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C7×#5132 #13056166

order1344
automorphism?
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23

Aliases

direct productC7×#5132

Splittings

FittingC2×C3×C7×M16 ⋊ C2
HallC3×C7 ⋊ #413
SylowC3 ⋊ C7×#413

Subgroups

subgroupG/H
centerC4×C7(C2×D3)⋊C4
commutatorC22×C3C22×C4×C7
FrattiniC2×C4C22×D3×C7
FittingC2×C3×C7×M16C2
socleC2×C3×C7C2×C22⋊C4
prime radicalC2×C7×#235C2
cosocleC2×C3×C4C23×C7
hypercenterC2×C7×M16D3
hypocenterC3C7×#413
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×M16

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

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, 5455);