C₅²×C₂²⋊Q₈ #11717282

order	800
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product	C₅²×C₂²⋊Q₈
unambiguous semidirect product	C₂²⋊(C₅²×Q₈)

	subgroup	G/H
center, socle	C₂²×C₅²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₅²
prime radical	C₂³×C₅²	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ G

296 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4,5)(12,14)(15,16,18,20)(17,21,22,19);(6,7,8,9,10)(15,17,18,22)(16,19,20,21);(11,12)(13,14)(15,16)(17,19)(18,20)(21,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(800, 941);