C₇×C₅⋊(#41⋊C₃) #13132591

order	1680
isoclinism	isoclinic class of C₅⋊(#41⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₅⋊(#41⋊C₃)
unambiguous semidirect product	C₅⋊(C₇×#41⋊C₃)
unambiguous semidirect product	#713⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₂×C₇	D₅×A₄
commutator, cosocle, hypocenter	C₅×Q₈	C₂×C₃×C₇
Frattini	C₂	C₇×D₅×A₄
Fitting	C₅×C₇×Q₈	C₂×C₃
socle	C₂×C₅×C₇	C₂×A₄
LCS	G ▷ C₅×Q₈
UCS	C₁ ◁ C₂×C₇

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)(8,9,12,15)(10,20,19,21,16,14,17,22,11,13,18,23)(25,26)(27,28);(8,18,12,16)(9,22,15,20)(10,11,17,19)(13,14,21,23)(24,25,27,28,26)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 611);