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C3×D3835 #37527653

order23010
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3835
unambiguous semidirect productC5⋊(C3×D767)
unambiguous semidirect productC59⋊(C3×D65)

Splittings

Fitting, socle, HallC3×C5×C13×C59 ⋊ C2
derived, HallC5×C13×C59 ⋊ C2×C3
HallC3×C13×C59 ⋊ D5
HallC3×C5×C59 ⋊ D13
HallC13×C59 ⋊ C3×D5
HallC5×C59 ⋊ C3×D13
HallC3×C5×C13 ⋊ D59
HallC3×C59 ⋊ D65
HallC5×C13 ⋊ C3×D59
SylowC59 ⋊ C3×D65
HallC3×C13 ⋊ D295
HallC3×C5 ⋊ D767
SylowC13 ⋊ C3×D295
SylowC5 ⋊ C3×D767

Subgroups

subgroupG/H
center, hypercenterC3D3835
commutator, cosocle, hypocenterC5×C13×C59C2×C3
FrattiniC1G
Fitting, socleC3×C5×C13×C59C2
LCSG ▷ C5×C13×C59
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 11505

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80);(5,8)(6,7)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Conjugacy classes & character table

5757 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23010, 9);