C₃×#453 #5327

order	192
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	Aut(G) = C₂²×#2196 (order 512) · Inn(G) = #120 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

168 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 886);