C₃×#459 #5333

order	192
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) = C₂²×D₄×#120 (order 1024) · Inn(G) = C₂×D₄ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

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

140 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)(5,6)(7,9)(8,11)(10,12)(13,14,15);(1,2,3,4);(6,9)(7,12)(10,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 892);