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AGL(1,𝔽7)×#498 #72229556

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4

Aliases

direct productAGL(1,𝔽7)×#498

Splittings

FittingC7×#498 ⋊ C2×C3
HallD7×#498 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#498
SylowC7 ⋊ C2×C3×#498

Subgroups

subgroupG/H
centerC22C2×D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC7×#498C2×C3
socleC22×C7C22×C3×D4
prime radicalC22×AGL(1,𝔽7)C2×D4
hypercenter#498AGL(1,𝔽7)
hypocenterC7C2×C3×#498
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #498

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#1715, AGL(1,𝔽7)×#3184, AGL(1,𝔽7)×#1705
maximal normal subgroup ofAGL(1,𝔽7)×#1715, AGL(1,𝔽7)×#3184, AGL(1,𝔽7)×#1705

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