C₃×#493 #5367

order	192
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	Aut(G) = C₂×#72217072 (order 8192) · Inn(G) = C₂×D₄ · Out(G) = C₂×#23900
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

162 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

(2,4)(6,9)(7,10)(11,12)(13,14,15);(5,6,10,12,8,11,7,9);(1,2,3,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 926);