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C5×D2499 #38735923

order24990
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D2499
unambiguous semidirect productC3⋊(C5×D833)
unambiguous semidirect productC17⋊(C5×D147)
unambiguous semidirect productC49⋊(C5×D51)

Splittings

Fitting, HallC3×C5×C17×C49 ⋊ C2
derived, HallC3×C17×C49 ⋊ C2×C5
HallC5×C17×C49 ⋊ D3
HallC17×C49 ⋊ C5×D3
HallC3×C5×C49 ⋊ D17
HallC3×C5×C17 ⋊ D49
HallC5×C49 ⋊ D51
HallC3×C49 ⋊ C5×D17
HallC5×C17 ⋊ D147
HallC3×C17 ⋊ C5×D49
SylowC49 ⋊ C5×D51
SylowC17 ⋊ C5×D147
HallC3×C5 ⋊ D833
SylowC3 ⋊ C5×D833

Subgroups

subgroupG/H
center, hypercenterC5D2499
commutator, cosocle, hypocenterC3×C17×C49C2×C5
FrattiniC7C5×D357
FittingC3×C5×C17×C49C2
socleC3×C5×C7×C17D7
LCSG ▷ C3×C17×C49
UCSC1 ◁ C5

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 74
transitive degreeμt(G) = 12495

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

Conjugacy classes & character table

6255 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24990, 13);