C₃×#2681 #84545

order	384
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

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

618 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17121);