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C5×D3741 #49153458

order37410
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3741
unambiguous semidirect productC3⋊(C5×D1247)
unambiguous semidirect productC29⋊(C5×D129)
unambiguous semidirect productC43⋊(C5×D87)

Splittings

Fitting, socle, HallC3×C5×C29×C43 ⋊ C2
derived, HallC3×C29×C43 ⋊ C2×C5
HallC5×C29×C43 ⋊ D3
HallC29×C43 ⋊ C5×D3
HallC3×C5×C43 ⋊ D29
HallC3×C5×C29 ⋊ D43
HallC5×C43 ⋊ D87
HallC5×C29 ⋊ D129
HallC3×C43 ⋊ C5×D29
HallC3×C29 ⋊ C5×D43
SylowC43 ⋊ C5×D87
SylowC29 ⋊ C5×D129
HallC3×C5 ⋊ D1247
SylowC3 ⋊ C5×D1247

Subgroups

subgroupG/H
center, hypercenterC5D3741
commutator, cosocle, hypocenterC3×C29×C43C2×C5
FrattiniC1G
Fitting, socleC3×C5×C29×C43C2
LCSG ▷ C3×C29×C43
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 18705

A minimal faithful permutation representation — 2 generators on 80 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80);(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)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Conjugacy classes & character table

9360 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37410, 13);