C₃×#2564 #84428

order	384
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

584 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17004);