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C3×D3565 #36339015

order21390
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3565
unambiguous semidirect productC5⋊(C3×D713)
unambiguous semidirect productC23⋊(C3×D155)

Splittings

Fitting, socle, HallC3×C5×C23×C31 ⋊ C2
derived, HallC5×C23×C31 ⋊ C2×C3
HallC3×C23×C31 ⋊ D5
HallC23×C31 ⋊ C3×D5
HallC3×C5×C31 ⋊ D23
HallC3×C5×C23 ⋊ D31
HallC5×C31 ⋊ C3×D23
HallC5×C23 ⋊ C3×D31
HallC3×C31 ⋊ D115
HallC3×C23 ⋊ D155
SylowC31 ⋊ C3×D115
SylowC23 ⋊ C3×D155
HallC3×C5 ⋊ D713
SylowC5 ⋊ C3×D713

Subgroups

subgroupG/H
center, hypercenterC3D3565
commutator, cosocle, hypocenterC5×C23×C31C2×C3
FrattiniC1G
Fitting, socleC3×C5×C23×C31C2
LCSG ▷ C5×C23×C31
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 62
transitive degreeμt(G) = 10695

A minimal faithful permutation representation — 2 generators on 62 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);(5,8)(6,7)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Conjugacy classes & character table

5352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21390, 11);