Finite Group Databasebeta

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AGL(1,𝔽7)×#235 #72244408

order2016
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productAGL(1,𝔽7)×#235

Splittings

FittingC2×C3×C4×C7 ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×#41
SylowC7 ⋊ C2×C3×#235

Subgroups

subgroupG/H
centerC4C2×D3×AGL(1,𝔽7)
commutator, socle, cosocleC2×C3×C7C24×C3
FrattiniC2C22×D3×AGL(1,𝔽7)
FittingC2×C3×C4×C7C22×C3
hypercenterC2×C4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×#41
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#1004, D3×#41×AGL(1,𝔽7), AGL(1,𝔽7)×#1009, …
maximal normal subgroup ofAGL(1,𝔽7)×#1004, D3×#41×AGL(1,𝔽7), AGL(1,𝔽7)×#1009, …
maximal quotient ofAGL(1,𝔽7)×#3804, AGL(1,𝔽7)×#3799, C3×AGL(1,𝔽7)×#235, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.