Finite Group DatabasebetaFinite Group Databasebeta| order | 31650 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C5⋊(C5⋊(D211⋊C3)) |
| unambiguous semidirect product | (C211⋊C3)⋊(C5⋊D5) |
| Fitting, socle, derived, Hall | C52×C211 ⋊ C2×C3 |
| Hall | C52×C211⋊C3 ⋊ C2 |
| Hall | C5⋊D1055 ⋊ C3 |
| Hall | C211⋊C3 ⋊ C5⋊D5 |
| Sylow | C211 ⋊ C3×C5⋊D5 |
| Sylow | C52 ⋊ D211⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C211 | C2×C3 |
| LCS | G ▷ C52×C211 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 5275 |
A minimal faithful permutation representation — 3 generators on 221 points:
(2,3)(4,5)(7,8)(9,10)(12,16,17,13,14,18)(15,24,25,19,20,34)(21,37,38,27,28,51)(22,36,32,30,26,23)(29,56,57,40,41,74)(31,54,55,43,39,48)(33,35,53,45,46,49)(42,79,80,59,60,103)(44,77,78,62,58,71)(47,52,76,64,65,72)(50,92,90,68,69,67)(61,108,109,82,83,138)(63,106,107,85,81,100)(66,75,105,87,88,101)(70,121,122,93,94,96)(73,127,119,97,98,91)(84,143,144,111,112,175)(86,141,142,114,110,135)(89,104,140,116,117,136)(95,152,153,123,124,131)(99,159,157,128,129,126)(102,164,150,132,133,120)(113,145,172,115,146,178)(118,139,177,147,148,173)(125,182,183,154,155,168)(130,186,187,160,161,163)(134,192,184,165,166,158)(137,197,180,169,170,151)(149,176,201,156,179,205)(162,208,209,188,189,196)(167,212,210,193,194,191)(171,214,206,198,199,185)(174,218,190,202,203,181)(195,219,220,207,204,213)(200,221,217,215,216,211);(6,7,9,10,8)(11,12,15,21,29,42,61,84,113,114,85,62,43,30,14,18,23,33,47,66,89,118,149,154,123,93,68,46,20,26,34,48,67,91,120,151,181,188,160,128,97,65,28,39,49,51,71,96,126,158,185,207,193,165,132,88,41,58,69,72,74,100,131,163,191,211,215,198,169,117,60,81,94,98,101,103,135,168,196,213,221,219,202,148,83,110,124,129,133,136,138,172,201,218,214,212,208,179,112,145,155,161,166,170,173,175,144,177,197,192,186,182,146,143,176,189,194,199,203,205,178,109,140,164,159,152,141,108,139,174,204,216,220,209,183,142,80,105,127,121,106,79,104,137,171,200,217,210,187,153,107,57,76,92,77,56,75,102,134,167,195,206,184,157,122,78,38,53,54,37,52,73,99,130,162,190,180,150,119,90,55,25,36,24,35,50,70,95,125,156,147,116,87,64,45,32,17,16,22,31,44,63,86,115,111,82,59,40,27,19,13);(1,2,4,5,3)Polycyclic presentation:
2,3,5^2,211|c^a=c^4,d^a=d^4,e^a=e^210,e^b=e^14917 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31650, 42);