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C7×#4821 #13055856

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

Aliases

direct productC7×#4821

Splittings

FittingC3×C7×#104 ⋊ C2
HallC3×C7 ⋊ #421
SylowC3 ⋊ C7×#421

Subgroups

subgroupG/H
centerC4×C7D3×D4
commutatorC3×C4C22×C4×C7
FrattiniC2×C4C22×D3×C7
FittingC3×C7×#104C2
socleC2×C3×C7C2×C22⋊C4
prime radicalD3×C7×#41C2
cosocleC2×C3×C4C23×C7
hypercenterC7×#104D3
hypocenterC3C7×#421
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C7×#104

308 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)(11,14,15,13,16,21,24,22)(12,18,19,25,20,23,26,17);(11,13)(12,17)(14,24)(15,21)(16,22)(18,19)(20,25)(23,26);(9,10)(11,12)(13,17)(14,23)(15,19)(16,20)(18,21)(22,25)(24,26)

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, 5145);