C₂²×C₃×#127 #87324

order	384
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

C₂²×C₃×#127

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

3000 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19900);