C₃×#495 #5369

order	192
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	Aut(G) = C₂×#72214076 (order 2048) · Inn(G) = C₂×D₄ · Out(G) = C₂²×#120
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₂×C₃×D₈	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

130 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 928);