Finite Group Databasebeta

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C2×C7×D1305 #49141310

order36540
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×D1305
unambiguous semidirect product(C2×C5)⋊(C7×D261)
unambiguous semidirect product(C2×C9)⋊(C7×D145)

Splittings

FittingC2×C5×C7×C9×C29 ⋊ C2
derived, HallC5×C9×C29 ⋊ C22×C7
HallC5×C7×C9×C29 ⋊ C22
HallC7×C9×C29 ⋊ C2×D5
HallC5×C7×C29 ⋊ C2×D9
HallC5×C7×C9 ⋊ C2×D29
HallC9×C29 ⋊ C2×C7×D5
HallC7×C29 ⋊ C2×D45
HallC5×C29 ⋊ C2×C7×D9
HallC7×C9 ⋊ C2×D145
HallC5×C9 ⋊ C2×C7×D29
HallC5×C7 ⋊ C2×D261
SylowC29 ⋊ C2×C7×D45
SylowC9 ⋊ C2×C7×D145
SylowC5 ⋊ C2×C7×D261

Subgroups

subgroupG/H
center, hypercenterC2×C7D1305
commutator, cosocle, hypocenterC5×C9×C29C22×C7
FrattiniC3C2×C7×D435
FittingC2×C5×C7×C9×C29C2
socleC2×C3×C5×C7×C29D3
LCSG ▷ C5×C9×C29
UCSC1 ◁ C2×C7

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 52
transitive degreeμt(G) = 18270

A minimal faithful permutation representation — 2 generators on 52 points:

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

Conjugacy classes & character table

9156 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 152);