Finite Group Databasebeta

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AGL(1,𝔽7)×#87831 #72243229

order16800
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productAGL(1,𝔽7)×#87831

Splittings

HallD7×#87831 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C2×C22⋊C4
HallC52×C7 ⋊ C2×C3×C22⋊C4
SylowC52 ⋊ C22⋊C4×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×#87831
SylowC7 ⋊ C2×C3×#87831

Subgroups

subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽7)×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C52×C7C22×C3×C4
prime radicalC2×D52×AGL(1,𝔽7)C2
cosocleC2×C7×C5⋊D5C23×C3
hypocenterC52×C7C2×C3×C22⋊C4
LCSG ▷ C2×C52×C7 ▷ C52×C7
UCSC1 ◁ C2

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#11717375
maximal normal subgroup ofAGL(1,𝔽7)×#11717375

GAP

Past the SmallGroups library coverage — no SmallGroups id.