Finite Group Databasebeta

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ΓL(1,𝔽8)×#805 #72247236

order2016
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#805

Splittings

FittingC3×C42×C7 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ #104
SylowC7 ⋊ C3×#805

Subgroups

subgroupG/H
centerC4ΓL(1,𝔽8)×D12
commutatorC3×C4×C7C2×C3×C4
FrattiniC2×C4C2×D3×ΓL(1,𝔽8)
FittingC3×C42×C7C2×C3
socleC2×C3×C7C3×C22⋊C4
prime radicalΓL(1,𝔽8)×#235C2
cosocleC2×C3×C4×C7C22×C3
hypercenterC42D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×#104
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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)×#104, AGL(1,𝔽7)×#805, #72254338, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)×#104, AGL(1,𝔽7)×#805, #72254338, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.