Finite Group Databasebeta

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AGL(1,𝔽7)×#6910 #72249136

order10206
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C36

Aliases

direct productAGL(1,𝔽7)×#6910

Splittings

FittingC7×#6910 ⋊ C2×C3
HallΓL(1,𝔽8)×#6910 ⋊ C2
HallD7 ⋊ C3×#6910
SylowC7 ⋊ C2×C3×#6910

Subgroups

subgroupG/H
centerC3C3×H(3,𝔽3)×AGL(1,𝔽7)
commutator, cosocleC32×C7C2×C34
FrattiniC32C33×AGL(1,𝔽7)
FittingC7×#6910C2×C3
socleC3×C7C2×C32×H(3,𝔽3)
prime radicalC3×H(3,𝔽3)×AGL(1,𝔽7)C3
hypercenter#6910AGL(1,𝔽7)
hypocenterC7C2×C3×#6910
LCSG ▷ C32×C7 ▷ C3×C7 ▷ C7
UCSC1 ◁ C3 ◁ C33 ◁ #6910

2176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 189

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#10624749, AGL(1,𝔽7)×#4184⋊C3
maximal normal subgroup ofAGL(1,𝔽7)×#10624749, AGL(1,𝔽7)×#4184⋊C3

GAP

Past the SmallGroups library coverage — no SmallGroups id.