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ΓL(1,𝔽8)2×AGL(1,ℤ/8) #72245372

order14112
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3

Aliases

direct productΓL(1,𝔽8)2×AGL(1,ℤ/8)

Splittings

Fitting, HallC72×AGL(1,ℤ/8) ⋊ C32
SylowC72 ⋊ C32×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC2C2×D4×ΓL(1,𝔽8)2
commutator, cosocleC4×C72C23×C32
FrattiniC4C23×ΓL(1,𝔽8)2
FittingC72×AGL(1,ℤ/8)C32
socleC2×C72C2×C32×D4
hypercenterAGL(1,ℤ/8)ΓL(1,𝔽8)2
hypocenterC72C32×AGL(1,ℤ/8)
LCSG ▷ C4×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,ℤ/8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), ΓL(1,𝔽8)2×#453, #72246400
maximal normal subgroup ofAGL(1,ℤ/8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), ΓL(1,𝔽8)2×#453, #72246400

GAP

Past the SmallGroups library coverage — no SmallGroups id.