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ΓL(1,𝔽8)×#556 #13060717

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productΓL(1,𝔽8)×#556
unambiguous semidirect product(C7×#556)⋊C3

Splittings

Fitting, HallC7×#556 ⋊ C3
SylowC7 ⋊ C3×#556

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#556C3
prime radicalC2×D4×ΓL(1,𝔽8)C22
hypercenter#556ΓL(1,𝔽8)
hypocenterC7C3×#556
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #556

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10006);