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AGL(1,𝔽7)×#538 #72260121

order2688
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productAGL(1,𝔽7)×#538

Splittings

FittingC7×#538 ⋊ C2×C3
HallD7×#538 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#538
SylowC7 ⋊ C2×C3×#538

Subgroups

subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C25×C3
FittingC7×#538C2×C3
hypercenter#538AGL(1,𝔽7)
hypocenterC7C2×C3×#538
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #538

2672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 5 generators on 23 points:

(1,2,7)(3,4,6)(10,15)(13,14)(17,20)(21,23);(8,10,11,15)(9,13,12,14)(16,18,19,22)(17,21,20,23);(9,12)(10,15)(16,17)(18,21)(19,20)(22,23);(8,9)(10,14)(11,12)(13,15)(17,20)(21,23);(2,3)(4,6)(5,7)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#3532, AGL(1,𝔽7)×#3490
maximal normal subgroup ofAGL(1,𝔽7)×#3532, AGL(1,𝔽7)×#3490

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,7)(3,4,6)(10,15)(13,14)(17,20)(21,23), (8,10,11,15)(9,13,12,14)(16,18,19,22)(17,21,20,23), (9,12)(10,15)(16,17)(18,21)(19,20)(22,23), (8,9)(10,14)(11,12)(13,15)(17,20)(21,23), (2,3)(4,6)(5,7));