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ΓL(1,𝔽8)×#63818 #72254141

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

Aliases

direct productΓL(1,𝔽8)×#63818

Splittings

HallC32×ΓL(1,𝔽8) ⋊ C2×M16
SylowC7 ⋊ C3×#63818

Subgroups

subgroupG/H
center, FrattiniC4C2×D32×ΓL(1,𝔽8)
commutator, socleC2×C32×C7C22×C3×C4
FittingC2×C32×C4×C7C22×C3
prime radicalC22×C3⋊D3×ΓL(1,𝔽8)C4
cosocleC32×C4×C7C23×C3
hypercenterC2×C4D32×ΓL(1,𝔽8)
hypocenterC32×C7C2×C3×M16
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C4 ◁ C2×C4

764 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)×#63818, ΓL(1,𝔽8)×#10595100, #72269350
maximal normal subgroup ofAGL(1,𝔽7)×#63818, ΓL(1,𝔽8)×#10595100, #72269350

GAP

Past the SmallGroups library coverage — no SmallGroups id.