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ΓL(1,𝔽8)×#64306 #72247103

order6048
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productΓL(1,𝔽8)×#64306

Splittings

FittingC2×C32×C4×C7 ⋊ C22×C3
HallC32×ΓL(1,𝔽8) ⋊ C2×#41
SylowC7 ⋊ C3×#64306

Subgroups

subgroupG/H
centerC4C2×D32×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C32×C7C24×C3
FrattiniC2C22×D32×ΓL(1,𝔽8)
FittingC2×C32×C4×C7C22×C3
hypercenterC2×C4D32×ΓL(1,𝔽8)
hypocenterC32×C7C2×C3×#41
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C4 ◁ C2×C4

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#64306, ΓL(1,𝔽8)×#10595100, #72269413, …
maximal normal subgroup ofAGL(1,𝔽7)×#64306, ΓL(1,𝔽8)×#10595100, #72269413, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.