C₇×D₅ #599

order	70
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₅) (order 120) · Inn(G) = D₅ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₅
unambiguous semidirect product	C₅⋊(C₂×C₇)

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

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

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(70, 1);