metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16:2F5, C5:C8.2D4, D4:D5:4C4, C8.7(C2xF5), C40:C2:2C4, C8:F5:5C2, C5:Q16:2C4, D4.F5:4C2, Q8.F5:2C2, D4.6(C2xF5), C2.22(D4xF5), Q8.2(C2xF5), C5:2(C8.26D4), C40.15(C2xC4), D4:F5:4C2, Q8:2F5:2C2, (C5xSD16):2C4, D20.4(C2xC4), C10.21(C4xD4), D10.Q8:5C2, C4.8(C22xF5), C20.8(C22xC4), D5:C8.4C22, (C4xF5).4C22, D10.4(C4oD4), C4.F5.4C22, Dic10.4(C2xC4), Dic5.75(C2xD4), (C4xD5).30C23, (C8xD5).26C22, SD16:3D5.1C2, D4:2D5.7C22, Q8:2D5.5C22, (C5xD4).6(C2xC4), (C5xQ8).2(C2xC4), C5:2C8.10(C2xC4), SmallGroup(320,1075)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16:2F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, dad-1=a5, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 402 in 104 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, D4, D4, Q8, Q8, D5, C10, C10, C42, C2xC8, M4(2), D8, SD16, SD16, Q16, C4oD4, Dic5, Dic5, C20, C20, F5, D10, D10, C2xC10, C8:C4, C4wrC2, C8.C4, C8oD4, C4oD8, C5:2C8, C40, C5:C8, C5:C8, Dic10, C4xD5, C4xD5, D20, D20, C2xDic5, C5:D4, C5xD4, C5xQ8, C2xF5, C8.26D4, C8xD5, C40:C2, D4:D5, C5:Q16, C5xSD16, D5:C8, D5:C8, C4.F5, C4.F5, C4xF5, C2xC5:C8, C22.F5, D4:2D5, Q8:2D5, C8:F5, D10.Q8, D4:F5, Q8:2F5, SD16:3D5, D4.F5, Q8.F5, SD16:2F5
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, F5, C4xD4, C2xF5, C8.26D4, C22xF5, D4xF5, SD16:2F5
Character table of SD16:2F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 20A | 20B | 40A | 40B | |
| size | 1 | 1 | 4 | 10 | 20 | 2 | 4 | 5 | 5 | 20 | 20 | 20 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 4 | 16 | 8 | 16 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -1 | 1 | 1 | -i | i | i | -i | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | -1 | 1 | -1 | i | -i | -i | i | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | -1 | 1 | 1 | i | -i | -i | i | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | -1 | 1 | -1 | -i | i | i | -i | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | 1 | 1 | 1 | i | -i | -i | i | -1 | -i | i | -i | i | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | 1 | 1 | -1 | -i | i | i | -i | 1 | -i | i | i | -i | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | 1 | 1 | 1 | -i | i | i | -i | -1 | i | -i | i | -i | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | 1 | 1 | -1 | i | -i | -i | i | 1 | i | -i | -i | i | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | 2 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2xF5 |
| ρ22 | 4 | 4 | -4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | -1 | orthogonal lifted from C2xF5 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from C2xF5 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.26D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.26D4 |
| ρ27 | 8 | 8 | 0 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4xF5 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -√-10 | √-10 | complex faithful |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | √-10 | -√-10 | complex faithful |
►On 80 points
Generators in S80
(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 59)(2 62)(3 57)(4 60)(5 63)(6 58)(7 61)(8 64)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 46)(18 41)(19 44)(20 47)(21 42)(22 45)(23 48)(24 43)(33 75)(34 78)(35 73)(36 76)(37 79)(38 74)(39 77)(40 80)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)
(1 12 43 79 71)(2 13 44 80 72)(3 14 45 73 65)(4 15 46 74 66)(5 16 47 75 67)(6 9 48 76 68)(7 10 41 77 69)(8 11 42 78 70)(17 38 56 60 30)(18 39 49 61 31)(19 40 50 62 32)(20 33 51 63 25)(21 34 52 64 26)(22 35 53 57 27)(23 36 54 58 28)(24 37 55 59 29)
(2 6)(4 8)(9 44 68 80)(10 41 69 77)(11 46 70 74)(12 43 71 79)(13 48 72 76)(14 45 65 73)(15 42 66 78)(16 47 67 75)(17 54 34 32)(18 51 35 29)(19 56 36 26)(20 53 37 31)(21 50 38 28)(22 55 39 25)(23 52 40 30)(24 49 33 27)(57 59 61 63)(58 64 62 60)
G:=sub<Sym(80)| (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), (1,59)(2,62)(3,57)(4,60)(5,63)(6,58)(7,61)(8,64)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(33,75)(34,78)(35,73)(36,76)(37,79)(38,74)(39,77)(40,80)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,12,43,79,71)(2,13,44,80,72)(3,14,45,73,65)(4,15,46,74,66)(5,16,47,75,67)(6,9,48,76,68)(7,10,41,77,69)(8,11,42,78,70)(17,38,56,60,30)(18,39,49,61,31)(19,40,50,62,32)(20,33,51,63,25)(21,34,52,64,26)(22,35,53,57,27)(23,36,54,58,28)(24,37,55,59,29), (2,6)(4,8)(9,44,68,80)(10,41,69,77)(11,46,70,74)(12,43,71,79)(13,48,72,76)(14,45,65,73)(15,42,66,78)(16,47,67,75)(17,54,34,32)(18,51,35,29)(19,56,36,26)(20,53,37,31)(21,50,38,28)(22,55,39,25)(23,52,40,30)(24,49,33,27)(57,59,61,63)(58,64,62,60)>;
G:=Group( (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), (1,59)(2,62)(3,57)(4,60)(5,63)(6,58)(7,61)(8,64)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(33,75)(34,78)(35,73)(36,76)(37,79)(38,74)(39,77)(40,80)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,12,43,79,71)(2,13,44,80,72)(3,14,45,73,65)(4,15,46,74,66)(5,16,47,75,67)(6,9,48,76,68)(7,10,41,77,69)(8,11,42,78,70)(17,38,56,60,30)(18,39,49,61,31)(19,40,50,62,32)(20,33,51,63,25)(21,34,52,64,26)(22,35,53,57,27)(23,36,54,58,28)(24,37,55,59,29), (2,6)(4,8)(9,44,68,80)(10,41,69,77)(11,46,70,74)(12,43,71,79)(13,48,72,76)(14,45,65,73)(15,42,66,78)(16,47,67,75)(17,54,34,32)(18,51,35,29)(19,56,36,26)(20,53,37,31)(21,50,38,28)(22,55,39,25)(23,52,40,30)(24,49,33,27)(57,59,61,63)(58,64,62,60) );
G=PermutationGroup([[(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)], [(1,59),(2,62),(3,57),(4,60),(5,63),(6,58),(7,61),(8,64),(9,28),(10,31),(11,26),(12,29),(13,32),(14,27),(15,30),(16,25),(17,46),(18,41),(19,44),(20,47),(21,42),(22,45),(23,48),(24,43),(33,75),(34,78),(35,73),(36,76),(37,79),(38,74),(39,77),(40,80),(49,69),(50,72),(51,67),(52,70),(53,65),(54,68),(55,71),(56,66)], [(1,12,43,79,71),(2,13,44,80,72),(3,14,45,73,65),(4,15,46,74,66),(5,16,47,75,67),(6,9,48,76,68),(7,10,41,77,69),(8,11,42,78,70),(17,38,56,60,30),(18,39,49,61,31),(19,40,50,62,32),(20,33,51,63,25),(21,34,52,64,26),(22,35,53,57,27),(23,36,54,58,28),(24,37,55,59,29)], [(2,6),(4,8),(9,44,68,80),(10,41,69,77),(11,46,70,74),(12,43,71,79),(13,48,72,76),(14,45,65,73),(15,42,66,78),(16,47,67,75),(17,54,34,32),(18,51,35,29),(19,56,36,26),(20,53,37,31),(21,50,38,28),(22,55,39,25),(23,52,40,30),(24,49,33,27),(57,59,61,63),(58,64,62,60)]])
Matrix representation of SD16:2F5 ►in GL8(F41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 37 |
| 0 | 0 | 0 | 0 | 38 | 0 | 32 | 20 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 6 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 39 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 32 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 19 | 1 | 23 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,2,38,2,9,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,37,20,6,39],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,3,32,19,0,0,0,0,0,0,0,1,0,0,0,0,5,16,13,23,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,40,26,1,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40] >;
SD16:2F5 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_2F_5 % in TeX
G:=Group("SD16:2F5"); // GroupNames label
G:=SmallGroup(320,1075);
// by ID
G=gap.SmallGroup(320,1075);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,136,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
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