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C5×D3441 #46806191

order34410
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3441
unambiguous semidirect productC3⋊(C5×D1147)
unambiguous semidirect productC37⋊(C5×D93)

Splittings

Fitting, socle, HallC3×C5×C31×C37 ⋊ C2
derived, HallC3×C31×C37 ⋊ C2×C5
HallC5×C31×C37 ⋊ D3
HallC31×C37 ⋊ C5×D3
HallC3×C5×C37 ⋊ D31
HallC3×C5×C31 ⋊ D37
HallC5×C37 ⋊ D93
HallC5×C31 ⋊ D111
HallC3×C37 ⋊ C5×D31
HallC3×C31 ⋊ C5×D37
SylowC37 ⋊ C5×D93
SylowC31 ⋊ C5×D111
HallC3×C5 ⋊ D1147
SylowC3 ⋊ C5×D1147

Subgroups

subgroupG/H
center, hypercenterC5D3441
commutator, cosocle, hypocenterC3×C31×C37C2×C5
FrattiniC1G
Fitting, socleC3×C5×C31×C37C2
LCSG ▷ C3×C31×C37
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 17205

A minimal faithful permutation representation — 2 generators on 76 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);(7,8)(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)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Conjugacy classes & character table

8610 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34410, 21);