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ΓL(1,𝔽8)×#383 #13054914

order1344
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productΓL(1,𝔽8)×#383

Splittings

Fitting, HallC7×#383 ⋊ C3
SylowC7 ⋊ C3×#383

Subgroups

subgroupG/H
center, FrattiniC23C23×ΓL(1,𝔽8)
commutatorC22×C7C22×C3×C4
FittingC7×#383C3
socle, cosocleC23×C7C23×C3
prime radicalC23×ΓL(1,𝔽8)C23
hypercenter#383ΓL(1,𝔽8)
hypocenterC7C3×#383
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #383

276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

(2,3,4)(5,7,6)(8,9,10,11)(14,18)(16,19)(21,24)(22,26);(1,2,3,5,4,6,7)(12,13,15,17)(14,16,18,19)(20,21)(22,25)(23,24)(26,27);(12,14)(13,16)(15,18)(17,19)(20,22,23,26)(21,25,24,27)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4203);