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C2×#10588712⋊C3 #72268933

order3456
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC2×#10588712⋊C3

Splittings

FittingC2×3−1+2×#120 ⋊ C2
Sylow3−1+2 ⋊ C2×B(4,𝔽2)

Subgroups

subgroupG/H
centerC22#63499⋊C3
commutator, cosocleC23×C9C24×C3
FrattiniC23×C3C23×C3×D3
FittingC2×3−1+2×#120C2
socleC22×C3C3×#937
hypercenterC2×#120AGL(1,ℤ/9)
hypocenterC9C2×C3×B(4,𝔽2)
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

3385 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:

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

Conjugacy classes & character table

208 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((3,5)(4,7)(6,9)(8,10)(12,18,19,14,15,16)(13,17), (7,10)(11,19,18,13,12,14,17,15,16), (1,2), (3,4)(5,7)(6,8)(9,10));