C₃×#494 #5368

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) = C₂×C₄

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

122 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)(7,13,16,20)(8,17,14,11)(10,18,19,15)(21,22,23);(5,7,14,18,9,16,8,15)(6,10,17,20,12,19,11,13);(1,2,3,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 927);