C₃×#358 #4605

order	192
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = C₂³×#442 (order 512) · Inn(G) = D₈ · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

58 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 164);