C₃×#3152 #85016

order	384
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

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

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

normal subgroups only

This group has 552 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) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(2,4)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(21,22,23);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20);(1,2,3,4);(7,14)(10,19)(13,20)(15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17592);