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C7×#5236 #13056274

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

Aliases

direct productC7×#5236

Splittings

HallC3×C7 ⋊ #421
SylowC3 ⋊ C7×#421

Subgroups

subgroupG/H
centerC4×C7C2×#67
commutatorC3×C4C22×C4×C7
FrattiniC2×C4C22×D3×C7
Fitting, prime radicalC2×C3×C7×#41C2
socleC2×C3×C7C2×C22⋊C4
cosocleC2×C3×C4C23×C7
hypercenterC2×C7×#41D3
hypocenterC3C7×#421
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#41

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

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, 5563);