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ΓL(1,𝔽8)×#536 #13060697

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productΓL(1,𝔽8)×#536
unambiguous semidirect product(C7×#536)⋊C3

Splittings

Fitting, HallC7×#536 ⋊ C3
SylowC7 ⋊ C3×#536

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#536C3
prime radicalC2×#41×ΓL(1,𝔽8)C2
hypercenter#536ΓL(1,𝔽8)
hypocenterC7C3×#536
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #536

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9986);