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C7×#5768 #13061484

order1344
automorphism?
homologyAbelianization (H1) = C24×C7 · Schur multiplier (H2) = C25

Aliases

direct productC7×#5768

Splittings

FittingC3×C7×#135 ⋊ C2
HallC3×C7 ⋊ #575
SylowC3 ⋊ C7×#575

Subgroups

subgroupG/H
centerC4×C7D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×#135C2
socleC2×C3×C7C22×D4
hypercenterC7×#135D3
hypocenterC3C7×#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C7×#135

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:

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10773);