Finite Group Databasebeta

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ΓL(1,𝔽8)×#5077 #72254349

order4032
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productΓL(1,𝔽8)×#5077

Splittings

FittingC3×C7×#127 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ #453
SylowC7 ⋊ C3×#5077

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×#937
commutator, cosocleC2×C3×C4×C7C23×C3
FrattiniC2×C4C22×D3×ΓL(1,𝔽8)
FittingC3×C7×#127C2×C3
socleC2×C3×C7C3×#120
hypercenter#127D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×#453
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

780 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)×#453, AGL(1,𝔽7)×#5077, #72254460
maximal normal subgroup ofD3×ΓL(1,𝔽8)×#453, AGL(1,𝔽7)×#5077, #72254460

GAP

Past the SmallGroups library coverage — no SmallGroups id.