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C3×#20269520 #34117709

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

Aliases

direct productC3×#20269520

Splittings

Fitting, socle, HallC3×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C2×C3×C5
HallC3×#16809824 ⋊ C2
HallC3×D11×C61 ⋊ C5
Hall#16809824 ⋊ C2×C3
HallD11×C61 ⋊ C3×C5
HallC3×C61 ⋊ AGL(1,𝔽11)
HallC3×D11 ⋊ C61⋊C5
SylowC61 ⋊ C3×AGL(1,𝔽11)
HallC3×C11 ⋊ C2×C61⋊C5
HallD11 ⋊ C3×C61⋊C5
SylowC11 ⋊ C2×C3×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC3#20269520
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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 38);