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C5×D31×D51 #44539357

order31620
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D31×D51

Splittings

Fitting, socle, HallC3×C5×C17×C31 ⋊ C22
derived, HallC3×C17×C31 ⋊ C22×C5
HallC5×C17×C31 ⋊ C2×D3
HallC17×C31 ⋊ C2×C5×D3
HallC3×C5×C31 ⋊ C2×D17
HallC3×C5×C17 ⋊ C2×D31
HallC5×C31 ⋊ C2×D51
HallC3×C31 ⋊ C2×C5×D17
HallC5×C17 ⋊ D3×D31
HallC3×C17 ⋊ C2×C5×D31
SylowC31 ⋊ C2×C5×D51
SylowC17 ⋊ C5×D3×D31
HallC3×C5 ⋊ D17×D31
SylowC3 ⋊ C5×D17×D31

Subgroups

subgroupG/H
center, hypercenterC5D31×D51
commutator, cosocle, hypocenterC3×C17×C31C22×C5
FrattiniC1G
Fitting, socleC3×C5×C17×C31C22
LCSG ▷ C3×C17×C31
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 7905

A minimal faithful permutation representation — 2 generators on 56 points:

(2,3)(4,5,6,7,8)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28);(1,2,3)(9,10,12,14,16,18,20,22,24,25,23,21,19,17,15,13,11)(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)

Conjugacy classes & character table

2295 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31620, 114);