Abstract group 104976.mi: $C_3^4.S_3^2:S_3^2$

• Label: 104976.mi
• Order: \( 2^{4} \cdot 3^{8} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^4.S_3^2:S_3^2$
Order:	\(104976\)\(\medspace = 2^{4} \cdot 3^{8} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\He_3:(C_3:S_3).C_6^2.C_6.C_2^5$, of order \(3359232\)\(\medspace = 2^{9} \cdot 3^{8} \)
Composition factors:	$C_2$ x 4, $C_3$ x 8
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	12
Elements	1	1323	6560	2916	70848	23328	104976
Conjugacy classes	1	7	132	2	208	10	360
Divisions	1	7	102	2	138	6	256
Autjugacy classes	1	6	23	1	52	3	86

Minimal presentations

Permutation degree:	$24$
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	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{6}=c^{6}=d^{6}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,3,8,4,10,17)(2,6,13,7,15,9)(5,11,16,12,18,14)(19,20,23,22,24,21), (1,2,5) \!\cdots\! \rangle$
Transitive group:	36T20175				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_3^5.S_3^3)$ . $C_2$	$C_3^5$ . $(S_3^3:C_2)$	$\He_3^2$ . $(D_6:D_6)$	$C_3^4$ . $(S_3^2:S_3^2)$	all 29

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 74 normal subgroups (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3.C_3^5.D_{12}.C_2$
Commutator:	$G' \simeq$ $C_3^4.C_3^4.C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^6:(C_2\times D_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_3^4.C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2\times D_4$
Radical:	$R \simeq$ $C_3^4.S_3^2:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_2\times C_3^4:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times C_3^4.C_3^2$

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 36 (order at least 2916) are shown, as well as all normal subgroups of any index.

Order 104976: $C_3^4.S_3^2:S_3^2$

Order 52488: $C_3^4.C_3^2.C_6^2.C_2$ x 2, $C_3^6.\SOPlus(4,2)$ x 2, $C_3^4.C_3^4.C_2^3$, $\He_3^2:C_2.S_3^2$, $C_3^5.S_3^3$

Order 34992: $S_3\times \He_3^2:D_4$ x 2

Order 26244: $C_3^3.C_3^5.C_2^2$ x 2, $C_3^5.C_3^3.C_4$ x 2, $C_3^4.C_3^4.C_2^2$, $C_3^3.C_3^4.D_6$, $C_3^3.C_3^4.D_6$, $C_3^6.C_3.D_6$, $C_3.C_3^5.C_6^2$, $\He_3^2.S_3^2$, $C_3^6.S_3^2$

Order 17496: $C_3^5.C_6^2.C_2$ x 4, $C_3^4:S_3^3$ x 4, $C_3^4.C_3^3.D_4$ x 2, $C_3^4.C_3:S_3.C_6.C_2$ x 2, $C_3^3.C_3^4.C_2^3$ x 2, $S_3\times \He_3^2:C_4$ x 2, $C_3^4:S_3^3$ x 2, $\He_3^2:D_{12}$ x 2, $C_3^5:\SOPlus(4,2)$ x 2, $C_3^4.S_3^3$

Order 13122: $C_3^3.C_3^5.C_2$ x 2, $C_3^5.C_3^3.C_2$, $C_3^4.C_3^4.C_2$, $\He_3.C_3^4.C_6$, $C_3^4.C_3^4.C_2$, $C_3^6.C_3.C_6$

Order 11664: $C_3^5:(C_6\times D_4)$ x 4, $C_2\times \He_3^2:D_4$

Order 8748: $C_3^4.C_3^2.D_6$ x 10, $C_3^5:S_3^2$ x 10, $C_3^3.C_3^3.D_6$ x 8, $C_3^6:D_6$ x 8, $C_3^6:D_6$ x 8, $C_3^4.C_3^2.D_6$ x 6, $C_3.C_3^5.C_6.C_2$ x 4, $C_3.C_3^5.C_6.C_2$ x 4, $C_3^6.D_6$ x 4, $C_3^5.C_6^2$ x 4, $C_3.C_3^5.C_6.C_2$ x 4, $C_3.C_3^5.C_6.C_2$ x 4, $C_3.C_3^5.C_6.C_2$ x 4, $C_3^5.(C_6\times S_3)$ x 4, $C_3.C_3^5.C_6.C_2$ x 4, $C_3^6:D_6$ x 4, $C_3^5.C_6^2$ x 4, $C_3^5.C_3^2.C_4$ x 2, $C_3^3.C_3^3.D_6$ x 2, $C_3^4.C_3^3.C_2^2$ x 2, $C_3^4.C_3^3.C_2^2$ x 2, $C_3^3.C_3^3.D_6$ x 2, $C_3^4.C_3^3.C_4$ x 2, $C_3^4.C_3^2.D_6$ x 2, $C_3^5.S_3^2$ x 2, $\He_3^2:D_6$ x 2, $C_3^3.C_3^4.C_2^2$, $C_3^5.C_3.D_6$, $C_3.C_3^5.D_6$

Order 6561: $C_3^2\times C_3^4.C_3^2$

Order 5832: $C_3.C_3^5.C_2^3$ x 16, $C_3^6.C_2^3$ x 10, $C_3^6:D_4$ x 8, $C_3^3:S_3^3$ x 8, $C_3^5:D_{12}$ x 8, $C_3^3:S_3^3$ x 6, $C_3.C_3^5.C_2^3$ x 4, $C_3^3:S_3^3$ x 4, $C_3^4:(S_3\times C_{12})$ x 4, $\He_3^2:D_4$ x 4, $C_3^5:C_6:C_4$ x 3, $C_3.C_3^5.C_2^3$, $C_3^3.C_3^3.C_2^3$, $C_3^3:S_3^3$, $\He_3^2:C_2^3$, $C_2\times \He_3^2:C_4$

Order 4374: $C_3^4.C_3^3.C_2$ x 54, $C_3.C_3^5.C_6$ x 20, $C_3.C_3^5.C_6$ x 20, $C_3^4.C_3^3.C_2$ x 10, $C_3^3.C_3^4.C_2$ x 10, $C_3^3.C_3^4.C_2$ x 10, $C_3^3.C_3^4.C_2$ x 10, $C_3.C_3^5.C_6$ x 8, $C_3^6:C_6$ x 8, $C_3^5.C_3.C_6$ x 6, $C_3.C_3^5.C_6$ x 4, $C_3^6.C_6$ x 4, $C_3.C_3^5.C_6$ x 4, $C_3^6.C_6$ x 4, $C_3.C_3^5.C_6$ x 4, $C_3^6.C_6$ x 4, $C_3^6.C_6$ x 4, $C_3.C_3^5.C_6$ x 4, $C_3^4.C_3^3.C_2$ x 3, $C_3^5.C_3.C_6$ x 3, $C_3^5.C_3.S_3$

Order 3888: $C_3\wr D_4\times S_3$ x 8, $(C_3\times S_3^2):S_3^2$ x 4, $C_3\times S_3^2:S_3^2$ x 4

Order 2916: $C_3.C_3^5.C_2^2$ x 96, $C_3.C_3^5.C_2^2$ x 75, $C_3.C_3^5.C_2^2$ x 52, $C_3^4:S_3^2$ x 43, $C_3^5:D_6$ x 40, $(C_3^2\times \He_3).D_6$ x 34, $(C_3^2\times \He_3).D_6$ x 32, $C_3^4:C_6^2$ x 22, $C_3^4.C_6^2$ x 20, $C_3^6.C_2^2$ x 16, $C_3^6.C_2^2$ x 16, $C_3.C_3^5.C_2^2$ x 16, $C_3^4.C_6^2$ x 16, $C_3^5:D_6$ x 16, $C_3^5:D_6$ x 12, $C_3^5:D_6$ x 12, $C_3^4:C_6^2$ x 12, $C_3^6.C_2^2$ x 10, $C_3^4:C_6^2$ x 10, $C_3^4.S_3^2$ x 9, $C_3^4.(C_6\times S_3)$ x 8, $C_3^5.D_6$ x 8, $C_3^5:C_{12}$ x 8, $C_3^5:D_6$ x 8, $C_3^4.C_6^2$ x 8, $(C_3^2\times \He_3):C_{12}$ x 6, $C_3.C_3^5.C_2^2$ x 4, $C_3^4.S_3^2$ x 4, $C_3.C_3^5.C_2^2$ x 4, $C_3^5.D_6$ x 4, $C_3^5.D_6$ x 4, $C_3^5:D_6$ x 4, $C_3^5:D_6$ x 3, $\He_3^2:C_4$ x 2, $(C_3^2\times \He_3).D_6$, $C_3.C_3^5.C_2^2$, $C_3.C_3^5.C_2^2$, $C_3^4.C_6^2$, $\He_3.\He_3.C_2^2$, $C_3^4.C_6^2$, $\He_3^2:C_2^2$

Order 2187: $C_3\times C_3^4.C_3^2$ x 2

Order 1458: $C_3^5:C_6$ x 4, $\He_3^2:C_2$

Order 729: $C_3^6$ x 4, $\He_3^2$

Order 243: $C_3^5$ x 8

Order 162: $C_3^2\wr C_2$

Order 81: $C_3^4$ x 5

Order 54: $C_3^2:C_6$ x 2

Order 27: $C_3^3$ x 4

Order 18: $C_3:S_3$

Order 16: $C_2\times D_4$

Order 9: $C_3^2$ x 6

Order 3: $C_3$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 2916) are shown, as well as all normal subgroups of any index.

Order 104976: $C_3^4.S_3^2:S_3^2$

Order 52488: $C_3^4.C_3^2.C_6^2.C_2$, $C_3^4.C_3^4.C_2^3$, $\He_3^2:C_2.S_3^2$, $C_3^5.S_3^3$, $C_3^6.\SOPlus(4,2)$

Order 34992: $S_3\times \He_3^2:D_4$

Order 26244: $C_3^4.C_3^4.C_2^2$, $C_3^3.C_3^4.D_6$, $C_3^3.C_3^5.C_2^2$, $C_3^3.C_3^4.D_6$, $C_3^6.C_3.D_6$, $C_3.C_3^5.C_6^2$, $C_3^5.C_3^3.C_4$, $\He_3^2.S_3^2$, $C_3^6.S_3^2$

Order 17496: $C_3^4.C_3^3.D_4$, $C_3^5.C_6^2.C_2$, $C_3^4.C_3:S_3.C_6.C_2$, $C_3^4.S_3^3$, $C_3^3.C_3^4.C_2^3$, $S_3\times \He_3^2:C_4$, $C_3^4:S_3^3$, $\He_3^2:D_{12}$, $C_3^5:\SOPlus(4,2)$, $C_3^4:S_3^3$

Order 13122: $C_3^5.C_3^3.C_2$, $C_3^4.C_3^4.C_2$, $C_3^3.C_3^5.C_2$, $\He_3.C_3^4.C_6$, $C_3^4.C_3^4.C_2$, $C_3^6.C_3.C_6$

Order 11664: $C_2\times \He_3^2:D_4$, $C_3^5:(C_6\times D_4)$

Order 8748: $C_3.C_3^5.C_6.C_2$ x 2, $C_3^4.C_3^2.D_6$ x 2, $C_3^5:S_3^2$ x 2, $C_3^5.C_3^2.C_4$, $C_3.C_3^5.C_6.C_2$, $C_3.C_3^5.C_6.C_2$, $C_3^6.D_6$, $C_3^4.C_3^2.D_6$, $C_3^3.C_3^3.D_6$, $C_3^5.C_6^2$, $C_3^4.C_3^3.C_2^2$, $C_3^3.C_3^4.C_2^2$, $C_3^4.C_3^3.C_2^2$, $C_3^5.C_3.D_6$, $C_3^3.C_3^3.D_6$, $C_3.C_3^5.C_6.C_2$, $C_3^4.C_3^3.C_4$, $C_3^4.C_3^2.D_6$, $C_3.C_3^5.D_6$, $C_3.C_3^5.C_6.C_2$, $C_3^5.(C_6\times S_3)$, $C_3.C_3^5.C_6.C_2$, $C_3^5.S_3^2$, $C_3^3.C_3^3.D_6$, $C_3^6:D_6$, $\He_3^2:D_6$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^5.C_6^2$

Order 6561: $C_3^2\times C_3^4.C_3^2$

Order 5832: $C_3^6.C_2^3$ x 2, $C_3.C_3^5.C_2^3$ x 2, $\He_3^2:D_4$ x 2, $C_3.C_3^5.C_2^3$, $C_3^3.C_3^3.C_2^3$, $C_3.C_3^5.C_2^3$, $C_3^6:D_4$, $C_3^5:C_6:C_4$, $C_3^3:S_3^3$, $C_3^3:S_3^3$, $C_3^3:S_3^3$, $\He_3^2:C_2^3$, $C_2\times \He_3^2:C_4$, $C_3^5:D_{12}$, $C_3^3:S_3^3$, $C_3^4:(S_3\times C_{12})$

Order 4374: $C_3^4.C_3^3.C_2$ x 7, $C_3.C_3^5.C_6$ x 4, $C_3.C_3^5.C_6$ x 4, $C_3^4.C_3^3.C_2$ x 2, $C_3^4.C_3^3.C_2$ x 2, $C_3^3.C_3^4.C_2$ x 2, $C_3^3.C_3^4.C_2$ x 2, $C_3^5.C_3.C_6$ x 2, $C_3^3.C_3^4.C_2$ x 2, $C_3.C_3^5.C_6$, $C_3^6.C_6$, $C_3.C_3^5.C_6$, $C_3.C_3^5.C_6$, $C_3^6.C_6$, $C_3.C_3^5.C_6$, $C_3^6.C_6$, $C_3^6.C_6$, $C_3^5.C_3.C_6$, $C_3.C_3^5.C_6$, $C_3^5.C_3.S_3$, $C_3^6:C_6$

Order 3888: $(C_3\times S_3^2):S_3^2$, $C_3\wr D_4\times S_3$, $C_3\times S_3^2:S_3^2$

Order 2916: $C_3.C_3^5.C_2^2$ x 17, $C_3.C_3^5.C_2^2$ x 8, $C_3.C_3^5.C_2^2$ x 7, $C_3^4:S_3^2$ x 5, $(C_3^2\times \He_3).D_6$ x 4, $C_3^4:C_6^2$ x 4, $C_3^4.C_6^2$ x 3, $(C_3^2\times \He_3).D_6$ x 3, $C_3^5:D_6$ x 3, $C_3^6.C_2^2$ x 2, $C_3^6.C_2^2$ x 2, $C_3^4.S_3^2$ x 2, $C_3^6.C_2^2$ x 2, $C_3.C_3^5.C_2^2$ x 2, $C_3^4.C_6^2$ x 2, $C_3^4:C_6^2$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3.C_3^5.C_2^2$, $(C_3^2\times \He_3).D_6$, $C_3^4.S_3^2$, $C_3.C_3^5.C_2^2$, $C_3^4.(C_6\times S_3)$, $C_3.C_3^5.C_2^2$, $C_3.C_3^5.C_2^2$, $C_3^5.D_6$, $C_3^5.D_6$, $C_3^4.C_6^2$, $C_3^5.D_6$, $\He_3.\He_3.C_2^2$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:D_6$, $C_3^5:C_{12}$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^4:C_6^2$, $\He_3^2:C_2^2$, $\He_3^2:C_4$

Order 2187: $C_3\times C_3^4.C_3^2$

Order 1458: $C_3^5:C_6$, $\He_3^2:C_2$

Order 729: $C_3^6$, $\He_3^2$

Order 243: $C_3^5$

Order 162: $C_3^2\wr C_2$

Order 81: $C_3^4$ x 2

Order 54: $C_3^2:C_6$ x 2

Order 27: $C_3^3$ x 3

Order 18: $C_3:S_3$

Order 16: $C_2\times D_4$

Order 9: $C_3^2$ x 4

Order 3: $C_3$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 104976: $C_3^4.S_3^2:S_3^2$ ($C_1$)

Order 52488: $C_3^4.C_3^2.C_6^2.C_2$ ($C_2$) x 2, $C_3^6.\SOPlus(4,2)$ ($C_2$) x 2, $C_3^4.C_3^4.C_2^3$ ($C_2$), $\He_3^2:C_2.S_3^2$ ($C_2$), $C_3^5.S_3^3$ ($C_2$)

Order 26244: $C_3^3.C_3^5.C_2^2$ ($C_2^2$) x 2, $C_3^5.C_3^3.C_4$ ($C_2^2$) x 2, $C_3^4.C_3^4.C_2^2$ ($C_2^2$), $C_3^3.C_3^4.D_6$ ($C_2^2$), $C_3^6.C_3.D_6$ ($C_2^2$)

Order 17496: $C_3^4.C_3^3.D_4$ ($S_3$) x 2

Order 13122: $C_3^5.C_3^3.C_2$ ($D_4$), $C_3^4.C_3^4.C_2$ ($C_2^3$), $C_3^6.C_3.C_6$ ($D_4$)

Order 8748: $C_3^5.C_3^2.C_4$ ($D_6$) x 2, $C_3^3.C_3^3.D_6$ ($D_6$) x 2, $C_3^4.C_3^3.C_2^2$ ($D_6$) x 2

Order 6561: $C_3^2\times C_3^4.C_3^2$ ($C_2\times D_4$)

Order 4374: $C_3^4.C_3^3.C_2$ ($C_2\times D_6$) x 2

Order 2916: $(C_3^2\times \He_3).D_6$ ($S_3^2$)

Order 2187: $C_3\times C_3^4.C_3^2$ ($S_3\times D_4$) x 2

Order 1458: $C_3^5:C_6$ ($\SOPlus(4,2)$) x 4, $\He_3^2:C_2$ ($S_3\times D_6$)

Order 729: $C_3^6$ ($S_3^2:C_2^2$) x 4, $\He_3^2$ ($D_6:D_6$)

Order 243: $C_3^5$ ($S_3^3:C_2$) x 8

Order 162: $C_3^2\wr C_2$ ($C_3^4:D_4$)

Order 81: $C_3^4$ ($S_3^2:S_3^2$) x 4, $C_3^4$ ($C_2\times C_3^4:D_4$)

Order 54: $C_3^2:C_6$ ($C_3^3:\SOPlus(4,2)$), $C_3^2:C_6$ ($C_3^3:\SOPlus(4,2)$)

Order 27: $C_3^3$ ($C_3^5.C_2^3.C_2$) x 2, $C_3^3$ ($C_3^3.S_3\wr C_2.C_2$), $C_3^3$ ($C_3^3.S_3^2.C_2^2$)

Order 18: $C_3:S_3$ ($\He_3^2:D_4$)

Order 9: $C_3^2$ ($C_3.C_3^5.D_4.C_2$) x 2, $C_3^2$ ($C_3^3.S_3^3.C_2$) x 2, $C_3^2$ ($C_2\times \He_3^2:D_4$), $C_3^2$ ($C_3^6:(C_2\times D_4)$)

Order 3: $C_3$ ($S_3\times \He_3^2:D_4$) x 2, $C_3$ ($C_3^3.C_3^4.D_4.C_2$), $C_3$ ($C_3.C_3^5.D_{12}.C_2$)

Order 1: $C_1$ ($C_3^4.S_3^2:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 104976: $C_3^4.S_3^2:S_3^2$ ($C_1$)

Order 52488: $C_3^4.C_3^2.C_6^2.C_2$ ($C_2$), $C_3^4.C_3^4.C_2^3$ ($C_2$), $\He_3^2:C_2.S_3^2$ ($C_2$), $C_3^5.S_3^3$ ($C_2$), $C_3^6.\SOPlus(4,2)$ ($C_2$)

Order 26244: $C_3^4.C_3^4.C_2^2$ ($C_2^2$), $C_3^3.C_3^4.D_6$ ($C_2^2$), $C_3^3.C_3^5.C_2^2$ ($C_2^2$), $C_3^6.C_3.D_6$ ($C_2^2$), $C_3^5.C_3^3.C_4$ ($C_2^2$)

Order 17496: $C_3^4.C_3^3.D_4$ ($S_3$)

Order 13122: $C_3^5.C_3^3.C_2$ ($D_4$), $C_3^4.C_3^4.C_2$ ($C_2^3$), $C_3^6.C_3.C_6$ ($D_4$)

Order 8748: $C_3^5.C_3^2.C_4$ ($D_6$), $C_3^3.C_3^3.D_6$ ($D_6$), $C_3^4.C_3^3.C_2^2$ ($D_6$)

Order 6561: $C_3^2\times C_3^4.C_3^2$ ($C_2\times D_4$)

Order 4374: $C_3^4.C_3^3.C_2$ ($C_2\times D_6$)

Order 2916: $(C_3^2\times \He_3).D_6$ ($S_3^2$)

Order 2187: $C_3\times C_3^4.C_3^2$ ($S_3\times D_4$)

Order 1458: $C_3^5:C_6$ ($\SOPlus(4,2)$), $\He_3^2:C_2$ ($S_3\times D_6$)

Order 729: $C_3^6$ ($S_3^2:C_2^2$), $\He_3^2$ ($D_6:D_6$)

Order 243: $C_3^5$ ($S_3^3:C_2$)

Order 162: $C_3^2\wr C_2$ ($C_3^4:D_4$)

Order 81: $C_3^4$ ($C_2\times C_3^4:D_4$), $C_3^4$ ($S_3^2:S_3^2$)

Order 54: $C_3^2:C_6$ ($C_3^3:\SOPlus(4,2)$), $C_3^2:C_6$ ($C_3^3:\SOPlus(4,2)$)

Order 27: $C_3^3$ ($C_3^3.S_3\wr C_2.C_2$), $C_3^3$ ($C_3^5.C_2^3.C_2$), $C_3^3$ ($C_3^3.S_3^2.C_2^2$)

Order 18: $C_3:S_3$ ($\He_3^2:D_4$)

Order 9: $C_3^2$ ($C_3.C_3^5.D_4.C_2$), $C_3^2$ ($C_3^3.S_3^3.C_2$), $C_3^2$ ($C_2\times \He_3^2:D_4$), $C_3^2$ ($C_3^6:(C_2\times D_4)$)

Order 3: $C_3$ ($C_3^3.C_3^4.D_4.C_2$), $C_3$ ($C_3.C_3^5.D_{12}.C_2$), $C_3$ ($S_3\times \He_3^2:D_4$)

Order 1: $C_1$ ($C_3^4.S_3^2:S_3^2$)

Series

Derived series

$C_3^4.S_3^2:S_3^2$

$\rhd$

$C_3^4.C_3^4.C_2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3^4.S_3^2:S_3^2$

$\rhd$

$C_3^4.C_3^4.C_2^3$

$\rhd$

$C_3^6.C_3.D_6$

$\rhd$

$C_3^4.C_3^4.C_2$

$\rhd$

$C_3^2\times C_3^4.C_3^2$

$\rhd$

$C_3\times C_3^4.C_3^2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^4.S_3^2:S_3^2$

$\rhd$

$C_3^4.C_3^4.C_2$

$\rhd$

$C_3^2\times C_3^4.C_3^2$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

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

Character theory

Complex character table

The $360 \times 360$ character table is not available for this group.

Rational character table

The $256 \times 256$ rational character table is not available for this group.