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 |
(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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