order	400
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₄×C₅×#120 (order 640) · Inn(G) = C₂² · Out(G) = C₄×C₅×D₄
homology	Abelianization (H₁) = C₂×C₄×C₂₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₄⋊C₄×C₂₅
unambiguous semidirect product	C₄⋊(C₄×C₂₅)

Subgroups

	subgroup	G/H
center	C₂²×C₂₅	C₂²
commutator	C₂	C₂×C₄×C₂₅
Frattini, socle, prime radical, cosocle	C₂²×C₅	C₂²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅ ◁ G

39 conjugacy classes of subgroups.

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2,3,4)(6,8)(9,10,12,15,19,11,13,16,20,24,14,17,21,25,28,18,22,26,29,31,23,27,30,32,33);(5,6,7,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 22);