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C5×D3243 #44569866

order32430
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3243
unambiguous semidirect productC3⋊(C5×D1081)
unambiguous semidirect productC23⋊(C5×D141)
unambiguous semidirect productC47⋊(C5×D69)

Splittings

Fitting, socle, HallC3×C5×C23×C47 ⋊ C2
derived, HallC3×C23×C47 ⋊ C2×C5
HallC5×C23×C47 ⋊ D3
HallC23×C47 ⋊ C5×D3
HallC3×C5×C47 ⋊ D23
HallC3×C5×C23 ⋊ D47
HallC5×C47 ⋊ D69
HallC3×C47 ⋊ C5×D23
HallC5×C23 ⋊ D141
HallC3×C23 ⋊ C5×D47
SylowC47 ⋊ C5×D69
SylowC23 ⋊ C5×D141
HallC3×C5 ⋊ D1081
SylowC3 ⋊ C5×D1081

Subgroups

subgroupG/H
center, hypercenterC5D3243
commutator, cosocle, hypocenterC3×C23×C47C2×C5
FrattiniC1G
Fitting, socleC3×C5×C23×C47C2
LCSG ▷ C3×C23×C47
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 78
transitive degreeμt(G) = 16215

A minimal faithful permutation representation — 2 generators on 78 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);(7,8)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Conjugacy classes & character table

8115 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32430, 13);