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ΓL(1,𝔽8)×#551 #13060712

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

Aliases

direct productΓL(1,𝔽8)×#551
unambiguous semidirect product(C7×#551)⋊C3

Splittings

Fitting, HallC7×#551 ⋊ C3
SylowC7 ⋊ C3×#551

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#551C3
prime radicalΓL(1,𝔽8)×#120C2
hypercenter#551ΓL(1,𝔽8)
hypocenterC7C3×#551
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #551

496 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)(8,9,11,15)(10,16,17,13)(12,14,19,21)(18,22,23,20);(1,2,3,5,4,6,7)(8,10,12,18)(9,13,14,20)(11,17,19,23)(15,16,21,22);(9,14)(13,20)(15,21)(16,22);(10,17)(13,16)(18,23)(20,22)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#3532, ΓL(1,𝔽8)×#3490, AGL(1,𝔽7)×#551, …
maximal normal subgroup ofΓL(1,𝔽8)×#3532, ΓL(1,𝔽8)×#3490, AGL(1,𝔽7)×#551, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10001);