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ΓL(1,𝔽8)×#63825 #72269299

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

Aliases

direct productΓL(1,𝔽8)×#63825

Splittings

HallC32×ΓL(1,𝔽8) ⋊ AGL(1,ℤ/8)
SylowC7 ⋊ C3×#63825

Subgroups

subgroupG/H
centerC2C2×ΓL(1,𝔽8)×#626
commutator, cosocleC32×C4×C7C23×C3
FrattiniC4C2×D32×ΓL(1,𝔽8)
FittingC2×C32×C4×C7C22×C3
socleC2×C32×C7C2×C3×D4
hypercenterC2×C4D32×ΓL(1,𝔽8)
hypocenterC32×C7C3×AGL(1,ℤ/8)
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C2×C4

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#10595100
maximal normal subgroup ofΓL(1,𝔽8)×#10595100

GAP

Past the SmallGroups library coverage — no SmallGroups id.