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C13×C17×ΓL(1,𝔽16) #27210565

order13260
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C13×C17 · Schur multiplier (H2) = C1

Aliases

direct productC13×C17×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C13×C17×AGL(1,𝔽5))

Splittings

Fitting, socle, HallC3×C5×C13×C17 ⋊ C4
derived, HallC3×C5 ⋊ C4×C13×C17
HallC5×C13×C17 ⋊ Dic3
HallC3×C13×C17 ⋊ AGL(1,𝔽5)
HallC3×C5×C17 ⋊ C4×C13
HallC3×C5×C13 ⋊ C4×C17
HallC5×C17 ⋊ Dic3×C13
HallC5×C13 ⋊ Dic3×C17
HallC3×C17 ⋊ C13×AGL(1,𝔽5)
HallC3×C13 ⋊ C17×AGL(1,𝔽5)
SylowC5 ⋊ Dic3×C13×C17
SylowC3 ⋊ C13×C17×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC13×C17ΓL(1,𝔽16)
commutator, hypocenterC3×C5C4×C13×C17
FrattiniC1G
Fitting, socleC3×C5×C13×C17C4
prime radicalC3×D5×C13×C17C2
cosocleC3×D5C2×C13×C17
LCSG ▷ C3×C5
UCSC1 ◁ C13×C17

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)(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);(1,2,5,3,4)(6,7,8)

Conjugacy classes & character table

1989 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13260, 115);