Finite Group Databasebeta

← Prev (#72229444) Next (#72229446) →

C2×ΓL(1,𝔽8)×#453 #72229445

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC2×ΓL(1,𝔽8)×#453

Splittings

Fitting, HallC2×C7×#453 ⋊ C3
SylowC7 ⋊ C2×C3×#453

Subgroups

subgroupG/H
centerC22ΓL(1,𝔽8)×#120
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C24×ΓL(1,𝔽8)
FittingC2×C7×#453C3
socleC22×C7C3×#120
hypercenterC2×#453ΓL(1,𝔽8)
hypocenterC7C2×C3×#453
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#453

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×ΓL(1,𝔽8)×#5935, C2×AGL(1,𝔽7)×#453, D3×ΓL(1,𝔽8)×#453
maximal normal subgroup ofC2×AGL(1,𝔽7)×#453
maximal quotient ofD3×ΓL(1,𝔽8)×#453

GAP

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

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