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ΓL(1,𝔽8)×#553 #13060714

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

Aliases

direct productΓL(1,𝔽8)×#553
unambiguous semidirect product(C7×#553)⋊C3

Splittings

Fitting, HallC7×#553 ⋊ C3
SylowC7 ⋊ C3×#553

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#553C3
hypercenter#553ΓL(1,𝔽8)
hypocenterC7C3×#553
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #553

476 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)(10,11)(12,13)(14,15)(17,20)(21,22);(1,2,3,5,4,6,7)(9,13)(11,14)(16,17,19,20)(18,22,23,21);(8,10)(9,14)(11,13)(12,15);(8,11,12,14)(9,10,13,15)(16,18,19,23)(17,21,20,22)

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, 10003);