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C3×#20269515 #34117704

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

Aliases

direct productC3×#20269515

Splittings

Fitting, socle, HallC3×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC3×#16809822 ⋊ C2
HallC3×C11×D61 ⋊ C5
Hall#16809822 ⋊ 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#20269515
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:

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