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C7×#5775 #13061491

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

Aliases

direct productC7×#5775

Splittings

FittingC3×C7×AGL(1,ℤ/8) ⋊ C2
HallC3×C7 ⋊ #577
SylowC3 ⋊ C7×#577

Subgroups

subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×AGL(1,ℤ/8)C2
socleC2×C3×C7C22×D4
hypercenterC7×AGL(1,ℤ/8)D3
hypocenterC3C7×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×AGL(1,ℤ/8)

516 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)(11,13,15,19)(12,17,16,23)(14,22,21,20)(18,26,24,25);(13,20)(14,21)(17,25)(18,24)(19,22)(23,26);(12,16)(13,19)(18,24)(20,22);(9,10)(11,12)(13,17)(14,18)(15,16)(19,23)(20,25)(21,24)(22,26)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10780);