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AGL(1,𝔽7)×#390 #72259778

order2688
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26×C4

Aliases

direct productAGL(1,𝔽7)×#390

Splittings

FittingC7×#390 ⋊ C2×C3
HallD7×#390 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#390
SylowC7 ⋊ C2×C3×#390

Subgroups

subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽7)
commutatorC22×C7C23×C3×C4
FittingC7×#390C2×C3
socle, cosocleC23×C7C24×C3
prime radicalC22×D4×AGL(1,𝔽7)C2
hypercenter#390AGL(1,𝔽7)
hypocenterC7C2×C3×#390
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

2720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#2679, AGL(1,𝔽7)×#1393, AGL(1,𝔽7)×#1792
maximal normal subgroup ofAGL(1,𝔽7)×#2679, AGL(1,𝔽7)×#1393, AGL(1,𝔽7)×#1792
maximal quotient ofAGL(1,𝔽7)×#1438

GAP

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

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