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AGL(1,𝔽7)×#5241 #72254398

order8064
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productAGL(1,𝔽7)×#5241

Splittings

FittingC3×C7×H(5,𝔽2) ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×#453
SylowC7 ⋊ C2×C3×#5241

Subgroups

subgroupG/H
centerC2AGL(1,𝔽7)×#953
commutator, cosocleC2×C3×C4×C7C24×C3
FrattiniC2×C4C22×D3×AGL(1,𝔽7)
FittingC3×C7×H(5,𝔽2)C22×C3
socleC2×C3×C7C2×C3×#120
hypercenterH(5,𝔽2)D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×#453
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

4274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽7)×#453
maximal normal subgroup ofD3×AGL(1,𝔽7)×#453

GAP

Past the SmallGroups library coverage — no SmallGroups id.