C₂²×D₅ #180

order	40
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = AGL(1,𝔽₅)×S₄ (order 480) · Inn(G) = D₅ · Out(G) = C₂×S₄
holomorph	G ⋊ Aut(G) = #72214138
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×D₅
unambiguous semidirect product	C₅⋊C₂³
unambiguous semidirect product	(C₂×C₅)⋊C₂²

Fitting, socle	C₂²×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	D₅
commutator, cosocle, hypocenter	C₅	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 3 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 13);