metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊6D4, (C5×D4)⋊6D4, (C2×D8)⋊6D5, C4.60(D4×D5), (C10×D8)⋊14C2, C20⋊2D4⋊5C2, C5⋊5(D4⋊D4), D4⋊3(C5⋊D4), (C2×C8).35D10, (C2×D4).65D10, C20.167(C2×D4), D10⋊1C8⋊28C2, C10.56C22≀C2, C10.35(C4○D8), D4⋊Dic5⋊30C2, (C22×D5).43D4, C22.258(D4×D5), C2.30(D8⋊D5), C2.19(D8⋊3D5), C20.44D4⋊29C2, C10.51(C8⋊C22), (C2×C20).435C23, (C2×C40).249C22, (C2×Dic5).237D4, (D4×C10).84C22, C2.24(C23⋊D10), C4⋊Dic5.166C22, (C2×Dic10).126C22, C4.37(C2×C5⋊D4), (C2×D4⋊2D5)⋊2C2, (C2×D4.D5)⋊19C2, (C2×C4×D5).49C22, (C2×C10).348(C2×D4), (C2×C4).525(C22×D5), (C2×C5⋊2C8).149C22, SmallGroup(320,785)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, dad=a9, cbc-1=a5b, dbd=a10b, dcd=c-1 >
Subgroups: 686 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×2], D4 [×9], Q8 [×3], C23 [×3], D5, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×4], C20 [×2], D10 [×3], C2×C10, C2×C10 [×7], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, Dic10 [×2], Dic10, C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20, C5×D4 [×2], C5×D4 [×3], C22×D5, C22×C10 [×2], D4⋊D4, C2×C5⋊2C8, C4⋊Dic5, D4.D5 [×2], C23.D5, C2×C40, C5×D8 [×2], C2×Dic10, C2×C4×D5, D4⋊2D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4×C10 [×2], C20.44D4, D10⋊1C8, D4⋊Dic5, C2×D4.D5, C20⋊2D4, C10×D8, C2×D4⋊2D5, Dic10⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4⋊D4, D4×D5 [×2], C2×C5⋊D4, D8⋊D5, D8⋊3D5, C23⋊D10, Dic10⋊D4
►On 160 points
Generators in S160
(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)
(1 147 11 157)(2 146 12 156)(3 145 13 155)(4 144 14 154)(5 143 15 153)(6 142 16 152)(7 141 17 151)(8 160 18 150)(9 159 19 149)(10 158 20 148)(21 91 31 81)(22 90 32 100)(23 89 33 99)(24 88 34 98)(25 87 35 97)(26 86 36 96)(27 85 37 95)(28 84 38 94)(29 83 39 93)(30 82 40 92)(41 73 51 63)(42 72 52 62)(43 71 53 61)(44 70 54 80)(45 69 55 79)(46 68 56 78)(47 67 57 77)(48 66 58 76)(49 65 59 75)(50 64 60 74)(101 137 111 127)(102 136 112 126)(103 135 113 125)(104 134 114 124)(105 133 115 123)(106 132 116 122)(107 131 117 121)(108 130 118 140)(109 129 119 139)(110 128 120 138)
(1 90 56 112)(2 89 57 111)(3 88 58 110)(4 87 59 109)(5 86 60 108)(6 85 41 107)(7 84 42 106)(8 83 43 105)(9 82 44 104)(10 81 45 103)(11 100 46 102)(12 99 47 101)(13 98 48 120)(14 97 49 119)(15 96 50 118)(16 95 51 117)(17 94 52 116)(18 93 53 115)(19 92 54 114)(20 91 55 113)(21 64 135 153)(22 63 136 152)(23 62 137 151)(24 61 138 150)(25 80 139 149)(26 79 140 148)(27 78 121 147)(28 77 122 146)(29 76 123 145)(30 75 124 144)(31 74 125 143)(32 73 126 142)(33 72 127 141)(34 71 128 160)(35 70 129 159)(36 69 130 158)(37 68 131 157)(38 67 132 156)(39 66 133 155)(40 65 134 154)
(1 56)(2 45)(3 54)(4 43)(5 52)(6 41)(7 50)(8 59)(9 48)(10 57)(11 46)(12 55)(13 44)(14 53)(15 42)(16 51)(17 60)(18 49)(19 58)(20 47)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)(61 144)(62 153)(63 142)(64 151)(65 160)(66 149)(67 158)(68 147)(69 156)(70 145)(71 154)(72 143)(73 152)(74 141)(75 150)(76 159)(77 148)(78 157)(79 146)(80 155)(81 89)(82 98)(83 87)(84 96)(86 94)(88 92)(91 99)(93 97)(101 113)(103 111)(104 120)(105 109)(106 118)(108 116)(110 114)(115 119)(121 131)(122 140)(123 129)(124 138)(125 127)(126 136)(128 134)(130 132)(133 139)(135 137)
G:=sub<Sym(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), (1,147,11,157)(2,146,12,156)(3,145,13,155)(4,144,14,154)(5,143,15,153)(6,142,16,152)(7,141,17,151)(8,160,18,150)(9,159,19,149)(10,158,20,148)(21,91,31,81)(22,90,32,100)(23,89,33,99)(24,88,34,98)(25,87,35,97)(26,86,36,96)(27,85,37,95)(28,84,38,94)(29,83,39,93)(30,82,40,92)(41,73,51,63)(42,72,52,62)(43,71,53,61)(44,70,54,80)(45,69,55,79)(46,68,56,78)(47,67,57,77)(48,66,58,76)(49,65,59,75)(50,64,60,74)(101,137,111,127)(102,136,112,126)(103,135,113,125)(104,134,114,124)(105,133,115,123)(106,132,116,122)(107,131,117,121)(108,130,118,140)(109,129,119,139)(110,128,120,138), (1,90,56,112)(2,89,57,111)(3,88,58,110)(4,87,59,109)(5,86,60,108)(6,85,41,107)(7,84,42,106)(8,83,43,105)(9,82,44,104)(10,81,45,103)(11,100,46,102)(12,99,47,101)(13,98,48,120)(14,97,49,119)(15,96,50,118)(16,95,51,117)(17,94,52,116)(18,93,53,115)(19,92,54,114)(20,91,55,113)(21,64,135,153)(22,63,136,152)(23,62,137,151)(24,61,138,150)(25,80,139,149)(26,79,140,148)(27,78,121,147)(28,77,122,146)(29,76,123,145)(30,75,124,144)(31,74,125,143)(32,73,126,142)(33,72,127,141)(34,71,128,160)(35,70,129,159)(36,69,130,158)(37,68,131,157)(38,67,132,156)(39,66,133,155)(40,65,134,154), (1,56)(2,45)(3,54)(4,43)(5,52)(6,41)(7,50)(8,59)(9,48)(10,57)(11,46)(12,55)(13,44)(14,53)(15,42)(16,51)(17,60)(18,49)(19,58)(20,47)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(61,144)(62,153)(63,142)(64,151)(65,160)(66,149)(67,158)(68,147)(69,156)(70,145)(71,154)(72,143)(73,152)(74,141)(75,150)(76,159)(77,148)(78,157)(79,146)(80,155)(81,89)(82,98)(83,87)(84,96)(86,94)(88,92)(91,99)(93,97)(101,113)(103,111)(104,120)(105,109)(106,118)(108,116)(110,114)(115,119)(121,131)(122,140)(123,129)(124,138)(125,127)(126,136)(128,134)(130,132)(133,139)(135,137)>;
G:=Group( (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), (1,147,11,157)(2,146,12,156)(3,145,13,155)(4,144,14,154)(5,143,15,153)(6,142,16,152)(7,141,17,151)(8,160,18,150)(9,159,19,149)(10,158,20,148)(21,91,31,81)(22,90,32,100)(23,89,33,99)(24,88,34,98)(25,87,35,97)(26,86,36,96)(27,85,37,95)(28,84,38,94)(29,83,39,93)(30,82,40,92)(41,73,51,63)(42,72,52,62)(43,71,53,61)(44,70,54,80)(45,69,55,79)(46,68,56,78)(47,67,57,77)(48,66,58,76)(49,65,59,75)(50,64,60,74)(101,137,111,127)(102,136,112,126)(103,135,113,125)(104,134,114,124)(105,133,115,123)(106,132,116,122)(107,131,117,121)(108,130,118,140)(109,129,119,139)(110,128,120,138), (1,90,56,112)(2,89,57,111)(3,88,58,110)(4,87,59,109)(5,86,60,108)(6,85,41,107)(7,84,42,106)(8,83,43,105)(9,82,44,104)(10,81,45,103)(11,100,46,102)(12,99,47,101)(13,98,48,120)(14,97,49,119)(15,96,50,118)(16,95,51,117)(17,94,52,116)(18,93,53,115)(19,92,54,114)(20,91,55,113)(21,64,135,153)(22,63,136,152)(23,62,137,151)(24,61,138,150)(25,80,139,149)(26,79,140,148)(27,78,121,147)(28,77,122,146)(29,76,123,145)(30,75,124,144)(31,74,125,143)(32,73,126,142)(33,72,127,141)(34,71,128,160)(35,70,129,159)(36,69,130,158)(37,68,131,157)(38,67,132,156)(39,66,133,155)(40,65,134,154), (1,56)(2,45)(3,54)(4,43)(5,52)(6,41)(7,50)(8,59)(9,48)(10,57)(11,46)(12,55)(13,44)(14,53)(15,42)(16,51)(17,60)(18,49)(19,58)(20,47)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(61,144)(62,153)(63,142)(64,151)(65,160)(66,149)(67,158)(68,147)(69,156)(70,145)(71,154)(72,143)(73,152)(74,141)(75,150)(76,159)(77,148)(78,157)(79,146)(80,155)(81,89)(82,98)(83,87)(84,96)(86,94)(88,92)(91,99)(93,97)(101,113)(103,111)(104,120)(105,109)(106,118)(108,116)(110,114)(115,119)(121,131)(122,140)(123,129)(124,138)(125,127)(126,136)(128,134)(130,132)(133,139)(135,137) );
G=PermutationGroup([(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)], [(1,147,11,157),(2,146,12,156),(3,145,13,155),(4,144,14,154),(5,143,15,153),(6,142,16,152),(7,141,17,151),(8,160,18,150),(9,159,19,149),(10,158,20,148),(21,91,31,81),(22,90,32,100),(23,89,33,99),(24,88,34,98),(25,87,35,97),(26,86,36,96),(27,85,37,95),(28,84,38,94),(29,83,39,93),(30,82,40,92),(41,73,51,63),(42,72,52,62),(43,71,53,61),(44,70,54,80),(45,69,55,79),(46,68,56,78),(47,67,57,77),(48,66,58,76),(49,65,59,75),(50,64,60,74),(101,137,111,127),(102,136,112,126),(103,135,113,125),(104,134,114,124),(105,133,115,123),(106,132,116,122),(107,131,117,121),(108,130,118,140),(109,129,119,139),(110,128,120,138)], [(1,90,56,112),(2,89,57,111),(3,88,58,110),(4,87,59,109),(5,86,60,108),(6,85,41,107),(7,84,42,106),(8,83,43,105),(9,82,44,104),(10,81,45,103),(11,100,46,102),(12,99,47,101),(13,98,48,120),(14,97,49,119),(15,96,50,118),(16,95,51,117),(17,94,52,116),(18,93,53,115),(19,92,54,114),(20,91,55,113),(21,64,135,153),(22,63,136,152),(23,62,137,151),(24,61,138,150),(25,80,139,149),(26,79,140,148),(27,78,121,147),(28,77,122,146),(29,76,123,145),(30,75,124,144),(31,74,125,143),(32,73,126,142),(33,72,127,141),(34,71,128,160),(35,70,129,159),(36,69,130,158),(37,68,131,157),(38,67,132,156),(39,66,133,155),(40,65,134,154)], [(1,56),(2,45),(3,54),(4,43),(5,52),(6,41),(7,50),(8,59),(9,48),(10,57),(11,46),(12,55),(13,44),(14,53),(15,42),(16,51),(17,60),(18,49),(19,58),(20,47),(21,23),(22,32),(24,30),(25,39),(26,28),(27,37),(29,35),(31,33),(34,40),(36,38),(61,144),(62,153),(63,142),(64,151),(65,160),(66,149),(67,158),(68,147),(69,156),(70,145),(71,154),(72,143),(73,152),(74,141),(75,150),(76,159),(77,148),(78,157),(79,146),(80,155),(81,89),(82,98),(83,87),(84,96),(86,94),(88,92),(91,99),(93,97),(101,113),(103,111),(104,120),(105,109),(106,118),(108,116),(110,114),(115,119),(121,131),(122,140),(123,129),(124,138),(125,127),(126,136),(128,134),(130,132),(133,139),(135,137)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 20 | 2 | 2 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 | D8⋊3D5 |
| kernel | Dic10⋊D4 | C20.44D4 | D10⋊1C8 | D4⋊Dic5 | C2×D4.D5 | C20⋊2D4 | C10×D8 | C2×D4⋊2D5 | Dic10 | C2×Dic5 | C5×D4 | C22×D5 | C2×D8 | C2×C8 | C2×D4 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10⋊D4 ►in GL6(𝔽41)
| 7 | 1 | 0 | 0 | 0 | 0 |
| 33 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 33 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 26 |
| 0 | 0 | 0 | 0 | 28 | 3 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 9 | 0 | 0 |
| 0 | 0 | 5 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,40,32],[7,33,0,0,0,0,6,34,0,0,0,0,0,0,40,26,0,0,0,0,0,1,0,0,0,0,0,0,38,28,0,0,0,0,26,3],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,35,5,0,0,0,0,9,6,0,0,0,0,0,0,32,2,0,0,0,0,0,9],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,40,26,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40] >;
Dic10⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes D_4 % in TeX
G:=Group("Dic10:D4"); // GroupNames label
G:=SmallGroup(320,785);
// by ID
G=gap.SmallGroup(320,785);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations