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C3×#20269517 #34117706

order20130
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×#20269517

Splittings

Fitting, socle, HallC3×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC3×#16809821 ⋊ C2
HallC3×C11×D61 ⋊ C5
Hall#16809821 ⋊ C2×C3
HallC11×D61 ⋊ C3×C5
HallC3×D61 ⋊ C11⋊C5
HallC3×C61 ⋊ C2×C11⋊C5
HallD61 ⋊ C3×C11⋊C5
SylowC61 ⋊ C2×C3×C11⋊C5
HallC3×C11 ⋊ D61⋊C5
SylowC11 ⋊ C3×D61⋊C5

Subgroups

subgroupG/H
center, hypercenterC3#20269517
commutator, cosocle, hypocenterC11×C61C2×C3×C5
FrattiniC1G
Fitting, socleC3×C11×C61C2×C5
LCSG ▷ C11×C61
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 75
transitive degreeμt(G) = 2013

A minimal faithful permutation representation — 2 generators on 75 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 35);