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C3×C41×D55 #27214259

order13530
isoclinismisoclinic class of D55
automorphism?
homologyAbelianization (H1) = C2×C3×C41 · Schur multiplier (H2) = C1

Aliases

direct productC3×C41×D55
unambiguous semidirect productC5⋊(C3×D11×C41)
unambiguous semidirect productC11⋊(C3×D5×C41)

Splittings

Fitting, socle, HallC3×C5×C11×C41 ⋊ C2
derived, HallC5×C11 ⋊ C2×C3×C41
HallC5×C11×C41 ⋊ C2×C3
HallC3×C11×C41 ⋊ D5
HallC3×C5×C41 ⋊ D11
HallC11×C41 ⋊ C3×D5
HallC5×C41 ⋊ C3×D11
HallC3×C5×C11 ⋊ C2×C41
HallC3×C11 ⋊ D5×C41
HallC3×C5 ⋊ D11×C41
SylowC11 ⋊ C3×D5×C41
SylowC5 ⋊ C3×D11×C41

Subgroups

subgroupG/H
center, hypercenterC3×C41D55
commutator, cosocle, hypocenterC5×C11C2×C3×C41
FrattiniC1G
Fitting, socleC3×C5×C11×C41C2
LCSG ▷ C5×C11
UCSC1 ◁ C3×C41

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)(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);(1,2,4,6,8,10,11,9,7,5,3)(12,13,15,16,14)

Conjugacy classes & character table

3567 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×D41×D55, C3×D5×D11×C41, C3×D55⋊D41, …
maximal normal subgroup ofC3×D41×D55, C3×D5×D11×C41, C3×D55⋊D41, …
maximal quotient ofC3×C41×Dic55, C2×C3×C41×D55

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13530, 13);