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C3×#20269516 #34117705

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

Aliases

direct productC3×#20269516

Splittings

Fitting, socle, HallC3×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC3×#16809823 ⋊ C2
HallC3×C11×D61 ⋊ C5
Hall#16809823 ⋊ 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#20269516
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:

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