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ΓL(1,𝔽8)×#504 #13055035

order1344
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×#504

Splittings

Fitting, HallC7×#504 ⋊ C3
SylowC7 ⋊ C3×#504

Subgroups

subgroupG/H
centerC16C22×ΓL(1,𝔽8)
commutator, socleC2×C7C22×C3×C8
FrattiniC8C23×ΓL(1,𝔽8)
FittingC7×#504C3
prime radical#41×ΓL(1,𝔽8)C4
cosocleC7×C8C23×C3
hypercenter#504ΓL(1,𝔽8)
hypocenterC7C3×#504
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C16 ◁ #504

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(2,3,4)(5,7,6)(8,9,11,15,12,16,21,27,13,17,22,28,23,29,33,37)(10,14,19,25,20,26,32,36,18,24,30,34,31,35,38,39);(1,2,3,5,4,6,7)(8,10)(9,14)(11,19)(12,20)(13,18)(15,25)(16,26)(17,24)(21,32)(22,30)(23,31)(27,36)(28,34)(29,35)(33,38)(37,39);(10,18)(14,24)(19,30)(20,31)(25,34)(26,35)(32,38)(36,39)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4324);