C₃×#497 #5371

order	192
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) = C₂×#72210600 (order 2048) · Inn(G) = C₂×D₄ · Out(G) = C₂×B(4,𝔽₂)
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×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

124 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)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(21,22,23);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20);(1,2,3,4)(7,14)(10,19)(13,20)(15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 930);