C₂²×C₃×#124 #87320

order	384
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

C₂²×C₃×#124

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

1880 conjugacy classes of subgroups; only the 952 normal subgroups are shown.

normal subgroups only

This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 952 normal subgroups are shown.

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

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

A minimal faithful permutation representation — 5 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19896);