Finite Group DatabasebetaFinite Group Databasebeta| order | 31710 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C7⋊(D151⋊C5)) |
| unambiguous semidirect product | C7⋊(C3⋊(D151⋊C5)) |
| unambiguous semidirect product | (C151⋊C5)⋊D21 |
| unambiguous semidirect product | D3171⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C151 | C2×C5 |
| LCS | G ▷ C3×C7×C151 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 3171 |
A minimal faithful permutation representation — 2 generators on 161 points:
(2,3)(5,6)(7,8)(9,10)(12,16,17,35,40,13,14,22,24,63)(15,29,30,94,98,19,20,51,53,134)(18,45,46,47,74,26,27,84,85,87)(21,58,59,39,112,32,33,91,71,72)(23,68,69,126,93,37,38,83,115,110)(25,78,79,75,76,42,43,80,41,86)(28,89,90,152,139,48,49,131,132,121)(31,103,104,34,107,55,56,143,60,61)(36,111,73,64,62,65,66,118,102,108)(44,124,123,140,133,81,82,154,99,159)(50,96,97,146,147,92,52,137,106,161)(54,141,95,120,105,100,101,136,77,57)(67,70,149,135,88,109,113,116,160,129)(114,155,117,138,148,150,130,142,151,128)(119,122,156,157,158,153,144,125,127,145);(1,2,3)(4,5,7,9,10,8,6)(11,12,15,21,34,62,109,65,17,25,44,83,127,156,124,136,30,54,102,24,41,77,121,134,59,89,33,46,86,128,146,105,55,29,52,90,133,160,53,99,66,112,147,97,138,126,104,145,116,152,150,38,70,117,141,153,118,71,37,14,18,28,50,93,40,74,107,27,47,88,130,154,79,78,122,56,103,144,43,80,123,155,129,85,45,61,87,63,110,92,48,26,16,23,39,73,119,101,142,113,68,114,132,149,158,143,115,151,137,161,72,111,140,94,135,159,131,96,20,31,57,106,148,76,84,58,49,91,98,139,120,75,35,64,100,51,95,82,125,157,69,81,42,22,36,67,108,60,32,19,13)Polycyclic presentation:
2,3,5,7,151|b^a=b^2,d^a=d^6,e^a=e^150,e^c=e^59375 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31710, 32);