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C5×D3689 #49146261

order36890
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3689
unambiguous semidirect productC7⋊(C5×D527)
unambiguous semidirect productC17⋊(C5×D217)

Splittings

Fitting, socle, HallC5×C7×C17×C31 ⋊ C2
derived, HallC7×C17×C31 ⋊ C2×C5
HallC5×C17×C31 ⋊ D7
HallC5×C7×C31 ⋊ D17
HallC5×C7×C17 ⋊ D31
HallC17×C31 ⋊ C5×D7
HallC7×C31 ⋊ C5×D17
HallC5×C31 ⋊ D119
HallC7×C17 ⋊ C5×D31
HallC5×C17 ⋊ D217
HallC5×C7 ⋊ D527
SylowC31 ⋊ C5×D119
SylowC17 ⋊ C5×D217
SylowC7 ⋊ C5×D527

Subgroups

subgroupG/H
center, hypercenterC5D3689
commutator, cosocle, hypocenterC7×C17×C31C2×C5
FrattiniC1G
Fitting, socleC5×C7×C17×C31C2
LCSG ▷ C7×C17×C31
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 18445

A minimal faithful permutation representation — 2 generators on 60 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);(7,12)(8,11)(9,10)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Conjugacy classes & character table

9230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36890, 9);