C₃×#462 #5336

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) = C₂²

Aliases

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

108 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(2,4)(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(21,22,23);(1,2,3,4);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 895);