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AGL(1,𝔽7)×#365 #72228656

order2688
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productAGL(1,𝔽7)×#365

Splittings

FittingC7×#365 ⋊ C2×C3
HallD7×#365 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#365
SylowC7 ⋊ C2×C3×#365

Subgroups

subgroupG/H
centerC2#106×AGL(1,𝔽7)
commutatorC7×C8C22×C3×C4
FrattiniC2×C8C22×AGL(1,𝔽7)
FittingC7×#365C2×C3
socleC2×C7C2×C3×#106
prime radicalC22×AGL(1,𝔽7)SD16
cosocleC2×C7×C8C23×C3
hypercenter#365AGL(1,𝔽7)
hypocenterC7C2×C3×#365
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #365

304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×AGL(1,ℤ/16)
maximal normal subgroup ofAGL(1,𝔽7)×AGL(1,ℤ/16)
maximal quotient ofAGL(1,𝔽7)×#1368

GAP

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

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