Finite Group Databasebeta

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C7×#4952 #13055985

order1344
isoclinismisoclinic class of #4656
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C23

Aliases

direct productC7×#4952

Splittings

HallC3×C7 ⋊ #397
SylowC3 ⋊ C7×#397

Subgroups

subgroupG/H
centerC23×C7C22×D3
commutator, cosocleC23×C3C23×C7
FrattiniC23C22×D3×C7
FittingC2×C3×C7×C22⋊C4C2
socleC23×C3×C7C23
prime radicalC24×C3×C7C22
hypercenterC2×C7×C22⋊C4D3
hypocenterC3C7×#397
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C7 ◁ C2×C7×C22⋊C4

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5274);