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C3×D6355 #49185758

order38130
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D6355
unambiguous semidirect productC5⋊(C3×D1271)
unambiguous semidirect productC41⋊(C3×D155)

Splittings

Fitting, socle, HallC3×C5×C31×C41 ⋊ C2
derived, HallC5×C31×C41 ⋊ C2×C3
HallC3×C31×C41 ⋊ D5
HallC31×C41 ⋊ C3×D5
HallC3×C5×C41 ⋊ D31
HallC3×C5×C31 ⋊ D41
HallC5×C41 ⋊ C3×D31
HallC5×C31 ⋊ C3×D41
HallC3×C41 ⋊ D155
HallC3×C31 ⋊ D205
SylowC41 ⋊ C3×D155
SylowC31 ⋊ C3×D205
HallC3×C5 ⋊ D1271
SylowC5 ⋊ C3×D1271

Subgroups

subgroupG/H
center, hypercenterC3D6355
commutator, cosocle, hypocenterC5×C31×C41C2×C3
FrattiniC1G
Fitting, socleC3×C5×C31×C41C2
LCSG ▷ C5×C31×C41
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 19065

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2,3)(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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80);(5,8)(6,7)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Conjugacy classes & character table

9537 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38130, 21);