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AGL(1,𝔽7)×#543 #72277480

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

Aliases

direct productAGL(1,𝔽7)×#543

Splittings

FittingC7×#543 ⋊ C2×C3
HallD7×#543 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#543
SylowC7 ⋊ C2×C3×#543

Subgroups

subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C25×C3
FittingC7×#543C2×C3
prime radicalC24×AGL(1,𝔽7)C22
hypercenter#543AGL(1,𝔽7)
hypocenterC7C2×C3×#543
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #543

2592 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)(4,5,6)(17,21)(20,23);(18,22)(20,23);(8,10,11,15)(9,12,13,14)(16,18)(17,20)(19,22)(21,23);(8,9,11,13)(10,14,15,12)(16,17)(18,20)(19,21)(22,23);(2,3)(4,5)(6,7)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#3490
maximal normal subgroup ofAGL(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)(4,5,6)(17,21)(20,23), (18,22)(20,23), (8,10,11,15)(9,12,13,14)(16,18)(17,20)(19,22)(21,23), (8,9,11,13)(10,14,15,12)(16,17)(18,20)(19,21)(22,23), (2,3)(4,5)(6,7));