C₃×#537 #5867

order	192
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	Aut(G) = C₂×#72215688 (order 4096) · Inn(G) = C₂⁴ · Out(G) = C₂×#2196
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

294 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1426);