C₂×D₅×AGL(1,𝔽₅) #87915

order	400
isoclinism	isoclinic class of D₅×AGL(1,𝔽₅)
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅)² (order 1600) · Inn(G) = D₅×AGL(1,𝔽₅) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₅×AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊(C₄×D₅)
unambiguous semidirect product	(C₅×AGL(1,𝔽₅))⋊C₂²
unambiguous semidirect product	#1042⋊C₂²

Fitting, socle	C₂×C₅² ⋊ C₂×C₄
derived, Sylow	C₅² ⋊ C₂²×C₄

113 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 209);