C₃×#496 #5370

order	192
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) = C₂×#72210600 (order 2048) · Inn(G) = C₂×D₄ · Out(G) = C₂×B(4,𝔽₂)
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×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

144 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)(7,11)(10,12)(13,14,15);(5,6,7,9,8,10,11,12);(1,2,3,4)(6,10)(9,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 929);