metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:4D10, SD16:2D10, C40.1C23, M4(2):8D10, C20.20C24, Dic20:1C22, D20.13C23, Dic10.13C23, C8:C22:6D5, (C2xD4):30D10, D8:3D5:1C2, D8:D5:2C2, D4:D5:6C22, (C4xD5).99D4, C4.190(D4xD5), (D4xD5):9C22, (C5xD8):2C22, (C8xD5):2C22, C8.1(C22xD5), C4oD4.28D10, D10.88(C2xD4), SD16:D5:1C2, C20.241(C2xD4), C8.D10:1C2, C40:C2:2C22, C8:D5:2C22, (D5xM4(2)):2C2, D4.D5:5C22, (Q8xD5):10C22, C5:Q16:3C22, C4.20(C23xD5), C22.47(D4xD5), SD16:3D5:1C2, D4.9D10:9C2, (D4xC10):22C22, C5:3(D8:C22), C5:2C8.10C23, (C5xSD16):2C22, (C22xD5).51D4, D4.13(C22xD5), (C5xD4).13C23, (C4xD5).66C23, D4.D10:10C2, Q8.13(C22xD5), (C5xQ8).13C23, D4:2D5:10C22, (C2xC20).111C23, (C2xDic5).251D4, Dic5.100(C2xD4), Q8:2D5:10C22, C4oD20.28C22, C10.121(C22xD4), (C5xM4(2)):2C22, C4.Dic5:13C22, (C2xDic10):39C22, C2.94(C2xD4xD5), (D5xC4oD4):4C2, (C5xC8:C22):2C2, (C2xC10).66(C2xD4), (C2xD4:2D5):26C2, (C2xC4xD5).170C22, (C2xC4).95(C22xD5), (C5xC4oD4).24C22, SmallGroup(320,1445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16:D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a3, cac-1=a-1, cbc-1=a6b, dbd=a2b, dcd=c-1 >
Subgroups: 974 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2xC8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xM4(2), C4oD8, C8:C22, C8:C22, C8.C22, C2xC4oD4, C5:2C8, C40, Dic10, Dic10, Dic10, C4xD5, C4xD5, D20, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xD4, C5xQ8, C22xD5, C22xD5, C22xC10, D8:C22, C8xD5, C8:D5, C40:C2, Dic20, C4.Dic5, D4:D5, D4.D5, C5:Q16, C5xM4(2), C5xD8, C5xSD16, C2xDic10, C2xC4xD5, C2xC4xD5, C4oD20, C4oD20, D4xD5, D4xD5, D4:2D5, D4:2D5, D4:2D5, Q8xD5, Q8:2D5, C22xDic5, C2xC5:D4, D4xC10, C5xC4oD4, D5xM4(2), C8.D10, D8:D5, D8:3D5, SD16:D5, SD16:3D5, D4.D10, D4.9D10, C5xC8:C22, C2xD4:2D5, D5xC4oD4, SD16:D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, C22xD5, D8:C22, D4xD5, C23xD5, C2xD4xD5, SD16:D10
►On 80 points
(1 33 74 56 8 51 79 38)(2 39 80 52 9 57 75 34)(3 35 76 58 10 53 71 40)(4 31 72 54 6 59 77 36)(5 37 78 60 7 55 73 32)(11 64 47 23 17 28 42 69)(12 70 43 29 18 24 48 65)(13 66 49 25 19 30 44 61)(14 62 45 21 20 26 50 67)(15 68 41 27 16 22 46 63)
(1 65)(2 61)(3 67)(4 63)(5 69)(6 27)(7 23)(8 29)(9 25)(10 21)(11 60)(12 56)(13 52)(14 58)(15 54)(16 36)(17 32)(18 38)(19 34)(20 40)(22 72)(24 74)(26 76)(28 78)(30 80)(31 41)(33 43)(35 45)(37 47)(39 49)(42 55)(44 57)(46 59)(48 51)(50 53)(62 71)(64 73)(66 75)(68 77)(70 79)
(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)
(1 5)(2 4)(6 9)(7 8)(11 12)(13 15)(16 19)(17 18)(21 62)(22 61)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 52)(32 51)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 44)(42 43)(45 50)(46 49)(47 48)(71 76)(72 75)(73 74)(77 80)(78 79)
G:=sub<Sym(80)| (1,33,74,56,8,51,79,38)(2,39,80,52,9,57,75,34)(3,35,76,58,10,53,71,40)(4,31,72,54,6,59,77,36)(5,37,78,60,7,55,73,32)(11,64,47,23,17,28,42,69)(12,70,43,29,18,24,48,65)(13,66,49,25,19,30,44,61)(14,62,45,21,20,26,50,67)(15,68,41,27,16,22,46,63), (1,65)(2,61)(3,67)(4,63)(5,69)(6,27)(7,23)(8,29)(9,25)(10,21)(11,60)(12,56)(13,52)(14,58)(15,54)(16,36)(17,32)(18,38)(19,34)(20,40)(22,72)(24,74)(26,76)(28,78)(30,80)(31,41)(33,43)(35,45)(37,47)(39,49)(42,55)(44,57)(46,59)(48,51)(50,53)(62,71)(64,73)(66,75)(68,77)(70,79), (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), (1,5)(2,4)(6,9)(7,8)(11,12)(13,15)(16,19)(17,18)(21,62)(22,61)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,52)(32,51)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,44)(42,43)(45,50)(46,49)(47,48)(71,76)(72,75)(73,74)(77,80)(78,79)>;
G:=Group( (1,33,74,56,8,51,79,38)(2,39,80,52,9,57,75,34)(3,35,76,58,10,53,71,40)(4,31,72,54,6,59,77,36)(5,37,78,60,7,55,73,32)(11,64,47,23,17,28,42,69)(12,70,43,29,18,24,48,65)(13,66,49,25,19,30,44,61)(14,62,45,21,20,26,50,67)(15,68,41,27,16,22,46,63), (1,65)(2,61)(3,67)(4,63)(5,69)(6,27)(7,23)(8,29)(9,25)(10,21)(11,60)(12,56)(13,52)(14,58)(15,54)(16,36)(17,32)(18,38)(19,34)(20,40)(22,72)(24,74)(26,76)(28,78)(30,80)(31,41)(33,43)(35,45)(37,47)(39,49)(42,55)(44,57)(46,59)(48,51)(50,53)(62,71)(64,73)(66,75)(68,77)(70,79), (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), (1,5)(2,4)(6,9)(7,8)(11,12)(13,15)(16,19)(17,18)(21,62)(22,61)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,52)(32,51)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,44)(42,43)(45,50)(46,49)(47,48)(71,76)(72,75)(73,74)(77,80)(78,79) );
G=PermutationGroup([[(1,33,74,56,8,51,79,38),(2,39,80,52,9,57,75,34),(3,35,76,58,10,53,71,40),(4,31,72,54,6,59,77,36),(5,37,78,60,7,55,73,32),(11,64,47,23,17,28,42,69),(12,70,43,29,18,24,48,65),(13,66,49,25,19,30,44,61),(14,62,45,21,20,26,50,67),(15,68,41,27,16,22,46,63)], [(1,65),(2,61),(3,67),(4,63),(5,69),(6,27),(7,23),(8,29),(9,25),(10,21),(11,60),(12,56),(13,52),(14,58),(15,54),(16,36),(17,32),(18,38),(19,34),(20,40),(22,72),(24,74),(26,76),(28,78),(30,80),(31,41),(33,43),(35,45),(37,47),(39,49),(42,55),(44,57),(46,59),(48,51),(50,53),(62,71),(64,73),(66,75),(68,77),(70,79)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,12),(13,15),(16,19),(17,18),(21,62),(22,61),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,52),(32,51),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,44),(42,43),(45,50),(46,49),(47,48),(71,76),(72,75),(73,74),(77,80),(78,79)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 5 | 5 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D8:C22 | D4xD5 | D4xD5 | SD16:D10 |
| kernel | SD16:D10 | D5xM4(2) | C8.D10 | D8:D5 | D8:3D5 | SD16:D5 | SD16:3D5 | D4.D10 | D4.9D10 | C5xC8:C22 | C2xD4:2D5 | D5xC4oD4 | C4xD5 | C2xDic5 | C22xD5 | C8:C22 | M4(2) | D8 | SD16 | C2xD4 | C4oD4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of SD16:D10 ►in GL8(F41)
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 7 | 1 | 2 | 7 | 0 | 0 | 0 | 0 |
| 3 | 18 | 40 | 0 | 0 | 0 | 0 | 0 |
| 4 | 18 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 7 | 1 | 2 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 6 | 34 | 34 | 0 | 0 | 0 | 0 |
| 16 | 13 | 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 35 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 6 | 34 | 34 | 0 | 0 | 0 | 0 |
| 16 | 13 | 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(8,GF(41))| [0,7,3,4,0,0,0,0,0,1,18,18,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,7,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0],[0,35,8,16,0,0,0,0,7,6,6,13,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[35,36,8,16,0,0,0,0,7,6,6,13,0,0,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0] >;
SD16:D10 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes D_{10} % in TeX
G:=Group("SD16:D10"); // GroupNames label
G:=SmallGroup(320,1445);
// by ID
G=gap.SmallGroup(320,1445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=d*a*d=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations