direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8×Dic5, C5⋊6(C4×D8), (C5×D8)⋊8C4, C2.5(D5×D8), C40⋊16(C2×C4), (C2×D8).7D5, C8⋊4(C2×Dic5), (C8×Dic5)⋊4C2, D4⋊1(C2×Dic5), (D4×Dic5)⋊3C2, (C10×D8).4C2, C40⋊5C4⋊22C2, C10.42(C2×D8), C10.123(C4×D4), (C2×C8).235D10, C2.10(D4×Dic5), (C2×D4).139D10, C20.89(C4○D4), C10.31(C4○D8), D4⋊Dic5⋊24C2, C2.5(D8⋊3D5), (C2×C40).87C22, C22.114(D4×D5), C4.26(D4⋊2D5), C4.1(C22×Dic5), C20.130(C22×C4), (C2×C20).426C23, (C2×Dic5).280D4, (D4×C10).76C22, C4⋊Dic5.161C22, (C4×Dic5).268C22, (C5×D4)⋊16(C2×C4), (C2×C10).339(C2×D4), (C2×C4).516(C22×D5), (C2×C5⋊2C8).276C22, SmallGroup(320,776)
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, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 470 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×8], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C5⋊2C8, C40 [×2], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], C4×D8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×4], C22×Dic5 [×2], D4×C10 [×2], C8×Dic5, C40⋊5C4, D4⋊Dic5 [×2], D4×Dic5 [×2], C10×D8, D8×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×D8, C4○D8, C2×Dic5 [×6], C22×D5, C4×D8, D4×D5, D4⋊2D5, C22×Dic5, D5×D8, D8⋊3D5, D4×Dic5, D8×Dic5
►On 160 points
Generators in S160
(1 90 12 74 39 61 48 55)(2 81 13 75 40 62 49 56)(3 82 14 76 31 63 50 57)(4 83 15 77 32 64 41 58)(5 84 16 78 33 65 42 59)(6 85 17 79 34 66 43 60)(7 86 18 80 35 67 44 51)(8 87 19 71 36 68 45 52)(9 88 20 72 37 69 46 53)(10 89 11 73 38 70 47 54)(21 104 156 111 149 127 140 95)(22 105 157 112 150 128 131 96)(23 106 158 113 141 129 132 97)(24 107 159 114 142 130 133 98)(25 108 160 115 143 121 134 99)(26 109 151 116 144 122 135 100)(27 110 152 117 145 123 136 91)(28 101 153 118 146 124 137 92)(29 102 154 119 147 125 138 93)(30 103 155 120 148 126 139 94)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 123 16 128)(12 122 17 127)(13 121 18 126)(14 130 19 125)(15 129 20 124)(21 61 26 66)(22 70 27 65)(23 69 28 64)(24 68 29 63)(25 67 30 62)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 95 39 100)(35 94 40 99)(41 106 46 101)(42 105 47 110)(43 104 48 109)(44 103 49 108)(45 102 50 107)(51 155 56 160)(52 154 57 159)(53 153 58 158)(54 152 59 157)(55 151 60 156)(71 138 76 133)(72 137 77 132)(73 136 78 131)(74 135 79 140)(75 134 80 139)(81 143 86 148)(82 142 87 147)(83 141 88 146)(84 150 89 145)(85 149 90 144)
G:=sub<Sym(160)| (1,90,12,74,39,61,48,55)(2,81,13,75,40,62,49,56)(3,82,14,76,31,63,50,57)(4,83,15,77,32,64,41,58)(5,84,16,78,33,65,42,59)(6,85,17,79,34,66,43,60)(7,86,18,80,35,67,44,51)(8,87,19,71,36,68,45,52)(9,88,20,72,37,69,46,53)(10,89,11,73,38,70,47,54)(21,104,156,111,149,127,140,95)(22,105,157,112,150,128,131,96)(23,106,158,113,141,129,132,97)(24,107,159,114,142,130,133,98)(25,108,160,115,143,121,134,99)(26,109,151,116,144,122,135,100)(27,110,152,117,145,123,136,91)(28,101,153,118,146,124,137,92)(29,102,154,119,147,125,138,93)(30,103,155,120,148,126,139,94), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,123,16,128)(12,122,17,127)(13,121,18,126)(14,130,19,125)(15,129,20,124)(21,61,26,66)(22,70,27,65)(23,69,28,64)(24,68,29,63)(25,67,30,62)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,95,39,100)(35,94,40,99)(41,106,46,101)(42,105,47,110)(43,104,48,109)(44,103,49,108)(45,102,50,107)(51,155,56,160)(52,154,57,159)(53,153,58,158)(54,152,59,157)(55,151,60,156)(71,138,76,133)(72,137,77,132)(73,136,78,131)(74,135,79,140)(75,134,80,139)(81,143,86,148)(82,142,87,147)(83,141,88,146)(84,150,89,145)(85,149,90,144)>;
G:=Group( (1,90,12,74,39,61,48,55)(2,81,13,75,40,62,49,56)(3,82,14,76,31,63,50,57)(4,83,15,77,32,64,41,58)(5,84,16,78,33,65,42,59)(6,85,17,79,34,66,43,60)(7,86,18,80,35,67,44,51)(8,87,19,71,36,68,45,52)(9,88,20,72,37,69,46,53)(10,89,11,73,38,70,47,54)(21,104,156,111,149,127,140,95)(22,105,157,112,150,128,131,96)(23,106,158,113,141,129,132,97)(24,107,159,114,142,130,133,98)(25,108,160,115,143,121,134,99)(26,109,151,116,144,122,135,100)(27,110,152,117,145,123,136,91)(28,101,153,118,146,124,137,92)(29,102,154,119,147,125,138,93)(30,103,155,120,148,126,139,94), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,123,16,128)(12,122,17,127)(13,121,18,126)(14,130,19,125)(15,129,20,124)(21,61,26,66)(22,70,27,65)(23,69,28,64)(24,68,29,63)(25,67,30,62)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,95,39,100)(35,94,40,99)(41,106,46,101)(42,105,47,110)(43,104,48,109)(44,103,49,108)(45,102,50,107)(51,155,56,160)(52,154,57,159)(53,153,58,158)(54,152,59,157)(55,151,60,156)(71,138,76,133)(72,137,77,132)(73,136,78,131)(74,135,79,140)(75,134,80,139)(81,143,86,148)(82,142,87,147)(83,141,88,146)(84,150,89,145)(85,149,90,144) );
G=PermutationGroup([(1,90,12,74,39,61,48,55),(2,81,13,75,40,62,49,56),(3,82,14,76,31,63,50,57),(4,83,15,77,32,64,41,58),(5,84,16,78,33,65,42,59),(6,85,17,79,34,66,43,60),(7,86,18,80,35,67,44,51),(8,87,19,71,36,68,45,52),(9,88,20,72,37,69,46,53),(10,89,11,73,38,70,47,54),(21,104,156,111,149,127,140,95),(22,105,157,112,150,128,131,96),(23,106,158,113,141,129,132,97),(24,107,159,114,142,130,133,98),(25,108,160,115,143,121,134,99),(26,109,151,116,144,122,135,100),(27,110,152,117,145,123,136,91),(28,101,153,118,146,124,137,92),(29,102,154,119,147,125,138,93),(30,103,155,120,148,126,139,94)], [(1,60),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(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)], [(1,116,6,111),(2,115,7,120),(3,114,8,119),(4,113,9,118),(5,112,10,117),(11,123,16,128),(12,122,17,127),(13,121,18,126),(14,130,19,125),(15,129,20,124),(21,61,26,66),(22,70,27,65),(23,69,28,64),(24,68,29,63),(25,67,30,62),(31,98,36,93),(32,97,37,92),(33,96,38,91),(34,95,39,100),(35,94,40,99),(41,106,46,101),(42,105,47,110),(43,104,48,109),(44,103,49,108),(45,102,50,107),(51,155,56,160),(52,154,57,159),(53,153,58,158),(54,152,59,157),(55,151,60,156),(71,138,76,133),(72,137,77,132),(73,136,78,131),(74,135,79,140),(75,134,80,139),(81,143,86,148),(82,142,87,147),(83,141,88,146),(84,150,89,145),(85,149,90,144)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 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 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | D8 | C4○D4 | D10 | Dic5 | D10 | C4○D8 | D4⋊2D5 | D4×D5 | D5×D8 | D8⋊3D5 |
| kernel | D8×Dic5 | C8×Dic5 | C40⋊5C4 | D4⋊Dic5 | D4×Dic5 | C10×D8 | C5×D8 | C2×Dic5 | C2×D8 | Dic5 | C20 | C2×C8 | D8 | C2×D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 8 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of D8×Dic5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 12 | 17 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 0 | 12 | 0 |
| 35 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 18 | 20 | 0 | 0 |
| 35 | 23 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,17,17],[40,0,0,0,0,40,0,0,0,0,0,12,0,0,24,0],[35,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[18,35,0,0,20,23,0,0,0,0,32,0,0,0,0,32] >;
D8×Dic5 in GAP, Magma, Sage, TeX
D_8\times {\rm Dic}_5 % in TeX
G:=Group("D8xDic5"); // GroupNames label
G:=SmallGroup(320,776);
// by ID
G=gap.SmallGroup(320,776);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,851,438,102,12550]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations