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C5×D3009 #44473859

order30090
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3009
unambiguous semidirect productC3⋊(C5×D1003)
unambiguous semidirect productC17⋊(C5×D177)
unambiguous semidirect productC59⋊(C5×D51)

Splittings

Fitting, socle, HallC3×C5×C17×C59 ⋊ C2
derived, HallC3×C17×C59 ⋊ C2×C5
HallC5×C17×C59 ⋊ D3
HallC17×C59 ⋊ C5×D3
HallC3×C5×C59 ⋊ D17
HallC5×C59 ⋊ D51
HallC3×C5×C17 ⋊ D59
HallC3×C59 ⋊ C5×D17
HallC5×C17 ⋊ D177
SylowC59 ⋊ C5×D51
HallC3×C17 ⋊ C5×D59
SylowC17 ⋊ C5×D177
HallC3×C5 ⋊ D1003
SylowC3 ⋊ C5×D1003

Subgroups

subgroupG/H
center, hypercenterC5D3009
commutator, cosocle, hypocenterC3×C17×C59C2×C5
FrattiniC1G
Fitting, socleC3×C5×C17×C59C2
LCSG ▷ C3×C17×C59
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 84
transitive degreeμt(G) = 15045

A minimal faithful permutation representation — 2 generators on 84 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84);(7,8)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Conjugacy classes & character table

7530 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30090, 12);