C₃×#2862 #84726

order	384
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

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

504 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17302);