C₂×C₃×#529 #87323

order	384
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂×C₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

1232 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19899);