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C5×C3⋊#11725130 #27210556

order13260
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊#11725130
unambiguous semidirect productC3⋊(C5×#11725130)

Splittings

Fitting, socle, HallC3×C5×C13×C17 ⋊ C4
derived, HallC3×C13×C17 ⋊ C4×C5
HallC5×C13×C17 ⋊ Dic3
HallC3×C5×C17 ⋊ #261
HallC13×C17 ⋊ C5×Dic3
HallC3×C5×C13 ⋊ #595
HallC5×C17 ⋊ C3⋊#261
HallC5×C13 ⋊ C3⋊#595
HallC3×C17 ⋊ C5×#261
HallC3×C13 ⋊ C5×#595
SylowC17 ⋊ C5×C3⋊#261
HallC3×C5 ⋊ #11725130
SylowC13 ⋊ C5×C3⋊#595
SylowC3 ⋊ C5×#11725130

Subgroups

subgroupG/H
center, hypercenterC5C3⋊#11725130
commutator, hypocenterC3×C13×C17C4×C5
FrattiniC1G
Fitting, socleC3×C5×C13×C17C4
prime radicalC3×C5×D221C2
cosocleC3×D221C2×C5
LCSG ▷ C3×C13×C17
UCSC1 ◁ C5

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 3315

A minimal faithful permutation representation — 2 generators on 38 points:

(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(19,20,21,23)(22,25,26,27)(24,29,30,28)(32,33)(34,35,36,37,38);(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)(18,19,22,27,29,20,24,30,23,28,25,26,21)(31,32,33)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13260, 106);