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C7×#5769 #13061485

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

Aliases

direct productC7×#5769

Splittings

FittingC3×C7×#135 ⋊ C2
HallC3×C7 ⋊ #576
SylowC3 ⋊ C7×#576

Subgroups

subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×#135C2
socleC2×C3×C7C22×D4
hypercenterC7×#135D3
hypocenterC3C7×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#135

536 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)(12,16)(13,20)(14,23)(17,25)(19,26)(21,22);(11,13)(12,17)(14,20)(15,22)(16,19)(18,26)(21,23)(24,25);(11,12)(13,19)(14,18)(15,16)(17,22)(20,25)(21,26)(23,24);(9,10)(12,16)(17,19)(18,24)(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, 10774);