C₃×#470 #5344

order	192
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) = C₂×#6628842 (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	C₂²×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₂³×C₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

110 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 903);