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C7×#5964 #13062361

order1344
automorphism?
homologyAbelianization (H1) = C25×C7 · Schur multiplier (H2) = C29

Aliases

direct productC7×#5964

Splittings

FittingC2×C3×C7×#41 ⋊ C2
HallC3×C7 ⋊ #585
SylowC3 ⋊ C7×#585

Subgroups

subgroupG/H
centerC4×C7C23×D3
commutator, cosocleC2×C3C25×C7
FrattiniC2C24×D3×C7
FittingC2×C3×C7×#41C2
socleC2×C3×C7C25
hypercenterC2×C7×#41D3
hypocenterC3C7×#585
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C2×C7×#41

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 5 generators on 26 points:

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

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, 11650);