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AGL(1,𝔽7)×#450 #72276544

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productAGL(1,𝔽7)×#450

Splittings

FittingC7×#450 ⋊ C2×C3
HallD7×#450 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#450
SylowC7 ⋊ C2×C3×#450

Subgroups

subgroupG/H
centerC22C2×D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC7×#450C2×C3
socleC22×C7C22×C3×D4
prime radicalC22×D4×AGL(1,𝔽7)C2
hypercenter#450AGL(1,𝔽7)
hypocenterC7C2×C3×#450
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

1992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 112

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

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

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#3003
maximal normal subgroup ofAGL(1,𝔽7)×#3003

GAP

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

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