order	192
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) = C₂²×#1020⋊C₂ (order 768) · Inn(G) = #120 · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ G

198 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 890);