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C3×D3355 #34117692

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

Aliases

direct productC3×D3355
unambiguous semidirect productC5⋊(C3×D671)
unambiguous semidirect productC11⋊(C3×D305)

Splittings

Fitting, socle, HallC3×C5×C11×C61 ⋊ C2
derived, HallC5×C11×C61 ⋊ C2×C3
HallC3×C11×C61 ⋊ D5
HallC3×C5×C61 ⋊ D11
HallC11×C61 ⋊ C3×D5
HallC5×C61 ⋊ C3×D11
HallC3×C61 ⋊ D55
HallC3×C5×C11 ⋊ D61
SylowC61 ⋊ C3×D55
HallC5×C11 ⋊ C3×D61
HallC3×C11 ⋊ D305
HallC3×C5 ⋊ D671
SylowC11 ⋊ C3×D305
SylowC5 ⋊ C3×D671

Subgroups

subgroupG/H
center, hypercenterC3D3355
commutator, cosocle, hypocenterC5×C11×C61C2×C3
FrattiniC1G
Fitting, socleC3×C5×C11×C61C2
LCSG ▷ C5×C11×C61
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 10065

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

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

Conjugacy classes & character table

5037 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 21);