C₃²×C₇×D₃₁ #16840912

order	3906
isoclinism	isoclinic class of D₃₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×C₇×D₃₁

Splittings

Fitting, socle, Hall	C₃²×C₇×C₃₁ ⋊ C₂
derived, Sylow	C₃₁ ⋊ C₂×C₃²×C₇
Hall	C₃²×C₃₁ ⋊ C₂×C₇
Hall	C₇×C₃₁ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₇	D₃₁
commutator, cosocle, hypocenter	C₃₁	C₂×C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₃₁	C₂
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₃²×C₇

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 1953

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2,3)(7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(4,5,6)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Conjugacy classes & character table

1071 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃²×C₇×D₃₁⋊C₅, C₃²×C₇×D₁₅₅, C₃²×C₅×C₇×D₃₁, …
maximal normal subgroup of	C₃²×C₇×D₃₁⋊C₅, C₃²×C₅×C₇×D₃₁, C₃²×C₄₉×D₃₁, …
maximal quotient of	C₃²×C₇×D₁₅₅, C₃²×C₅×C₇×D₃₁, C₃²×C₄₉×D₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3906, 62);

C32×C7×D31 #16840912