Finite Group Databasebeta

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ΓL(1,𝔽8)×#64149 #72246439

order6048
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3×C4

Aliases

direct productΓL(1,𝔽8)×#64149

Splittings

FittingC2×C32×C7×D4 ⋊ C2×C3
HallC32×ΓL(1,𝔽8) ⋊ #127
SylowC7 ⋊ C3×#64149

Subgroups

subgroupG/H
center, FrattiniC22C22×C3⋊D3×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C32×C7C23×C3
FittingC2×C32×C7×D4C2×C3
hypercenterC2×D4C3⋊D3×ΓL(1,𝔽8)
hypocenterC32×C7C3×#127
LCSG ▷ C22×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C2×D4

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1008

Conjugacy classes & character table

270 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3⋊D3×ΓL(1,𝔽8)×#127, AGL(1,𝔽7)×#64149, #72246558
maximal normal subgroup ofC3⋊D3×ΓL(1,𝔽8)×#127, AGL(1,𝔽7)×#64149, #72246558

GAP

Past the SmallGroups library coverage — no SmallGroups id.