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C3×D2255 #27214260

order13530
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D2255
unambiguous semidirect productC5⋊(C3×D451)
unambiguous semidirect productC11⋊(C3×D205)
unambiguous semidirect productC41⋊(C3×D55)

Splittings

Fitting, socle, HallC3×C5×C11×C41 ⋊ C2
derived, HallC5×C11×C41 ⋊ C2×C3
HallC3×C11×C41 ⋊ D5
HallC3×C5×C41 ⋊ D11
HallC11×C41 ⋊ C3×D5
HallC5×C41 ⋊ C3×D11
HallC3×C5×C11 ⋊ D41
HallC3×C41 ⋊ D55
HallC5×C11 ⋊ C3×D41
SylowC41 ⋊ C3×D55
HallC3×C11 ⋊ D205
HallC3×C5 ⋊ D451
SylowC11 ⋊ C3×D205
SylowC5 ⋊ C3×D451

Subgroups

subgroupG/H
center, hypercenterC3D2255
commutator, cosocle, hypocenterC5×C11×C41C2×C3
FrattiniC1G
Fitting, socleC3×C5×C11×C41C2
LCSG ▷ C5×C11×C41
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 6765

A minimal faithful permutation representation — 2 generators on 60 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)(43,44)(45,46)(47,48)(49,50)(51,52)(54,55)(56,57)(58,59,60);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(42,43,45,47,49,51,52,50,48,46,44)(53,54,56,57,55)

Conjugacy classes & character table

3387 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×D41×D55, C3×D11×D205, C3×D5×D451, …
maximal normal subgroup ofC3×D41×D55, C3×D11×D205, C3×D5×D451, …
maximal quotient ofC3×Dic2255, C2×C3×D2255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13530, 14);