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C5×D3111 #44527442

order31110
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3111
unambiguous semidirect productC3⋊(C5×D1037)
unambiguous semidirect productC17⋊(C5×D183)

Splittings

Fitting, socle, HallC3×C5×C17×C61 ⋊ C2
derived, HallC3×C17×C61 ⋊ C2×C5
HallC5×C17×C61 ⋊ D3
HallC17×C61 ⋊ C5×D3
HallC3×C5×C61 ⋊ D17
HallC5×C61 ⋊ D51
HallC3×C5×C17 ⋊ D61
HallC3×C61 ⋊ C5×D17
HallC5×C17 ⋊ D183
SylowC61 ⋊ C5×D51
HallC3×C17 ⋊ C5×D61
SylowC17 ⋊ C5×D183
HallC3×C5 ⋊ D1037
SylowC3 ⋊ C5×D1037

Subgroups

subgroupG/H
center, hypercenterC5D3111
commutator, cosocle, hypocenterC3×C17×C61C2×C5
FrattiniC1G
Fitting, socleC3×C5×C17×C61C2
LCSG ▷ C3×C17×C61
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 86
transitive degreeμt(G) = 15555

A minimal faithful permutation representation — 2 generators on 86 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86);(7,8)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Conjugacy classes & character table

7785 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31110, 15);