#108×PSU(3,𝔽₂)⋊C₃ #72245220

order	6912
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₃

solvable

Aliases

direct product

#108×PSU(3,𝔽₂)⋊C₃

Splittings

Fitting

C₃²×#108 ⋊ SL(2,𝔽₃)

Subgroups

	subgroup	G/H
center	C₄	D₄×PSU(3,𝔽₂)⋊C₃
commutator	C₄×PSU(3,𝔽₂)	C₂×C₃×C₄
Frattini	C₂×C₄	C₂²×PSU(3,𝔽₂)⋊C₃
Fitting	C₃²×#108	SL(2,𝔽₃)
socle	C₂×C₃²	C₄⋊C₄×SL(2,𝔽₃)
prime radical	C₂²×PSU(3,𝔽₂)⋊C₃	Q₈
cosocle	C₂×C₄×PSU(3,𝔽₂)	C₂²×C₃
hypercenter	#108	PSU(3,𝔽₂)⋊C₃
hypocenter	PSU(3,𝔽₂)	C₃×#108
LCS	G ▷ C₄×PSU(3,𝔽₂) ▷ C₂×PSU(3,𝔽₂) ▷ PSU(3,𝔽₂)
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

582 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,4)(3,8,9)(5,7,6)(10,12,14,21,15,22,13,20)(11,16,17,19,18,24,23,25);(2,8,6,3)(4,5,7,9)(10,11,13,17,15,18,14,23)(12,19,20,24,22,25,21,16)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((1,2,4)(3,8,9)(5,7,6)(10,12,14,21,15,22,13,20)(11,16,17,19,18,24,23,25), (2,8,6,3)(4,5,7,9)(10,11,13,17,15,18,14,23)(12,19,20,24,22,25,21,16));