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C5×C22⋊#10620323 #27190274

order13140
automorphism?
homologyAbelianization (H1) = C5×C9 · Schur multiplier (H2) = C2

Aliases

direct productC5×C22⋊#10620323
unambiguous semidirect productC22⋊(C5×#10620323)
unambiguous semidirect product(C22×C5)⋊#10620323

Splittings

Fitting, socle, HallC22×C5×C73 ⋊ C9
derived, HallC22×C73 ⋊ C5×C9
HallC5×C73 ⋊ C22⋊C9
SylowC73 ⋊ C5×C22⋊C9
HallC22×C5 ⋊ #10620323
SylowC22 ⋊ C5×#10620323

Subgroups

subgroupG/H
center, hypercenterC5C22⋊#10620323
commutator, hypocenterC22×C73C5×C9
FrattiniC1G
Fitting, socleC22×C5×C73C9
prime radicalC22×C5×C73⋊C3C3
cosocleC22×C73⋊C3C3×C5
LCSG ▷ C22×C73
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 82
transitive degreeμt(G) = 1460

A minimal faithful permutation representation — 2 generators on 82 points:

(2,3,5,4,6,10,8,12,20)(7,14,26,15,27,46,29,35,38)(9,17,33,18,34,57,16,31,41)(11,19,36,22,28,49,40,23,42)(13,24,43,25,44,21,32,55,39)(30,51,66,52,45,63,56,64,48)(37,59,72,60,73,71,50,58,62)(47,53,68,65,69,67,61,54,70)(75,76,77)(78,79,80,81,82);(1,2,3,7,5,11,14,28,4,9,19,37,26,47,49,60,6,13,17,35,36,58,59,46,15,30,53,66,40,61,73,65,10,21,24,45,33,20,38,42,22,41,62,70,72,39,29,50,27,48,51,67,68,44,52,12,23,31,54,55,71,64,69,25,8,16,32,56,43,57,63,34,18)(74,77)(75,76)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 108);