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C2×#99×AGL(1,ℤ/9) #72268916

order3456
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#99×AGL(1,ℤ/9)

Splittings

FittingC2×3−1+2×#99 ⋊ C2
Sylow3−1+2 ⋊ C22×#99

Subgroups

subgroupG/H
centerC22C22⋊C4×AGL(1,ℤ/9)
commutatorC22×C9C23×C3×C4
FrattiniC23×C3C23×C3×D3
FittingC2×3−1+2×#99C2
socleC22×C3C3×D3×C22⋊C4
prime radicalC22×D4×AGL(1,ℤ/9)C2
cosocleC23×C9C24×C3
hypercenterC2×#99AGL(1,ℤ/9)
hypocenterC9C22×C3×#99
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

3441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,ℤ/9)×B(4,𝔽2)
maximal normal subgroup ofC2×AGL(1,ℤ/9)×B(4,𝔽2)

GAP

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

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