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C3⋊#17984462 #27210576

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

Aliases

unambiguous semidirect productC3⋊#17984462

Splittings

Fitting, socle, derived, HallC3×C5×C13×C17 ⋊ C4
HallC5×C13×C17 ⋊ Dic3
HallC3×C13×C17 ⋊ AGL(1,𝔽5)
HallC3×C5×C17 ⋊ #261
HallC13×C17 ⋊ ΓL(1,𝔽16)
HallC3×C5×C13 ⋊ Dic17
HallC5×C17 ⋊ C3⋊#261
HallC5×C13 ⋊ Dic51
HallC3×C17 ⋊ #63121
HallC3×C13 ⋊ #66774
SylowC17 ⋊ C3⋊#63121
HallC3×C5 ⋊ #11725132
SylowC13 ⋊ C3⋊#66774
SylowC5 ⋊ C3⋊#11725132
SylowC3 ⋊ #17984462

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C13×C17C4
prime radical, cosocleC3×C17×D65C2
LCSG ▷ C3×C5×C13×C17
UCSC1

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

Polycyclic presentation:

2^2,3,5,13,17|a^2=b,c^a=c^2,d^a=d^2,e^a=e^8,f^a=f^16,d^b=d^4,e^b=e^12

Conjugacy classes & character table

870 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13260, 126);