metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊5C2, Q16⋊3D5, D10.7D4, C8.10D10, Q8.5D10, C40.8C22, C20.10C23, Dic5.26D4, D20.5C22, (C8×D5)⋊3C2, C5⋊4(C4○D8), Q8⋊D5⋊4C2, (C5×Q16)⋊3C2, C2.24(D4×D5), Q8⋊2D5⋊3C2, C10.36(C2×D4), C5⋊2C8.8C22, C4.10(C22×D5), (C5×Q8).5C22, (C4×D5).20C22, SmallGroup(160,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.D10
G = < a,b,c,d | a4=d2=1, b2=c10=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=a2c9 >
Subgroups: 232 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×3], C10, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, D10 [×2], C4○D8, C5⋊2C8, C40, C4×D5, C4×D5 [×2], D20 [×2], D20 [×2], C5×Q8 [×2], C8×D5, D40, Q8⋊D5 [×2], C5×Q16, Q8⋊2D5 [×2], Q8.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C4○D8, C22×D5, D4×D5, Q8.D10
Character table of Q8.D10
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 20 | 20 | 2 | 4 | 4 | 5 | 5 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ15 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ16 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ17 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ18 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | 2 | √2 | -√2 | √-2 | -√-2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | complex lifted from C4○D8 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2 | 2 | √2 | -√2 | -√-2 | √-2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2 | 2 | -√2 | √2 | √-2 | -√-2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | 2 | -√2 | √2 | -√-2 | √-2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | complex lifted from C4○D8 |
| ρ23 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ24 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2√2 | 2√2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | orthogonal faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2√2 | -2√2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | orthogonal faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2√2 | 2√2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | orthogonal faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2√2 | -2√2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 | ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5 | -ζ83ζ54-ζ83ζ5+ζ8ζ54+ζ8ζ5 | orthogonal faithful |
►On 80 points
Generators in S80
(1 41 11 51)(2 52 12 42)(3 43 13 53)(4 54 14 44)(5 45 15 55)(6 56 16 46)(7 47 17 57)(8 58 18 48)(9 49 19 59)(10 60 20 50)(21 73 31 63)(22 64 32 74)(23 75 33 65)(24 66 34 76)(25 77 35 67)(26 68 36 78)(27 79 37 69)(28 70 38 80)(29 61 39 71)(30 72 40 62)
(1 37 11 27)(2 70 12 80)(3 39 13 29)(4 72 14 62)(5 21 15 31)(6 74 16 64)(7 23 17 33)(8 76 18 66)(9 25 19 35)(10 78 20 68)(22 56 32 46)(24 58 34 48)(26 60 36 50)(28 42 38 52)(30 44 40 54)(41 79 51 69)(43 61 53 71)(45 63 55 73)(47 65 57 75)(49 67 59 77)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 40)(18 39)(19 38)(20 37)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)
G:=sub<Sym(80)| (1,41,11,51)(2,52,12,42)(3,43,13,53)(4,54,14,44)(5,45,15,55)(6,56,16,46)(7,47,17,57)(8,58,18,48)(9,49,19,59)(10,60,20,50)(21,73,31,63)(22,64,32,74)(23,75,33,65)(24,66,34,76)(25,77,35,67)(26,68,36,78)(27,79,37,69)(28,70,38,80)(29,61,39,71)(30,72,40,62), (1,37,11,27)(2,70,12,80)(3,39,13,29)(4,72,14,62)(5,21,15,31)(6,74,16,64)(7,23,17,33)(8,76,18,66)(9,25,19,35)(10,78,20,68)(22,56,32,46)(24,58,34,48)(26,60,36,50)(28,42,38,52)(30,44,40,54)(41,79,51,69)(43,61,53,71)(45,63,55,73)(47,65,57,75)(49,67,59,77), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,40)(18,39)(19,38)(20,37)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)>;
G:=Group( (1,41,11,51)(2,52,12,42)(3,43,13,53)(4,54,14,44)(5,45,15,55)(6,56,16,46)(7,47,17,57)(8,58,18,48)(9,49,19,59)(10,60,20,50)(21,73,31,63)(22,64,32,74)(23,75,33,65)(24,66,34,76)(25,77,35,67)(26,68,36,78)(27,79,37,69)(28,70,38,80)(29,61,39,71)(30,72,40,62), (1,37,11,27)(2,70,12,80)(3,39,13,29)(4,72,14,62)(5,21,15,31)(6,74,16,64)(7,23,17,33)(8,76,18,66)(9,25,19,35)(10,78,20,68)(22,56,32,46)(24,58,34,48)(26,60,36,50)(28,42,38,52)(30,44,40,54)(41,79,51,69)(43,61,53,71)(45,63,55,73)(47,65,57,75)(49,67,59,77), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,40)(18,39)(19,38)(20,37)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69) );
G=PermutationGroup([(1,41,11,51),(2,52,12,42),(3,43,13,53),(4,54,14,44),(5,45,15,55),(6,56,16,46),(7,47,17,57),(8,58,18,48),(9,49,19,59),(10,60,20,50),(21,73,31,63),(22,64,32,74),(23,75,33,65),(24,66,34,76),(25,77,35,67),(26,68,36,78),(27,79,37,69),(28,70,38,80),(29,61,39,71),(30,72,40,62)], [(1,37,11,27),(2,70,12,80),(3,39,13,29),(4,72,14,62),(5,21,15,31),(6,74,16,64),(7,23,17,33),(8,76,18,66),(9,25,19,35),(10,78,20,68),(22,56,32,46),(24,58,34,48),(26,60,36,50),(28,42,38,52),(30,44,40,54),(41,79,51,69),(43,61,53,71),(45,63,55,73),(47,65,57,75),(49,67,59,77)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,40),(18,39),(19,38),(20,37),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69)])
Q8.D10 is a maximal subgroup of
D40⋊1C4 Q16.F5 C16⋊D10 SD32⋊3D5 Q32⋊D5 D80⋊5C2 Q16⋊5F5 Q16⋊F5 D20.30D4 D5×C4○D8 D8⋊15D10 D40⋊C22 C40.C23 D120⋊C2 D40⋊5S3 D20.D6 D20.16D6 D120⋊8C2 Dic5.7S4
Q8.D10 is a maximal quotient of
Dic5⋊7SD16 Q8.Dic10 Q8⋊Dic5⋊C2 Q8⋊2D5⋊C4 D10⋊2SD16 D20⋊4D4 (C2×C8).D10 D20.12D4 D40⋊12C4 C8.6Dic10 C8.27(C4×D5) C8⋊7D20 C2.D8⋊D5 D20.2Q8 Q16×Dic5 (C2×Q16)⋊D5 D20.17D4 D10⋊3Q16 C40.28D4 D120⋊C2 D40⋊5S3 D20.D6 D20.16D6 D120⋊8C2
Matrix representation of Q8.D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 37 |
| 0 | 0 | 21 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 36 |
| 0 | 0 | 0 | 9 |
| 1 | 6 | 0 | 0 |
| 35 | 6 | 0 | 0 |
| 0 | 0 | 11 | 19 |
| 0 | 0 | 13 | 30 |
| 1 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 |
| 0 | 0 | 0 | 34 |
| 0 | 0 | 35 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,37,40],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,36,9],[1,35,0,0,6,6,0,0,0,0,11,13,0,0,19,30],[1,35,0,0,0,40,0,0,0,0,0,35,0,0,34,0] >;
Q8.D10 in GAP, Magma, Sage, TeX
Q_8.D_{10} % in TeX
G:=Group("Q8.D10"); // GroupNames label
G:=SmallGroup(160,140);
// by ID
G=gap.SmallGroup(160,140);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,362,116,86,297,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^10=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=a^2*c^9>;
// generators/relations
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