3^3:S_3^2$ ($C_2^3$), $C_3^3:C_6^2$ ($D_4$), $C_3^3\times S_3^2$ ($D_4$)

Order 648: $C_3:S_3^3$ ($D_6$), $C_3:S_3^3$ ($D_6$), $C_3^3:(C_4\times S_3)$ ($D_6$)

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

Order 324: $C_3^2:S_3^2$ ($C_3:D_4$), $C_3^2:S_3^2$ ($C_2\times D_6$), $C_3^2\times S_3^2$ ($C_3:D_4$)

Order 243: $C_3^5$ ($C_2^2\wr C_2$)

Order 162: $C_3^3:S_3$ ($S_3\times D_4$), $C_3^2\wr C_2$ ($C_6:D_4$), $C_3^2\wr C_2$ ($S_3\times D_4$)

Order 108: $C_3:S_3^2$ ($\SOPlus(4,2)$), $C_3\times S_3^2$ ($\SOPlus(4,2)$)

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

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

Order 36: $S_3^2$ ($S_3^2:S_3$)

Order 27: $C_3^3$ ($D_6\wr C_2$) x 2

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

Order 9: $C_3^2$ ($(S_3\times D_6):D_6$), $C_3^2$ ($D_6:\SOPlus(4,2)$)

Order 3: $C_3$ ($S_3^2\wr C_2$)

Order 1: $C_1$ ($(C_3^3\times S_3^2):D_4$)

Series

Derived series

$(C_3^3\times S_3^2):D_4$

$\rhd$

$(C_3^3\times S_3^2):D_4$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(C_3^3\times S_3^2):D_4$

$\rhd$

$(C_3^3\times S_3^2):D_4$

$\rhd$

$C_3:S_3^4$

$\rhd$

$C_3:S_3^4$

$\rhd$

$C_3^2:S_3^3$

$\rhd$

$C_3^2:S_3^3$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(C_3^3\times S_3^2):D_4$

$\rhd$

$(C_3^3\times S_3^2):D_4$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

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

Rational character table

See the $87 \times 87$ rational character table.