C₂²×H(3,𝔽₅) #92549

order	500
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅² · Schur multiplier (H₂) = C₂×C₅²

Aliases

direct product	C₂²×H(3,𝔽₅)
unambiguous semidirect product	(C₂²×C₅²)⋊C₅

	subgroup	G/H
center, socle	C₂²×C₅	C₅²
commutator, Frattini, cosocle	C₅	C₂²×C₅²
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

75 conjugacy classes of subgroups.

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 35);