Finite Group DatabasebetaFinite Group Databasebeta| order | 31710 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20263631 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C7×C151 | C2×C3 |
| LCS | G ▷ C5×C7×C151 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 5285 |
A minimal faithful permutation representation — 2 generators on 163 points:
(2,3)(4,5)(7,10,11,8,9,12)(14,18,19,15,16,20)(17,26,27,21,22,36)(23,39,40,29,30,53)(24,38,34,32,28,25)(31,58,59,42,43,76)(33,56,57,45,41,50)(35,37,55,47,48,51)(44,81,82,61,62,105)(46,79,80,64,60,73)(49,54,78,66,67,74)(52,94,92,70,71,69)(63,110,111,84,85,138)(65,108,109,87,83,102)(68,77,107,89,90,103)(72,122,123,95,96,98)(75,128,120,99,100,93)(86,143,144,113,114,162)(88,141,142,116,112,136)(91,106,140,118,115,137)(97,149,150,124,125,132)(101,154,152,129,130,127)(104,157,148,133,134,121)(117,158,153,135,145,160)(119,139,163,146,147,161)(126,155,156,131,151,159);(1,2,4,5,3)(6,7,9,12,11,10,8)(13,14,17,23,31,44,63,86,115,62,83,96,100,103,105,136,159,150,109,59,78,94,79,58,77,104,135,116,87,64,45,32,16,20,25,35,49,68,91,119,147,85,112,125,130,134,137,138,160,152,123,80,40,55,56,39,54,75,101,131,124,95,70,48,22,28,36,50,69,93,121,144,163,114,145,151,155,158,143,161,162,148,120,92,57,27,38,26,37,52,72,97,126,129,99,67,30,41,51,53,73,98,127,153,111,140,157,154,149,141,110,139,146,118,89,66,47,34,19,18,24,33,46,65,88,117,133,90,43,60,71,74,76,102,132,156,142,82,107,128,122,108,81,106,113,84,61,42,29,21,15)Polycyclic presentation:
2,3,5,7,151|c^a=c^4,d^a=d^6,e^a=e^150,d^b=d^2,e^b=e^32892 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31710, 59);