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C5×D3219 #44565720

order32190
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3219
unambiguous semidirect productC3⋊(C5×D1073)
unambiguous semidirect productC29⋊(C5×D111)
unambiguous semidirect productC37⋊(C5×D87)

Splittings

Fitting, socle, HallC3×C5×C29×C37 ⋊ C2
derived, HallC3×C29×C37 ⋊ C2×C5
HallC5×C29×C37 ⋊ D3
HallC29×C37 ⋊ C5×D3
HallC3×C5×C37 ⋊ D29
HallC3×C5×C29 ⋊ D37
HallC5×C37 ⋊ D87
HallC5×C29 ⋊ D111
HallC3×C37 ⋊ C5×D29
HallC3×C29 ⋊ C5×D37
SylowC37 ⋊ C5×D87
SylowC29 ⋊ C5×D111
HallC3×C5 ⋊ D1073
SylowC3 ⋊ C5×D1073

Subgroups

subgroupG/H
center, hypercenterC5D3219
commutator, cosocle, hypocenterC3×C29×C37C2×C5
FrattiniC1G
Fitting, socleC3×C5×C29×C37C2
LCSG ▷ C3×C29×C37
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 74
transitive degreeμt(G) = 16095

A minimal faithful permutation representation — 2 generators on 74 points:

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

Conjugacy classes & character table

8055 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32190, 13);