metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊3Dic5, C40⋊22(C2×C4), (C5×D8)⋊11C4, (C2×D8).6D5, C40⋊8C4⋊8C2, C8⋊2(C2×Dic5), C5⋊6(D8⋊C4), (D4×Dic5)⋊5C2, D4⋊2(C2×Dic5), (C10×D8).6C2, C40⋊6C4⋊20C2, (C2×C8).84D10, C10.124(C4×D4), C2.11(D4×Dic5), (C2×D4).141D10, C20.91(C4○D4), C2.7(D8⋊D5), D4⋊Dic5⋊26C2, C22.115(D4×D5), C4.27(D4⋊2D5), C4.2(C22×Dic5), C10.47(C8⋊C22), (C2×C40).146C22, (C2×C20).429C23, C20.131(C22×C4), (C2×Dic5).236D4, (D4×C10).79C22, C4⋊Dic5.163C22, (C4×Dic5).49C22, (C5×D4)⋊17(C2×C4), (C2×C10).342(C2×D4), (C2×C4).519(C22×D5), (C2×C5⋊2C8).146C22, SmallGroup(320,779)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 470 in 132 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, D8⋊C4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×D8, C22×Dic5, D4×C10, C40⋊8C4, C40⋊6C4, D4⋊Dic5, D4×Dic5, C10×D8, D8⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C8⋊C22, C2×Dic5, C22×D5, D8⋊C4, D4×D5, D4⋊2D5, C22×Dic5, D8⋊D5, D4×Dic5, D8⋊Dic5
►On 160 points
Generators in S160
(1 85 14 79 39 70 48 54)(2 86 15 80 40 61 49 55)(3 87 16 71 31 62 50 56)(4 88 17 72 32 63 41 57)(5 89 18 73 33 64 42 58)(6 90 19 74 34 65 43 59)(7 81 20 75 35 66 44 60)(8 82 11 76 36 67 45 51)(9 83 12 77 37 68 46 52)(10 84 13 78 38 69 47 53)(21 122 156 96 145 105 136 116)(22 123 157 97 146 106 137 117)(23 124 158 98 147 107 138 118)(24 125 159 99 148 108 139 119)(25 126 160 100 149 109 140 120)(26 127 151 91 150 110 131 111)(27 128 152 92 141 101 132 112)(28 129 153 93 142 102 133 113)(29 130 154 94 143 103 134 114)(30 121 155 95 144 104 135 115)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 51)(9 52)(10 53)(11 67)(12 68)(13 69)(14 70)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 129)(29 130)(30 121)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 88)(42 89)(43 90)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(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)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 96 6 91)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 108 16 103)(12 107 17 102)(13 106 18 101)(14 105 19 110)(15 104 20 109)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
G:=sub<Sym(160)| (1,85,14,79,39,70,48,54)(2,86,15,80,40,61,49,55)(3,87,16,71,31,62,50,56)(4,88,17,72,32,63,41,57)(5,89,18,73,33,64,42,58)(6,90,19,74,34,65,43,59)(7,81,20,75,35,66,44,60)(8,82,11,76,36,67,45,51)(9,83,12,77,37,68,46,52)(10,84,13,78,38,69,47,53)(21,122,156,96,145,105,136,116)(22,123,157,97,146,106,137,117)(23,124,158,98,147,107,138,118)(24,125,159,99,148,108,139,119)(25,126,160,100,149,109,140,120)(26,127,151,91,150,110,131,111)(27,128,152,92,141,101,132,112)(28,129,153,93,142,102,133,113)(29,130,154,94,143,103,134,114)(30,121,155,95,144,104,135,115), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,51)(9,52)(10,53)(11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,129)(29,130)(30,121)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,88)(42,89)(43,90)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,108,16,103)(12,107,17,102)(13,106,18,101)(14,105,19,110)(15,104,20,109)(21,90,26,85)(22,89,27,84)(23,88,28,83)(24,87,29,82)(25,86,30,81)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,139,56,134)(52,138,57,133)(53,137,58,132)(54,136,59,131)(55,135,60,140)(61,144,66,149)(62,143,67,148)(63,142,68,147)(64,141,69,146)(65,150,70,145)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;
G:=Group( (1,85,14,79,39,70,48,54)(2,86,15,80,40,61,49,55)(3,87,16,71,31,62,50,56)(4,88,17,72,32,63,41,57)(5,89,18,73,33,64,42,58)(6,90,19,74,34,65,43,59)(7,81,20,75,35,66,44,60)(8,82,11,76,36,67,45,51)(9,83,12,77,37,68,46,52)(10,84,13,78,38,69,47,53)(21,122,156,96,145,105,136,116)(22,123,157,97,146,106,137,117)(23,124,158,98,147,107,138,118)(24,125,159,99,148,108,139,119)(25,126,160,100,149,109,140,120)(26,127,151,91,150,110,131,111)(27,128,152,92,141,101,132,112)(28,129,153,93,142,102,133,113)(29,130,154,94,143,103,134,114)(30,121,155,95,144,104,135,115), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,51)(9,52)(10,53)(11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,129)(29,130)(30,121)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,88)(42,89)(43,90)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,108,16,103)(12,107,17,102)(13,106,18,101)(14,105,19,110)(15,104,20,109)(21,90,26,85)(22,89,27,84)(23,88,28,83)(24,87,29,82)(25,86,30,81)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,139,56,134)(52,138,57,133)(53,137,58,132)(54,136,59,131)(55,135,60,140)(61,144,66,149)(62,143,67,148)(63,142,68,147)(64,141,69,146)(65,150,70,145)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );
G=PermutationGroup([[(1,85,14,79,39,70,48,54),(2,86,15,80,40,61,49,55),(3,87,16,71,31,62,50,56),(4,88,17,72,32,63,41,57),(5,89,18,73,33,64,42,58),(6,90,19,74,34,65,43,59),(7,81,20,75,35,66,44,60),(8,82,11,76,36,67,45,51),(9,83,12,77,37,68,46,52),(10,84,13,78,38,69,47,53),(21,122,156,96,145,105,136,116),(22,123,157,97,146,106,137,117),(23,124,158,98,147,107,138,118),(24,125,159,99,148,108,139,119),(25,126,160,100,149,109,140,120),(26,127,151,91,150,110,131,111),(27,128,152,92,141,101,132,112),(28,129,153,93,142,102,133,113),(29,130,154,94,143,103,134,114),(30,121,155,95,144,104,135,115)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,51),(9,52),(10,53),(11,67),(12,68),(13,69),(14,70),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,122),(22,123),(23,124),(24,125),(25,126),(26,127),(27,128),(28,129),(29,130),(30,121),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,88),(42,89),(43,90),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(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),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,96,6,91),(2,95,7,100),(3,94,8,99),(4,93,9,98),(5,92,10,97),(11,108,16,103),(12,107,17,102),(13,106,18,101),(14,105,19,110),(15,104,20,109),(21,90,26,85),(22,89,27,84),(23,88,28,83),(24,87,29,82),(25,86,30,81),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,139,56,134),(52,138,57,133),(53,137,58,132),(54,136,59,131),(55,135,60,140),(61,144,66,149),(62,143,67,148),(63,142,68,147),(64,141,69,146),(65,150,70,145),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | Dic5 | D10 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 |
| kernel | D8⋊Dic5 | C40⋊8C4 | C40⋊6C4 | D4⋊Dic5 | D4×Dic5 | C10×D8 | C5×D8 | C2×Dic5 | C2×D8 | C20 | C2×C8 | D8 | C2×D4 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D8⋊Dic5 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 13 |
| 0 | 0 | 0 | 0 | 21 | 33 |
| 0 | 0 | 5 | 3 | 22 | 17 |
| 0 | 0 | 8 | 8 | 3 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 13 |
| 0 | 0 | 0 | 0 | 21 | 33 |
| 0 | 0 | 36 | 38 | 0 | 0 |
| 0 | 0 | 33 | 33 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 5 | 35 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 9 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 13 | 23 | 23 |
| 0 | 0 | 29 | 27 | 31 | 33 |
| 0 | 0 | 5 | 40 | 26 | 29 |
| 0 | 0 | 4 | 1 | 12 | 15 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,5,8,0,0,0,0,3,8,0,0,20,21,22,3,0,0,13,33,17,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,33,0,0,0,0,38,33,0,0,20,21,0,0,0,0,13,33,0,0],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,40,5,0,0,0,0,1,35],[13,9,0,0,0,0,13,28,0,0,0,0,0,0,14,29,5,4,0,0,13,27,40,1,0,0,23,31,26,12,0,0,23,33,29,15] >;
D8⋊Dic5 in GAP, Magma, Sage, TeX
D_8\rtimes {\rm Dic}_5 % in TeX
G:=Group("D8:Dic5"); // GroupNames label
G:=SmallGroup(320,779);
// by ID
G=gap.SmallGroup(320,779);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations