C₂²×C₃²×PSU(3,𝔽₂)⋊C₃ #72228430

order	7776
isoclinism	isoclinic class of PSU(3,𝔽₂)⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃⁴

Aliases

Fitting, socle	C₂²×C₃⁴ ⋊ SL(2,𝔽₃)
derived	PSU(3,𝔽₂) ⋊ C₂²×C₃³

	subgroup	G/H
center, hypercenter	C₂²×C₃²	PSU(3,𝔽₂)⋊C₃
commutator, cosocle, hypocenter	PSU(3,𝔽₂)	C₂²×C₃³
Frattini	C₁	G
Fitting, socle	C₂²×C₃⁴	SL(2,𝔽₃)
LCS	G ▷ PSU(3,𝔽₂)
UCS	C₁ ◁ C₂²×C₃²

2642 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 2642 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 288

A minimal faithful permutation representation — 3 generators on 19 points:

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

360 conjugacy classes. The character table is too large to display.

maximal subgroup of	C₂²×C₃⋊D₃×PSU(3,𝔽₂)⋊C₃, C₂²×C₃²×AGL(2,𝔽₃), C₂²×#72213922, …
maximal normal subgroup of	C₂²×C₃⋊D₃×PSU(3,𝔽₂)⋊C₃, C₂²×C₃²×AGL(2,𝔽₃), C₂²×#72213922, …

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((7,8,10)(9,14,15)(11,13,12)(18,19), (4,5,6)(8,14,12,9)(10,11,13,15)(16,17), (1,2,3));