C₃×#3182 #85046

order	384
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

402 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)(3,6)(4,5)(7,8)(10,13)(11,15)(14,16)(17,18,19);(2,5)(6,8)(10,14)(13,16);(9,10,11,13,12,14,15,16);(1,3,4,7)(2,6,5,8)(10,14)(13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17622);