C₅×C₇×D₃₁ #14524240

order	2170
isoclinism	isoclinic class of D₃₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×C₇×D₃₁

Splittings

Fitting, socle, Hall	C₅×C₇×C₃₁ ⋊ C₂
derived, Sylow	C₃₁ ⋊ C₂×C₅×C₇
Hall	C₇×C₃₁ ⋊ C₂×C₅
Hall	C₅×C₃₁ ⋊ C₂×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₇	D₃₁
commutator, cosocle, hypocenter	C₃₁	C₂×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₃₁	C₂
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₅×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 1085

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×D₇×D₃₁, C₇×D₅×D₃₁, D₃₁×D₃₅, …
maximal normal subgroup of	C₅×D₇×D₃₁, C₇×D₅×D₃₁, D₃₁×D₃₅, …
maximal quotient of	C₅×C₇×Dic₃₁, C₂×C₅×C₇×D₃₁, C₇×C₂₅×D₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2170, 3);

C5×C7×D31 #14524240