metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.11D4, C4.91(D4×D5), Q8⋊C4⋊9D5, C4⋊C4.146D10, C4.6(C4○D20), C20.115(C2×D4), (C2×C8).120D10, (C2×Q8).13D10, C5⋊2(Q8.D4), Dic5⋊3Q8⋊5C2, D20⋊6C4.2C2, C10.45(C4○D8), C20.17(C4○D4), C20.8Q8⋊13C2, (C2×Dic5).36D4, C22.192(D4×D5), C10.22(C4⋊D4), (C2×C20).238C23, (C2×C40).131C22, C20.23D4.5C2, (C2×D20).62C22, (Q8×C10).21C22, C2.25(D10⋊D4), C2.11(Q16⋊D5), C10.56(C8.C22), (C4×Dic5).28C22, C2.14(SD16⋊3D5), (C2×Dic10).71C22, (C2×C5⋊Q16)⋊2C2, (C5×Q8⋊C4)⋊9C2, (C2×C40⋊C2).3C2, (C2×C10).251(C2×D4), (C5×C4⋊C4).39C22, (C2×C5⋊2C8).33C22, (C2×C4).345(C22×D5), SmallGroup(320,425)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.11D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, bc=cb, dbd=a15b, dcd=a10c-1 >
Subgroups: 486 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×5], C23, D5, C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], D10 [×3], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C5⋊2C8, C40, Dic10 [×2], Dic10, D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Q8.D4, C40⋊C2 [×2], C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4 [×2], C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, Q8×C10, D20⋊6C4, C20.8Q8, C5×Q8⋊C4, Dic5⋊3Q8, C2×C40⋊C2, C2×C5⋊Q16, C20.23D4, Dic10.11D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8.C22, C22×D5, Q8.D4, C4○D20, D4×D5 [×2], D10⋊D4, SD16⋊3D5, Q16⋊D5, Dic10.11D4
►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 30 11 40)(2 29 12 39)(3 28 13 38)(4 27 14 37)(5 26 15 36)(6 25 16 35)(7 24 17 34)(8 23 18 33)(9 22 19 32)(10 21 20 31)(41 110 51 120)(42 109 52 119)(43 108 53 118)(44 107 54 117)(45 106 55 116)(46 105 56 115)(47 104 57 114)(48 103 58 113)(49 102 59 112)(50 101 60 111)(61 93 71 83)(62 92 72 82)(63 91 73 81)(64 90 74 100)(65 89 75 99)(66 88 76 98)(67 87 77 97)(68 86 78 96)(69 85 79 95)(70 84 80 94)(121 148 131 158)(122 147 132 157)(123 146 133 156)(124 145 134 155)(125 144 135 154)(126 143 136 153)(127 142 137 152)(128 141 138 151)(129 160 139 150)(130 159 140 149)
(1 100 55 143)(2 89 56 152)(3 98 57 141)(4 87 58 150)(5 96 59 159)(6 85 60 148)(7 94 41 157)(8 83 42 146)(9 92 43 155)(10 81 44 144)(11 90 45 153)(12 99 46 142)(13 88 47 151)(14 97 48 160)(15 86 49 149)(16 95 50 158)(17 84 51 147)(18 93 52 156)(19 82 53 145)(20 91 54 154)(21 63 107 135)(22 72 108 124)(23 61 109 133)(24 70 110 122)(25 79 111 131)(26 68 112 140)(27 77 113 129)(28 66 114 138)(29 75 115 127)(30 64 116 136)(31 73 117 125)(32 62 118 134)(33 71 119 123)(34 80 120 132)(35 69 101 121)(36 78 102 130)(37 67 103 139)(38 76 104 128)(39 65 105 137)(40 74 106 126)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 140)(76 139)(77 138)(78 137)(79 136)(80 135)(81 152)(82 151)(83 150)(84 149)(85 148)(86 147)(87 146)(88 145)(89 144)(90 143)(91 142)(92 141)(93 160)(94 159)(95 158)(96 157)(97 156)(98 155)(99 154)(100 153)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)
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), (1,30,11,40)(2,29,12,39)(3,28,13,38)(4,27,14,37)(5,26,15,36)(6,25,16,35)(7,24,17,34)(8,23,18,33)(9,22,19,32)(10,21,20,31)(41,110,51,120)(42,109,52,119)(43,108,53,118)(44,107,54,117)(45,106,55,116)(46,105,56,115)(47,104,57,114)(48,103,58,113)(49,102,59,112)(50,101,60,111)(61,93,71,83)(62,92,72,82)(63,91,73,81)(64,90,74,100)(65,89,75,99)(66,88,76,98)(67,87,77,97)(68,86,78,96)(69,85,79,95)(70,84,80,94)(121,148,131,158)(122,147,132,157)(123,146,133,156)(124,145,134,155)(125,144,135,154)(126,143,136,153)(127,142,137,152)(128,141,138,151)(129,160,139,150)(130,159,140,149), (1,100,55,143)(2,89,56,152)(3,98,57,141)(4,87,58,150)(5,96,59,159)(6,85,60,148)(7,94,41,157)(8,83,42,146)(9,92,43,155)(10,81,44,144)(11,90,45,153)(12,99,46,142)(13,88,47,151)(14,97,48,160)(15,86,49,149)(16,95,50,158)(17,84,51,147)(18,93,52,156)(19,82,53,145)(20,91,54,154)(21,63,107,135)(22,72,108,124)(23,61,109,133)(24,70,110,122)(25,79,111,131)(26,68,112,140)(27,77,113,129)(28,66,114,138)(29,75,115,127)(30,64,116,136)(31,73,117,125)(32,62,118,134)(33,71,119,123)(34,80,120,132)(35,69,101,121)(36,78,102,130)(37,67,103,139)(38,76,104,128)(39,65,105,137)(40,74,106,126), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,160)(94,159)(95,158)(96,157)(97,156)(98,155)(99,154)(100,153)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119)>;
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), (1,30,11,40)(2,29,12,39)(3,28,13,38)(4,27,14,37)(5,26,15,36)(6,25,16,35)(7,24,17,34)(8,23,18,33)(9,22,19,32)(10,21,20,31)(41,110,51,120)(42,109,52,119)(43,108,53,118)(44,107,54,117)(45,106,55,116)(46,105,56,115)(47,104,57,114)(48,103,58,113)(49,102,59,112)(50,101,60,111)(61,93,71,83)(62,92,72,82)(63,91,73,81)(64,90,74,100)(65,89,75,99)(66,88,76,98)(67,87,77,97)(68,86,78,96)(69,85,79,95)(70,84,80,94)(121,148,131,158)(122,147,132,157)(123,146,133,156)(124,145,134,155)(125,144,135,154)(126,143,136,153)(127,142,137,152)(128,141,138,151)(129,160,139,150)(130,159,140,149), (1,100,55,143)(2,89,56,152)(3,98,57,141)(4,87,58,150)(5,96,59,159)(6,85,60,148)(7,94,41,157)(8,83,42,146)(9,92,43,155)(10,81,44,144)(11,90,45,153)(12,99,46,142)(13,88,47,151)(14,97,48,160)(15,86,49,149)(16,95,50,158)(17,84,51,147)(18,93,52,156)(19,82,53,145)(20,91,54,154)(21,63,107,135)(22,72,108,124)(23,61,109,133)(24,70,110,122)(25,79,111,131)(26,68,112,140)(27,77,113,129)(28,66,114,138)(29,75,115,127)(30,64,116,136)(31,73,117,125)(32,62,118,134)(33,71,119,123)(34,80,120,132)(35,69,101,121)(36,78,102,130)(37,67,103,139)(38,76,104,128)(39,65,105,137)(40,74,106,126), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,160)(94,159)(95,158)(96,157)(97,156)(98,155)(99,154)(100,153)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119) );
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)], [(1,30,11,40),(2,29,12,39),(3,28,13,38),(4,27,14,37),(5,26,15,36),(6,25,16,35),(7,24,17,34),(8,23,18,33),(9,22,19,32),(10,21,20,31),(41,110,51,120),(42,109,52,119),(43,108,53,118),(44,107,54,117),(45,106,55,116),(46,105,56,115),(47,104,57,114),(48,103,58,113),(49,102,59,112),(50,101,60,111),(61,93,71,83),(62,92,72,82),(63,91,73,81),(64,90,74,100),(65,89,75,99),(66,88,76,98),(67,87,77,97),(68,86,78,96),(69,85,79,95),(70,84,80,94),(121,148,131,158),(122,147,132,157),(123,146,133,156),(124,145,134,155),(125,144,135,154),(126,143,136,153),(127,142,137,152),(128,141,138,151),(129,160,139,150),(130,159,140,149)], [(1,100,55,143),(2,89,56,152),(3,98,57,141),(4,87,58,150),(5,96,59,159),(6,85,60,148),(7,94,41,157),(8,83,42,146),(9,92,43,155),(10,81,44,144),(11,90,45,153),(12,99,46,142),(13,88,47,151),(14,97,48,160),(15,86,49,149),(16,95,50,158),(17,84,51,147),(18,93,52,156),(19,82,53,145),(20,91,54,154),(21,63,107,135),(22,72,108,124),(23,61,109,133),(24,70,110,122),(25,79,111,131),(26,68,112,140),(27,77,113,129),(28,66,114,138),(29,75,115,127),(30,64,116,136),(31,73,117,125),(32,62,118,134),(33,71,119,123),(34,80,120,132),(35,69,101,121),(36,78,102,130),(37,67,103,139),(38,76,104,128),(39,65,105,137),(40,74,106,126)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,134),(62,133),(63,132),(64,131),(65,130),(66,129),(67,128),(68,127),(69,126),(70,125),(71,124),(72,123),(73,122),(74,121),(75,140),(76,139),(77,138),(78,137),(79,136),(80,135),(81,152),(82,151),(83,150),(84,149),(85,148),(86,147),(87,146),(88,145),(89,144),(90,143),(91,142),(92,141),(93,160),(94,159),(95,158),(96,157),(97,156),(98,155),(99,154),(100,153),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(117,120),(118,119)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 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 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8.C22 | D4×D5 | D4×D5 | SD16⋊3D5 | Q16⋊D5 |
| kernel | Dic10.11D4 | D20⋊6C4 | C20.8Q8 | C5×Q8⋊C4 | Dic5⋊3Q8 | C2×C40⋊C2 | C2×C5⋊Q16 | C20.23D4 | Dic10 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic10.11D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 9 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,9,9,0,0,0,0,23,32,0,0,0,0,0,0,0,26,0,0,0,0,11,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
Dic10.11D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._{11}D_4 % in TeX
G:=Group("Dic10.11D4"); // GroupNames label
G:=SmallGroup(320,425);
// by ID
G=gap.SmallGroup(320,425);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,b*c=c*b,d*b*d=a^15*b,d*c*d=a^10*c^-1>;
// generators/relations