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C7×#564⋊C3 #13060802

order1344
isoclinismisoclinic class of #564⋊C3
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C24

Aliases

direct productC7×#564⋊C3
unambiguous semidirect product#564⋊(C3×C7)

Splittings

Fitting, HallC7×#564 ⋊ C3
derived, Sylow#564 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC7#564⋊C3
commutator, cosocle, hypocenter#564C3×C7
FrattiniC22C7×C22⋊A4
FittingC7×#564C3
socleC22×C7C22⋊A4
LCSG ▷ #564
UCSC1 ◁ C7

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,14)(10,16,35)(11,23,28)(12,18,39)(13,19,31)(15,29,41)(17,30,27)(20,34,51)(21,36,53)(22,37,47)(24,44,49)(25,45,48)(26,46,55)(32,50,40)(33,52,42)(38,54,43);(8,12,13,27)(9,19)(10,43,22,21)(11,26,25,24)(14,39,31,30)(15,34)(17,18)(20,40,41,42)(28,49,48,55)(29,52,51,50)(32,33)(35,53,47,54);(8,20,13,41)(9,45)(10,25,22,11)(12,42,27,40)(14,35,31,47)(15,16)(17,46)(18,44)(19,23)(21,24,43,26)(28,51,48,29)(30,54,39,53)(32,36)(33,38)(34,37)(49,50,55,52)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10091);