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ΓL(1,𝔽8)×#543 #13060704

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

Aliases

direct productΓL(1,𝔽8)×#543

Splittings

Fitting, HallC7×#543 ⋊ C3
SylowC7 ⋊ C3×#543

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#543C3
prime radicalC24×ΓL(1,𝔽8)C22
hypercenter#543ΓL(1,𝔽8)
hypocenterC7C3×#543
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #543

484 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)(8,9,12,15)(10,16,18,14)(11,13,19,21)(17,22,23,20);(1,2,3,5,4,6,7)(9,13)(10,17)(15,21)(18,23);(8,10,12,18)(9,14,15,16)(11,17,19,23)(13,20,21,22);(9,13)(14,20)(15,21)(16,22)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#543⋊C3, ΓL(1,𝔽8)×#3490, #72277456, …
maximal normal subgroup ofΓL(1,𝔽8)×#543⋊C3, ΓL(1,𝔽8)×#3490, #72277456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9993);