metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.1Q16, D10.4SD16, C4⋊C4⋊2F5, C10.1C4≀C2, (C2×Dic10)⋊5C4, D10⋊C8.1C2, C10.8(C23⋊C4), C2.4(D4⋊F5), D10⋊2Q8.1C2, C2.4(Q8⋊F5), (C2×Dic5).93D4, (C22×D5).55D4, C10.1(Q8⋊C4), C5⋊1(C23.31D4), D10.3Q8.1C2, C22.56(C22⋊F5), C2.11(D10.D4), (C5×C4⋊C4)⋊2C4, (C2×C4).9(C2×F5), (C2×C20).6(C2×C4), (C2×C4×D5).2C22, (C2×C10).19(C22⋊C4), SmallGroup(320,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.1Q16
G = < a,b,c,d | a10=b2=c8=1, d2=a5c4, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd-1=a5b, dcd-1=a4bc-1 >
Subgroups: 426 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, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic5, C20, F5, D10, D10, C2×C10, C2.C42, C22⋊C8, C22⋊Q8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C23.31D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8, C2×Dic10, C2×C4×D5, C22×F5, D10⋊C8, D10.3Q8, D10⋊2Q8, D10.1Q16
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, D10.D4, D4⋊F5, Q8⋊F5, D10.1Q16
Character table of D10.1Q16
| 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 | 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 |
| 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 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 | -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 | 2ζ4ζ52+2ζ4ζ5+ζ4 | -√5 | √5 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ43ζ53+2ζ43ζ5+ζ43 | orthogonal lifted from D10.D4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ43ζ54+2ζ43ζ52+ζ43 | √5 | -√5 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ4ζ52+2ζ4ζ5+ζ4 | orthogonal lifted from D10.D4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ43ζ53+2ζ43ζ5+ζ43 | √5 | -√5 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ53+ζ4 | orthogonal lifted from D10.D4 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ4ζ54+2ζ4ζ53+ζ4 | -√5 | √5 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ52+ζ43 | orthogonal lifted from D10.D4 |
| 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 | symplectic lifted from Q8⋊F5, 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 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 34)(22 33)(23 32)(24 31)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 80)
(1 53 33 73 18 48 28 68)(2 60 32 76 19 45 27 61)(3 57 31 79 20 42 26 64)(4 54 40 72 11 49 25 67)(5 51 39 75 12 46 24 70)(6 58 38 78 13 43 23 63)(7 55 37 71 14 50 22 66)(8 52 36 74 15 47 21 69)(9 59 35 77 16 44 30 62)(10 56 34 80 17 41 29 65)
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,34)(22,33)(23,32)(24,31)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80), (1,53,33,73,18,48,28,68)(2,60,32,76,19,45,27,61)(3,57,31,79,20,42,26,64)(4,54,40,72,11,49,25,67)(5,51,39,75,12,46,24,70)(6,58,38,78,13,43,23,63)(7,55,37,71,14,50,22,66)(8,52,36,74,15,47,21,69)(9,59,35,77,16,44,30,62)(10,56,34,80,17,41,29,65), (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)>;
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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,34)(22,33)(23,32)(24,31)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80), (1,53,33,73,18,48,28,68)(2,60,32,76,19,45,27,61)(3,57,31,79,20,42,26,64)(4,54,40,72,11,49,25,67)(5,51,39,75,12,46,24,70)(6,58,38,78,13,43,23,63)(7,55,37,71,14,50,22,66)(8,52,36,74,15,47,21,69)(9,59,35,77,16,44,30,62)(10,56,34,80,17,41,29,65), (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70) );
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)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(21,34),(22,33),(23,32),(24,31),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(50,60),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(70,80)], [(1,53,33,73,18,48,28,68),(2,60,32,76,19,45,27,61),(3,57,31,79,20,42,26,64),(4,54,40,72,11,49,25,67),(5,51,39,75,12,46,24,70),(6,58,38,78,13,43,23,63),(7,55,37,71,14,50,22,66),(8,52,36,74,15,47,21,69),(9,59,35,77,16,44,30,62),(10,56,34,80,17,41,29,65)], [(1,53,13,43),(2,54,14,44),(3,55,15,45),(4,56,16,46),(5,57,17,47),(6,58,18,48),(7,59,19,49),(8,60,20,50),(9,51,11,41),(10,52,12,42),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70)]])
Matrix representation of D10.1Q16 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 29 | 12 | 0 | 0 | 0 | 0 |
| 29 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 33 | 40 | 34 |
| 0 | 0 | 7 | 1 | 8 | 4 |
| 0 | 0 | 7 | 3 | 40 | 6 |
| 0 | 0 | 37 | 3 | 38 | 4 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 15 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 7 | 40 |
| 0 | 0 | 1 | 4 | 7 | 8 |
| 0 | 0 | 33 | 34 | 37 | 40 |
| 0 | 0 | 1 | 34 | 35 | 38 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[29,29,0,0,0,0,12,29,0,0,0,0,0,0,37,7,7,37,0,0,33,1,3,3,0,0,40,8,40,38,0,0,34,4,6,4],[15,15,0,0,0,0,15,26,0,0,0,0,0,0,3,1,33,1,0,0,6,4,34,34,0,0,7,7,37,35,0,0,40,8,40,38] >;
D10.1Q16 in GAP, Magma, Sage, TeX
D_{10}._1Q_{16} % in TeX
G:=Group("D10.1Q16"); // GroupNames label
G:=SmallGroup(320,207);
// by ID
G=gap.SmallGroup(320,207);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,675,794,192,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5*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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