C₃×#2675 #84539

order	384
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

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

804 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17115);