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C7×#4683 #13055718

order1344
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#4683

Splittings

HallC3×C7 ⋊ #499
SylowC3 ⋊ C7×#499

Subgroups

subgroupG/H
centerC22×C7C2×D12
commutator, cosocleC2×C3×C4C23×C7
FrattiniC2×C4C22×D3×C7
FittingC3×C4×C7×C8C2
socle, prime radicalC22×C3×C7C2×D4
hypercenterC4×C7×C8D3
hypocenterC3C7×#499
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C42×C7 ◁ C4×C7×C8

172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 1344

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

378 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5007);