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C5×D2001 #34093403

order20010
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D2001
unambiguous semidirect productC3⋊(C5×D667)
unambiguous semidirect productC23⋊(C5×D87)
unambiguous semidirect productC29⋊(C5×D69)

Splittings

Fitting, socle, HallC3×C5×C23×C29 ⋊ C2
derived, HallC3×C23×C29 ⋊ C2×C5
HallC5×C23×C29 ⋊ D3
HallC23×C29 ⋊ C5×D3
HallC3×C5×C29 ⋊ D23
HallC3×C5×C23 ⋊ D29
HallC5×C29 ⋊ D69
HallC5×C23 ⋊ D87
HallC3×C29 ⋊ C5×D23
HallC3×C23 ⋊ C5×D29
SylowC29 ⋊ C5×D69
SylowC23 ⋊ C5×D87
HallC3×C5 ⋊ D667
SylowC3 ⋊ C5×D667

Subgroups

subgroupG/H
center, hypercenterC5D2001
commutator, cosocle, hypocenterC3×C23×C29C2×C5
FrattiniC1G
Fitting, socleC3×C5×C23×C29C2
LCSG ▷ C3×C23×C29
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 10005

A minimal faithful permutation representation — 2 generators on 60 points:

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

Conjugacy classes & character table

5010 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20010, 12);