D₅×#170 #87826

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) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₅² ⋊ C₂×C₈

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×AGL(1,𝔽₅)
commutator, hypocenter	C₅²	C₂×C₈
Fitting, socle	C₂×C₅²	C₂×C₄
prime radical	C₂×C₅×D₅	C₄
cosocle	C₅×Dic₅	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂

47 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,5,4,6,7,8)(9,10,12,13,11)(15,16,17,18);(10,11)(12,13)(14,15,16,18,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 120);