C₅×#66662 #13132585

order	1680
isoclinism	isoclinic class of #66662
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, hypercenter	C₂×C₅	A₄×D₇
commutator, cosocle, hypocenter	C₇×Q₈	C₂×C₃×C₅
Frattini	C₂	C₅×A₄×D₇
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)(8,18,17,19,14,12,15,20,9,11,16,21)(23,24)(25,26)(27,28);(6,16,10,14)(7,20,13,18)(8,9,15,17)(11,12,19,21)(22,23,25,27,28,26,24)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 605);