D₅² #1045

order	100
isoclinism	isoclinic class of D₅²
automorphism	Aut(G) = AGL(1,𝔽₅)²⋊C₂ (order 800) · Inn(G) = G · Out(G) = D₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₅²
unambiguous semidirect product	(C₅×D₅)⋊C₂
unambiguous semidirect product	(C₅⋊D₅)⋊C₂

C₅² ⋊ C₂²

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²	C₂²
LCS	G ▷ C₅²
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 13);