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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, C4×D8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×D8, C22×Dic5, D4×C10, C8×Dic5, C40⋊5C4, D4⋊Dic5, D4×Dic5, C10×D8, D8×Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, D8, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×D8, C4○D8, C2×Dic5, 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 14 74 39 65 48 59)(2 81 15 75 40 66 49 60)(3 82 16 76 31 67 50 51)(4 83 17 77 32 68 41 52)(5 84 18 78 33 69 42 53)(6 85 19 79 34 70 43 54)(7 86 20 80 35 61 44 55)(8 87 11 71 36 62 45 56)(9 88 12 72 37 63 46 57)(10 89 13 73 38 64 47 58)(21 110 156 111 145 127 136 91)(22 101 157 112 146 128 137 92)(23 102 158 113 147 129 138 93)(24 103 159 114 148 130 139 94)(25 104 160 115 149 121 140 95)(26 105 151 116 150 122 131 96)(27 106 152 117 141 123 132 97)(28 107 153 118 142 124 133 98)(29 108 154 119 143 125 134 99)(30 109 155 120 144 126 135 100)
(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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 125 16 130)(12 124 17 129)(13 123 18 128)(14 122 19 127)(15 121 20 126)(21 65 26 70)(22 64 27 69)(23 63 28 68)(24 62 29 67)(25 61 30 66)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 102 46 107)(42 101 47 106)(43 110 48 105)(44 109 49 104)(45 108 50 103)(51 159 56 154)(52 158 57 153)(53 157 58 152)(54 156 59 151)(55 155 60 160)(71 134 76 139)(72 133 77 138)(73 132 78 137)(74 131 79 136)(75 140 80 135)(81 149 86 144)(82 148 87 143)(83 147 88 142)(84 146 89 141)(85 145 90 150)
G:=sub<Sym(160)| (1,90,14,74,39,65,48,59)(2,81,15,75,40,66,49,60)(3,82,16,76,31,67,50,51)(4,83,17,77,32,68,41,52)(5,84,18,78,33,69,42,53)(6,85,19,79,34,70,43,54)(7,86,20,80,35,61,44,55)(8,87,11,71,36,62,45,56)(9,88,12,72,37,63,46,57)(10,89,13,73,38,64,47,58)(21,110,156,111,145,127,136,91)(22,101,157,112,146,128,137,92)(23,102,158,113,147,129,138,93)(24,103,159,114,148,130,139,94)(25,104,160,115,149,121,140,95)(26,105,151,116,150,122,131,96)(27,106,152,117,141,123,132,97)(28,107,153,118,142,124,133,98)(29,108,154,119,143,125,134,99)(30,109,155,120,144,126,135,100), (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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,125,16,130)(12,124,17,129)(13,123,18,128)(14,122,19,127)(15,121,20,126)(21,65,26,70)(22,64,27,69)(23,63,28,68)(24,62,29,67)(25,61,30,66)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,102,46,107)(42,101,47,106)(43,110,48,105)(44,109,49,104)(45,108,50,103)(51,159,56,154)(52,158,57,153)(53,157,58,152)(54,156,59,151)(55,155,60,160)(71,134,76,139)(72,133,77,138)(73,132,78,137)(74,131,79,136)(75,140,80,135)(81,149,86,144)(82,148,87,143)(83,147,88,142)(84,146,89,141)(85,145,90,150)>;
G:=Group( (1,90,14,74,39,65,48,59)(2,81,15,75,40,66,49,60)(3,82,16,76,31,67,50,51)(4,83,17,77,32,68,41,52)(5,84,18,78,33,69,42,53)(6,85,19,79,34,70,43,54)(7,86,20,80,35,61,44,55)(8,87,11,71,36,62,45,56)(9,88,12,72,37,63,46,57)(10,89,13,73,38,64,47,58)(21,110,156,111,145,127,136,91)(22,101,157,112,146,128,137,92)(23,102,158,113,147,129,138,93)(24,103,159,114,148,130,139,94)(25,104,160,115,149,121,140,95)(26,105,151,116,150,122,131,96)(27,106,152,117,141,123,132,97)(28,107,153,118,142,124,133,98)(29,108,154,119,143,125,134,99)(30,109,155,120,144,126,135,100), (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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,125,16,130)(12,124,17,129)(13,123,18,128)(14,122,19,127)(15,121,20,126)(21,65,26,70)(22,64,27,69)(23,63,28,68)(24,62,29,67)(25,61,30,66)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,102,46,107)(42,101,47,106)(43,110,48,105)(44,109,49,104)(45,108,50,103)(51,159,56,154)(52,158,57,153)(53,157,58,152)(54,156,59,151)(55,155,60,160)(71,134,76,139)(72,133,77,138)(73,132,78,137)(74,131,79,136)(75,140,80,135)(81,149,86,144)(82,148,87,143)(83,147,88,142)(84,146,89,141)(85,145,90,150) );
G=PermutationGroup([[(1,90,14,74,39,65,48,59),(2,81,15,75,40,66,49,60),(3,82,16,76,31,67,50,51),(4,83,17,77,32,68,41,52),(5,84,18,78,33,69,42,53),(6,85,19,79,34,70,43,54),(7,86,20,80,35,61,44,55),(8,87,11,71,36,62,45,56),(9,88,12,72,37,63,46,57),(10,89,13,73,38,64,47,58),(21,110,156,111,145,127,136,91),(22,101,157,112,146,128,137,92),(23,102,158,113,147,129,138,93),(24,103,159,114,148,130,139,94),(25,104,160,115,149,121,140,95),(26,105,151,116,150,122,131,96),(27,106,152,117,141,123,132,97),(28,107,153,118,142,124,133,98),(29,108,154,119,143,125,134,99),(30,109,155,120,144,126,135,100)], [(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)], [(1,116,6,111),(2,115,7,120),(3,114,8,119),(4,113,9,118),(5,112,10,117),(11,125,16,130),(12,124,17,129),(13,123,18,128),(14,122,19,127),(15,121,20,126),(21,65,26,70),(22,64,27,69),(23,63,28,68),(24,62,29,67),(25,61,30,66),(31,94,36,99),(32,93,37,98),(33,92,38,97),(34,91,39,96),(35,100,40,95),(41,102,46,107),(42,101,47,106),(43,110,48,105),(44,109,49,104),(45,108,50,103),(51,159,56,154),(52,158,57,153),(53,157,58,152),(54,156,59,151),(55,155,60,160),(71,134,76,139),(72,133,77,138),(73,132,78,137),(74,131,79,136),(75,140,80,135),(81,149,86,144),(82,148,87,143),(83,147,88,142),(84,146,89,141),(85,145,90,150)]])
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