C₅×#66665 #13132588

order	1680
isoclinism	isoclinic class of #66665
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, hypercenter	C₂×C₅	D₇⋊A₄
commutator, cosocle, hypocenter	C₇×Q₈	C₂×C₃×C₅
Frattini	C₂	C₅×D₇⋊A₄
Fitting	C₅×C₇×Q₈	C₂×C₃
socle	C₂×C₅×C₇	C₂×A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂×C₅

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3,4,5)(6,7,10,13)(8,12,16,19,9,18,15,21,14,11,17,20)(23,27,26,24,25,28);(6,14,10,16)(7,18,13,20)(8,17,15,9)(11,21,19,12)(22,23,26,27,25,28,24)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 608);