Finite Group Databasebeta

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ΓL(1,𝔽8)×#823 #72244537

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

Aliases

direct productΓL(1,𝔽8)×#823

Splittings

FittingC3×C7×M16 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ #100
SylowC7 ⋊ C3×#823

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×(C2×D3)⋊C4
commutatorC22×C3×C7C2×C3×C4
FrattiniC2×C4C2×D3×ΓL(1,𝔽8)
FittingC3×C7×M16C2×C3
socleC2×C3×C7C3×C22⋊C4
prime radicalC2×ΓL(1,𝔽8)×D12C2
cosocleC2×C3×C4×C7C22×C3
hypercenterM16D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×#100
LCSG ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

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

normal subgroups only

This group has 204 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

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)×#100, AGL(1,𝔽7)×#823, #72244576, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)×#100, AGL(1,𝔽7)×#823, #72244576, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.