Finite Group Databasebeta

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AGL(1,𝔽7)×#837 #72254343

order4032
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productAGL(1,𝔽7)×#837

Splittings

HallC3×ΓL(1,𝔽8) ⋊ C2×#104
SylowC7 ⋊ C2×C3×#837

Subgroups

subgroupG/H
centerC4#67×AGL(1,𝔽7)
commutatorC3×C4×C7C22×C3×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽7)
FittingC3×C7×#41C22×C3
socleC2×C3×C7C2×C3×C22⋊C4
prime radicalC3×#41×AGL(1,𝔽7)C2
cosocleC2×C3×C4×C7C23×C3
hypercenter#41D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×#104
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#104×AGL(1,𝔽7), AGL(1,𝔽7)×#5241, AGL(1,𝔽7)×#5198
maximal normal subgroup ofD3×#104×AGL(1,𝔽7), AGL(1,𝔽7)×#5241, AGL(1,𝔽7)×#5198

GAP

Past the SmallGroups library coverage — no SmallGroups id.