Finite Group Databasebeta

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ΓL(1,𝔽8)×#932 #72245535

order2016
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×#932

Splittings

FittingC2×C3×C7×D4 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ AGL(1,ℤ/8)
SylowC7 ⋊ C3×#932

Subgroups

subgroupG/H
centerC2C2×ΓL(1,𝔽8)×#67
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C22×D3×ΓL(1,𝔽8)
FittingC2×C3×C7×D4C2×C3
socleC2×C3×C7C2×C3×D4
hypercenterC2×D4D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

326 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 87 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 ofAGL(1,𝔽7)×#932, D3×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72245644, …
maximal normal subgroup ofAGL(1,𝔽7)×#932, D3×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72245644, …
maximal quotient ofC3×ΓL(1,𝔽8)×#932

GAP

Past the SmallGroups library coverage — no SmallGroups id.