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AGL(1,𝔽7)×H(3,𝔽4) #72259781

order2688
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C26×C42

Aliases

direct productAGL(1,𝔽7)×H(3,𝔽4)

Splittings

FittingC7×H(3,𝔽4) ⋊ C2×C3
HallD7×H(3,𝔽4) ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×H(3,𝔽4)
SylowC7 ⋊ C2×C3×H(3,𝔽4)

Subgroups

subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C25×C3
FittingC7×H(3,𝔽4)C2×C3
hypercenterH(3,𝔽4)AGL(1,𝔽7)
hypocenterC7C2×C3×H(3,𝔽4)
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ H(3,𝔽4)

2816 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)(3,4,6)(9,13)(10,14)(17,21)(20,23);(9,13)(12,15)(18,22)(20,23);(8,10)(9,12)(11,14)(13,15)(16,18,19,22)(17,20,21,23);(8,9,11,13)(10,12,14,15)(16,17,19,21)(18,20,22,23);(2,3)(4,6)(5,7)

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#5464, AGL(1,𝔽7)×#2199, AGL(1,𝔽7)×#3532
maximal normal subgroup ofAGL(1,𝔽7)×#5464, AGL(1,𝔽7)×#2199, AGL(1,𝔽7)×#3532
maximal quotient ofAGL(1,𝔽7)×#2679

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