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ΓL(1,𝔽8)×#538 #13060699

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

Aliases

direct productΓL(1,𝔽8)×#538
unambiguous semidirect product(C7×#538)⋊C3

Splittings

Fitting, HallC7×#538 ⋊ C3
SylowC7 ⋊ C3×#538

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#538C3
hypercenter#538ΓL(1,𝔽8)
hypocenterC7C3×#538
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #538

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#3532, AGL(1,𝔽7)×#538, #72276683, …
maximal normal subgroup ofΓL(1,𝔽8)×#3532, AGL(1,𝔽7)×#538, #72276683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9988);