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C3×#90513 #13061167

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

Aliases

direct productC3×#90513

Splittings

FittingC2×C3×C7×#41 ⋊ C2
HallC3×C7 ⋊ #575
SylowC7 ⋊ C3×#575

Subgroups

subgroupG/H
centerC3×C4C2×C22⋊D7
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×C3×D7
FittingC2×C3×C7×#41C2
socleC2×C3×C7C22×D4
hypercenterC2×C3×#41D7
hypocenterC7C3×#575
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#41

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10456);