Finite Group Databasebeta

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AGL(1,𝔽7)×#1393 #72277249

order5376
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productAGL(1,𝔽7)×#1393

Splittings

FittingC7×#1393 ⋊ C2×C3
HallD7×#1393 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#1393
SylowC7 ⋊ C2×C3×#1393

Subgroups

subgroupG/H
centerC22#99×AGL(1,𝔽7)
commutatorC2×C4×C7C2×C3×C42
FrattiniC22×D4C22×AGL(1,𝔽7)
FittingC7×#1393C2×C3
socleC22×C7C2×C3×#99
prime radicalC22×D4×AGL(1,𝔽7)C22
cosocleC22×C7×D4C23×C3
hypercenter#1393AGL(1,𝔽7)
hypocenterC7C2×C3×#1393
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

2308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#13343
maximal normal subgroup ofAGL(1,𝔽7)×#13343

GAP

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

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