C₂×D₅² #6062

order	200
isoclinism	isoclinic class of D₅²
automorphism	Aut(G) = #72275679 (order 3200) · Inn(G) = D₅² · Out(G) = #120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₅²
unambiguous semidirect product	(C₅×D₅)⋊C₂²
unambiguous semidirect product	(C₅⋊D₅)⋊C₂²

Fitting, socle	C₂×C₅² ⋊ C₂²
derived, Sylow	C₅² ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂	D₅²
commutator, cosocle, hypocenter	C₅²	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₅²	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂

74 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 20

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 49);