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C29×D345 #34093406

order20010
automorphism?
homologyAbelianization (H1) = C2×C29 · Schur multiplier (H2) = C1

Aliases

direct productC29×D345
unambiguous semidirect productC3⋊(C29×D115)
unambiguous semidirect productC5⋊(C29×D69)
unambiguous semidirect productC23⋊(C29×D15)

Splittings

Fitting, socle, HallC3×C5×C23×C29 ⋊ C2
derived, HallC3×C5×C23 ⋊ C2×C29
HallC5×C23×C29 ⋊ D3
HallC3×C23×C29 ⋊ D5
HallC23×C29 ⋊ D15
HallC3×C5×C29 ⋊ D23
HallC5×C29 ⋊ D69
HallC5×C23 ⋊ D3×C29
HallC3×C29 ⋊ D115
HallC3×C23 ⋊ D5×C29
SylowC23 ⋊ C29×D15
HallC3×C5 ⋊ C29×D23
SylowC5 ⋊ C29×D69
SylowC3 ⋊ C29×D115

Subgroups

subgroupG/H
center, hypercenterC29D345
commutator, cosocle, hypocenterC3×C5×C23C2×C29
FrattiniC1G
Fitting, socleC3×C5×C23×C29C2
LCSG ▷ C3×C5×C23
UCSC1 ◁ C29

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)(25,26)(27,28)(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);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,25,27,28,26)(29,30,31)

Conjugacy classes & character table

5046 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20010, 15);