metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.4D8, D10.7SD16, (C2×D4)⋊1F5, C10.4C4≀C2, (D4×C10)⋊1C4, C4⋊Dic5⋊5C4, D10⋊C8⋊2C2, C20⋊2D4.1C2, C2.5(C23⋊F5), C2.6(D4⋊F5), D10.3Q8⋊2C2, (C22×D5).60D4, C5⋊2(C22.SD16), C10.14(C23⋊C4), C2.14(D20⋊C4), (C2×Dic5).105D4, C10.13(D4⋊C4), C22.59(C22⋊F5), (C2×C20).9(C2×C4), (C2×C4).14(C2×F5), (C2×C4×D5).3C22, (C2×C10).33(C22⋊C4), SmallGroup(320,258)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd=a5b, dcd=a4bc3 >
Subgroups: 490 in 90 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C2.C42, C22⋊C8, C4⋊D4, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C2×F5, C22×D5, C22×C10, C22.SD16, C4⋊Dic5, C23.D5, C2×C5⋊C8, C2×C4×D5, C2×C5⋊D4, D4×C10, C22×F5, D10⋊C8, D10.3Q8, C20⋊2D4, D10.SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, C23⋊C4, D4⋊C4, C4≀C2, C2×F5, C22.SD16, C22⋊F5, D20⋊C4, D4⋊F5, C23⋊F5, D10.SD16
Character table of D10.SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B | |
| size | 1 | 1 | 1 | 1 | 8 | 10 | 10 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | -i | i | i | -i | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -i | -i | i | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -i | i | i | -i | 1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | i | -i | -i | i | 1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | -1-i | 1-i | -1+i | 1+i | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | -1+i | 1+i | -1-i | 1-i | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√-2 | -√-2 | √-2 | √-2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √-2 | √-2 | -√-2 | -√-2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 1-i | -1-i | 1+i | -1+i | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 1+i | -1+i | 1-i | -1-i | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ19 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ20 | 4 | 4 | 4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -√5 | -√5 | √5 | √5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | √5 | √5 | -√5 | -√5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | -√5 | √5 | complex lifted from C23⋊F5 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | -√5 | √5 | complex lifted from C23⋊F5 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | √5 | -√5 | complex lifted from C23⋊F5 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | √5 | -√5 | complex lifted from C23⋊F5 |
| ρ28 | 8 | -8 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D20⋊C4, Schur index 2 |
| ρ29 | 8 | -8 | -8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊F5, Schur index 2 |
►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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 71)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)
(1 52 32 11 30 70 50 77)(2 59 31 14 21 67 49 80)(3 56 40 17 22 64 48 73)(4 53 39 20 23 61 47 76)(5 60 38 13 24 68 46 79)(6 57 37 16 25 65 45 72)(7 54 36 19 26 62 44 75)(8 51 35 12 27 69 43 78)(9 58 34 15 28 66 42 71)(10 55 33 18 29 63 41 74)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,71)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,65)(52,64)(53,63)(54,62)(55,61)(56,70)(57,69)(58,68)(59,67)(60,66), (1,52,32,11,30,70,50,77)(2,59,31,14,21,67,49,80)(3,56,40,17,22,64,48,73)(4,53,39,20,23,61,47,76)(5,60,38,13,24,68,46,79)(6,57,37,16,25,65,45,72)(7,54,36,19,26,62,44,75)(8,51,35,12,27,69,43,78)(9,58,34,15,28,66,42,71)(10,55,33,18,29,63,41,74), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,71)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,65)(52,64)(53,63)(54,62)(55,61)(56,70)(57,69)(58,68)(59,67)(60,66), (1,52,32,11,30,70,50,77)(2,59,31,14,21,67,49,80)(3,56,40,17,22,64,48,73)(4,53,39,20,23,61,47,76)(5,60,38,13,24,68,46,79)(6,57,37,16,25,65,45,72)(7,54,36,19,26,62,44,75)(8,51,35,12,27,69,43,78)(9,58,34,15,28,66,42,71)(10,55,33,18,29,63,41,74), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,71),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,65),(52,64),(53,63),(54,62),(55,61),(56,70),(57,69),(58,68),(59,67),(60,66)], [(1,52,32,11,30,70,50,77),(2,59,31,14,21,67,49,80),(3,56,40,17,22,64,48,73),(4,53,39,20,23,61,47,76),(5,60,38,13,24,68,46,79),(6,57,37,16,25,65,45,72),(7,54,36,19,26,62,44,75),(8,51,35,12,27,69,43,78),(9,58,34,15,28,66,42,71),(10,55,33,18,29,63,41,74)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,51),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,41),(18,42),(19,43),(20,44),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)]])
Matrix representation of D10.SD16 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 1 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 14 | 26 | 0 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 27 | 10 |
| 0 | 0 | 36 | 19 | 13 | 19 |
| 0 | 0 | 22 | 28 | 22 | 5 |
| 0 | 0 | 31 | 14 | 32 | 32 |
| 27 | 29 | 0 | 0 | 0 | 0 |
| 6 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 32 | 31 | 32 |
| 0 | 0 | 28 | 5 | 22 | 22 |
| 0 | 0 | 19 | 19 | 36 | 13 |
| 0 | 0 | 9 | 10 | 9 | 27 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40],[14,0,0,0,0,0,26,38,0,0,0,0,0,0,9,36,22,31,0,0,9,19,28,14,0,0,27,13,22,32,0,0,10,19,5,32],[27,6,0,0,0,0,29,14,0,0,0,0,0,0,14,28,19,9,0,0,32,5,19,10,0,0,31,22,36,9,0,0,32,22,13,27] >;
D10.SD16 in GAP, Magma, Sage, TeX
D_{10}.{\rm SD}_{16} % in TeX
G:=Group("D10.SD16"); // GroupNames label
G:=SmallGroup(320,258);
// by ID
G=gap.SmallGroup(320,258);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,136,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,d*b*d=a^5*b,d*c*d=a^4*b*c^3>;
// generators/relations
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