C₂×#1020 #5979

order	192
isoclinism	isoclinic class of #1020
automorphism	Aut(G) = C₂×(#1020⋊C₃)⋊C₂ (order 1152) · Inn(G) = #1020 · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

Fitting, socle	C₂⁵ ⋊ D₃
derived	C₂²⋊A₄ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	#1020
commutator, cosocle, hypocenter	C₂²⋊A₄	C₂²
Frattini	C₁	G
Fitting, socle	C₂⁵	D₃
LCS	G ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂

238 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 12

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1538);