C₅×D₇ #600

order	70
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₄×AGL(1,𝔽₇) (order 168) · Inn(G) = D₇ · Out(G) = C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×#64423⋊C₂
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₇
unambiguous semidirect product	C₇⋊(C₂×C₅)

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

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

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,3,4,5)(7,8)(9,10)(11,12);(6,7,9,11,12,10,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(70, 2);