metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.2Q16, D10.8SD16, (C2×Q8)⋊1F5, C4⋊Dic5⋊6C4, (Q8×C10)⋊1C4, C10.10C4≀C2, C2.8(C23⋊F5), D10⋊C8.2C2, D10⋊3Q8.1C2, C2.6(Q8⋊F5), C2.6(Q8⋊2F5), (C22×D5).61D4, C10.17(C23⋊C4), C10.5(Q8⋊C4), (C2×Dic5).109D4, C5⋊2(C23.31D4), D10.3Q8.2C2, C22.62(C22⋊F5), (C2×C4).17(C2×F5), (C2×C4×D5).4C22, (C2×C20).14(C2×C4), (C2×C10).39(C22⋊C4), SmallGroup(320,264)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd-1=a5b, dcd-1=a4bc-1 >
Subgroups: 410 in 80 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic5, C20, F5, D10, D10, C2×C10, C2.C42, C22⋊C8, C22⋊Q8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×F5, C22×D5, C23.31D4, C10.D4, C4⋊Dic5, D10⋊C4, C2×C5⋊C8, C2×C4×D5, Q8×C10, C22×F5, D10⋊C8, D10.3Q8, D10⋊3Q8, D10.Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, F5, C23⋊C4, Q8⋊C4, C4≀C2, C2×F5, C23.31D4, C22⋊F5, Q8⋊F5, Q8⋊2F5, C23⋊F5, D10.Q16
Character table of D10.Q16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 4 | 8 | 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 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | -√2 | √2 | √2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | √2 | -√2 | -√2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| 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 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 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 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 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 |
| ρ19 | 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 |
| ρ20 | 4 | 4 | 4 | 4 | 0 | 0 | 4 | 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 |
| ρ21 | 4 | 4 | 4 | 4 | 0 | 0 | 4 | -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 |
| ρ22 | 4 | 4 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | √5 | 1 | 1 | √5 | -√5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -√5 | 1 | 1 | -√5 | √5 | √5 | 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 | 2ζ53+2ζ5+1 | √5 | -√5 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 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+2ζ52+1 | √5 | -√5 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 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 | 2ζ52+2ζ5+1 | -√5 | √5 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 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 | 2ζ54+2ζ53+1 | -√5 | √5 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 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 Q8⋊2F5 |
| 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 Q8⋊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 75)(12 74)(13 73)(14 72)(15 71)(16 80)(17 79)(18 78)(19 77)(20 76)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 70)
(1 70 50 11 30 56 36 71)(2 67 49 14 21 53 35 74)(3 64 48 17 22 60 34 77)(4 61 47 20 23 57 33 80)(5 68 46 13 24 54 32 73)(6 65 45 16 25 51 31 76)(7 62 44 19 26 58 40 79)(8 69 43 12 27 55 39 72)(9 66 42 15 28 52 38 75)(10 63 41 18 29 59 37 78)
(1 70 30 56)(2 61 21 57)(3 62 22 58)(4 63 23 59)(5 64 24 60)(6 65 25 51)(7 66 26 52)(8 67 27 53)(9 68 28 54)(10 69 29 55)(11 45 71 31)(12 46 72 32)(13 47 73 33)(14 48 74 34)(15 49 75 35)(16 50 76 36)(17 41 77 37)(18 42 78 38)(19 43 79 39)(20 44 80 40)
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)(11,75)(12,74)(13,73)(14,72)(15,71)(16,80)(17,79)(18,78)(19,77)(20,76)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,70), (1,70,50,11,30,56,36,71)(2,67,49,14,21,53,35,74)(3,64,48,17,22,60,34,77)(4,61,47,20,23,57,33,80)(5,68,46,13,24,54,32,73)(6,65,45,16,25,51,31,76)(7,62,44,19,26,58,40,79)(8,69,43,12,27,55,39,72)(9,66,42,15,28,52,38,75)(10,63,41,18,29,59,37,78), (1,70,30,56)(2,61,21,57)(3,62,22,58)(4,63,23,59)(5,64,24,60)(6,65,25,51)(7,66,26,52)(8,67,27,53)(9,68,28,54)(10,69,29,55)(11,45,71,31)(12,46,72,32)(13,47,73,33)(14,48,74,34)(15,49,75,35)(16,50,76,36)(17,41,77,37)(18,42,78,38)(19,43,79,39)(20,44,80,40)>;
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)(11,75)(12,74)(13,73)(14,72)(15,71)(16,80)(17,79)(18,78)(19,77)(20,76)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,70), (1,70,50,11,30,56,36,71)(2,67,49,14,21,53,35,74)(3,64,48,17,22,60,34,77)(4,61,47,20,23,57,33,80)(5,68,46,13,24,54,32,73)(6,65,45,16,25,51,31,76)(7,62,44,19,26,58,40,79)(8,69,43,12,27,55,39,72)(9,66,42,15,28,52,38,75)(10,63,41,18,29,59,37,78), (1,70,30,56)(2,61,21,57)(3,62,22,58)(4,63,23,59)(5,64,24,60)(6,65,25,51)(7,66,26,52)(8,67,27,53)(9,68,28,54)(10,69,29,55)(11,45,71,31)(12,46,72,32)(13,47,73,33)(14,48,74,34)(15,49,75,35)(16,50,76,36)(17,41,77,37)(18,42,78,38)(19,43,79,39)(20,44,80,40) );
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),(11,75),(12,74),(13,73),(14,72),(15,71),(16,80),(17,79),(18,78),(19,77),(20,76),(31,44),(32,43),(33,42),(34,41),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,70)], [(1,70,50,11,30,56,36,71),(2,67,49,14,21,53,35,74),(3,64,48,17,22,60,34,77),(4,61,47,20,23,57,33,80),(5,68,46,13,24,54,32,73),(6,65,45,16,25,51,31,76),(7,62,44,19,26,58,40,79),(8,69,43,12,27,55,39,72),(9,66,42,15,28,52,38,75),(10,63,41,18,29,59,37,78)], [(1,70,30,56),(2,61,21,57),(3,62,22,58),(4,63,23,59),(5,64,24,60),(6,65,25,51),(7,66,26,52),(8,67,27,53),(9,68,28,54),(10,69,29,55),(11,45,71,31),(12,46,72,32),(13,47,73,33),(14,48,74,34),(15,49,75,35),(16,50,76,36),(17,41,77,37),(18,42,78,38),(19,43,79,39),(20,44,80,40)]])
Matrix representation of D10.Q16 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 39 | 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 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 37 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 22 |
| 0 | 0 | 0 | 0 | 38 | 19 | 38 | 0 |
| 0 | 0 | 0 | 0 | 19 | 22 | 22 | 19 |
| 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 0 | 38 | 38 |
| 0 | 0 | 0 | 0 | 3 | 22 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 22 | 3 |
| 0 | 0 | 0 | 0 | 38 | 38 | 0 | 19 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0],[1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[4,4,0,0,0,0,0,0,4,37,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,29,12,0,0,0,0,0,0,0,0,22,3,38,19,0,0,0,0,0,3,19,22,0,0,0,0,3,0,38,22,0,0,0,0,3,22,0,19],[5,3,0,0,0,0,0,0,5,36,0,0,0,0,0,0,0,0,15,15,0,0,0,0,0,0,15,26,0,0,0,0,0,0,0,0,19,3,0,38,0,0,0,0,0,22,3,38,0,0,0,0,38,3,22,0,0,0,0,0,38,0,3,19] >;
D10.Q16 in GAP, Magma, Sage, TeX
D_{10}.Q_{16} % in TeX
G:=Group("D10.Q16"); // GroupNames label
G:=SmallGroup(320,264);
// by ID
G=gap.SmallGroup(320,264);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,136,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,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^-1=a^5*b,d*c*d^-1=a^4*b*c^-1>;
// generators/relations
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