metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊11D4, C42.167D10, C10.782+ (1+4), C4⋊1D4⋊7D5, C4.72(D4×D5), (C4×D20)⋊50C2, C20⋊5(C4○D4), C5⋊4(Q8⋊6D4), C20.67(C2×D4), C20⋊2D4⋊37C2, C4⋊1(D4⋊2D5), C20⋊D4⋊27C2, (D4×Dic5)⋊35C2, (C4×Dic10)⋊51C2, (C2×D4).115D10, Dic5.53(C2×D4), C10.96(C22×D4), Dic5⋊D4⋊37C2, (C2×C10).262C24, (C2×C20).636C23, (C4×C20).204C22, C2.82(D4⋊6D10), C23.68(C22×D5), (C2×D20).278C22, (D4×C10).214C22, C4⋊Dic5.381C22, (C22×C10).76C23, C22.283(C23×D5), C23.D5.73C22, (C2×Dic5).279C23, (C4×Dic5).163C22, (C22×D5).116C23, D10⋊C4.149C22, (C2×Dic10).309C22, C10.D4.164C22, (C22×Dic5).158C22, C2.69(C2×D4×D5), (C5×C4⋊1D4)⋊9C2, C10.97(C2×C4○D4), (C2×D4⋊2D5)⋊22C2, C2.61(C2×D4⋊2D5), (C2×C4×D5).148C22, (C2×C4).598(C22×D5), (C2×C5⋊D4).78C22, SmallGroup(320,1390)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1158 in 312 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×18], C5, C2×C4 [×3], C2×C4 [×18], D4 [×24], Q8 [×4], C23 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×6], C2×D4 [×9], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], C20, D10 [×6], C2×C10, C2×C10 [×12], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4, C4⋊1D4 [×2], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×8], C5⋊D4 [×12], C2×C20 [×3], C5×D4 [×10], C22×D5 [×2], C22×C10 [×4], Q8⋊6D4, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C2×Dic10, C2×C4×D5 [×2], C2×D20, D4⋊2D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×8], D4×C10 [×6], C4×Dic10, C4×D20, D4×Dic5 [×2], C20⋊2D4 [×2], Dic5⋊D4 [×4], C20⋊D4 [×2], C5×C4⋊1D4, C2×D4⋊2D5 [×2], Dic10⋊11D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D5 [×7], Q8⋊6D4, D4×D5 [×2], D4⋊2D5 [×2], C23×D5, C2×D4×D5, C2×D4⋊2D5, D4⋊6D10, Dic10⋊11D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c-1 >
►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 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 67 31 77)(22 66 32 76)(23 65 33 75)(24 64 34 74)(25 63 35 73)(26 62 36 72)(27 61 37 71)(28 80 38 70)(29 79 39 69)(30 78 40 68)(81 146 91 156)(82 145 92 155)(83 144 93 154)(84 143 94 153)(85 142 95 152)(86 141 96 151)(87 160 97 150)(88 159 98 149)(89 158 99 148)(90 157 100 147)(101 134 111 124)(102 133 112 123)(103 132 113 122)(104 131 114 121)(105 130 115 140)(106 129 116 139)(107 128 117 138)(108 127 118 137)(109 126 119 136)(110 125 120 135)
(1 82 36 116)(2 83 37 117)(3 84 38 118)(4 85 39 119)(5 86 40 120)(6 87 21 101)(7 88 22 102)(8 89 23 103)(9 90 24 104)(10 91 25 105)(11 92 26 106)(12 93 27 107)(13 94 28 108)(14 95 29 109)(15 96 30 110)(16 97 31 111)(17 98 32 112)(18 99 33 113)(19 100 34 114)(20 81 35 115)(41 156 63 130)(42 157 64 131)(43 158 65 132)(44 159 66 133)(45 160 67 134)(46 141 68 135)(47 142 69 136)(48 143 70 137)(49 144 71 138)(50 145 72 139)(51 146 73 140)(52 147 74 121)(53 148 75 122)(54 149 76 123)(55 150 77 124)(56 151 78 125)(57 152 79 126)(58 153 80 127)(59 154 61 128)(60 155 62 129)
(1 139)(2 130)(3 121)(4 132)(5 123)(6 134)(7 125)(8 136)(9 127)(10 138)(11 129)(12 140)(13 131)(14 122)(15 133)(16 124)(17 135)(18 126)(19 137)(20 128)(21 160)(22 151)(23 142)(24 153)(25 144)(26 155)(27 146)(28 157)(29 148)(30 159)(31 150)(32 141)(33 152)(34 143)(35 154)(36 145)(37 156)(38 147)(39 158)(40 149)(41 117)(42 108)(43 119)(44 110)(45 101)(46 112)(47 103)(48 114)(49 105)(50 116)(51 107)(52 118)(53 109)(54 120)(55 111)(56 102)(57 113)(58 104)(59 115)(60 106)(61 81)(62 92)(63 83)(64 94)(65 85)(66 96)(67 87)(68 98)(69 89)(70 100)(71 91)(72 82)(73 93)(74 84)(75 95)(76 86)(77 97)(78 88)(79 99)(80 90)
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), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,67,31,77)(22,66,32,76)(23,65,33,75)(24,64,34,74)(25,63,35,73)(26,62,36,72)(27,61,37,71)(28,80,38,70)(29,79,39,69)(30,78,40,68)(81,146,91,156)(82,145,92,155)(83,144,93,154)(84,143,94,153)(85,142,95,152)(86,141,96,151)(87,160,97,150)(88,159,98,149)(89,158,99,148)(90,157,100,147)(101,134,111,124)(102,133,112,123)(103,132,113,122)(104,131,114,121)(105,130,115,140)(106,129,116,139)(107,128,117,138)(108,127,118,137)(109,126,119,136)(110,125,120,135), (1,82,36,116)(2,83,37,117)(3,84,38,118)(4,85,39,119)(5,86,40,120)(6,87,21,101)(7,88,22,102)(8,89,23,103)(9,90,24,104)(10,91,25,105)(11,92,26,106)(12,93,27,107)(13,94,28,108)(14,95,29,109)(15,96,30,110)(16,97,31,111)(17,98,32,112)(18,99,33,113)(19,100,34,114)(20,81,35,115)(41,156,63,130)(42,157,64,131)(43,158,65,132)(44,159,66,133)(45,160,67,134)(46,141,68,135)(47,142,69,136)(48,143,70,137)(49,144,71,138)(50,145,72,139)(51,146,73,140)(52,147,74,121)(53,148,75,122)(54,149,76,123)(55,150,77,124)(56,151,78,125)(57,152,79,126)(58,153,80,127)(59,154,61,128)(60,155,62,129), (1,139)(2,130)(3,121)(4,132)(5,123)(6,134)(7,125)(8,136)(9,127)(10,138)(11,129)(12,140)(13,131)(14,122)(15,133)(16,124)(17,135)(18,126)(19,137)(20,128)(21,160)(22,151)(23,142)(24,153)(25,144)(26,155)(27,146)(28,157)(29,148)(30,159)(31,150)(32,141)(33,152)(34,143)(35,154)(36,145)(37,156)(38,147)(39,158)(40,149)(41,117)(42,108)(43,119)(44,110)(45,101)(46,112)(47,103)(48,114)(49,105)(50,116)(51,107)(52,118)(53,109)(54,120)(55,111)(56,102)(57,113)(58,104)(59,115)(60,106)(61,81)(62,92)(63,83)(64,94)(65,85)(66,96)(67,87)(68,98)(69,89)(70,100)(71,91)(72,82)(73,93)(74,84)(75,95)(76,86)(77,97)(78,88)(79,99)(80,90)>;
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), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,67,31,77)(22,66,32,76)(23,65,33,75)(24,64,34,74)(25,63,35,73)(26,62,36,72)(27,61,37,71)(28,80,38,70)(29,79,39,69)(30,78,40,68)(81,146,91,156)(82,145,92,155)(83,144,93,154)(84,143,94,153)(85,142,95,152)(86,141,96,151)(87,160,97,150)(88,159,98,149)(89,158,99,148)(90,157,100,147)(101,134,111,124)(102,133,112,123)(103,132,113,122)(104,131,114,121)(105,130,115,140)(106,129,116,139)(107,128,117,138)(108,127,118,137)(109,126,119,136)(110,125,120,135), (1,82,36,116)(2,83,37,117)(3,84,38,118)(4,85,39,119)(5,86,40,120)(6,87,21,101)(7,88,22,102)(8,89,23,103)(9,90,24,104)(10,91,25,105)(11,92,26,106)(12,93,27,107)(13,94,28,108)(14,95,29,109)(15,96,30,110)(16,97,31,111)(17,98,32,112)(18,99,33,113)(19,100,34,114)(20,81,35,115)(41,156,63,130)(42,157,64,131)(43,158,65,132)(44,159,66,133)(45,160,67,134)(46,141,68,135)(47,142,69,136)(48,143,70,137)(49,144,71,138)(50,145,72,139)(51,146,73,140)(52,147,74,121)(53,148,75,122)(54,149,76,123)(55,150,77,124)(56,151,78,125)(57,152,79,126)(58,153,80,127)(59,154,61,128)(60,155,62,129), (1,139)(2,130)(3,121)(4,132)(5,123)(6,134)(7,125)(8,136)(9,127)(10,138)(11,129)(12,140)(13,131)(14,122)(15,133)(16,124)(17,135)(18,126)(19,137)(20,128)(21,160)(22,151)(23,142)(24,153)(25,144)(26,155)(27,146)(28,157)(29,148)(30,159)(31,150)(32,141)(33,152)(34,143)(35,154)(36,145)(37,156)(38,147)(39,158)(40,149)(41,117)(42,108)(43,119)(44,110)(45,101)(46,112)(47,103)(48,114)(49,105)(50,116)(51,107)(52,118)(53,109)(54,120)(55,111)(56,102)(57,113)(58,104)(59,115)(60,106)(61,81)(62,92)(63,83)(64,94)(65,85)(66,96)(67,87)(68,98)(69,89)(70,100)(71,91)(72,82)(73,93)(74,84)(75,95)(76,86)(77,97)(78,88)(79,99)(80,90) );
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)], [(1,50,11,60),(2,49,12,59),(3,48,13,58),(4,47,14,57),(5,46,15,56),(6,45,16,55),(7,44,17,54),(8,43,18,53),(9,42,19,52),(10,41,20,51),(21,67,31,77),(22,66,32,76),(23,65,33,75),(24,64,34,74),(25,63,35,73),(26,62,36,72),(27,61,37,71),(28,80,38,70),(29,79,39,69),(30,78,40,68),(81,146,91,156),(82,145,92,155),(83,144,93,154),(84,143,94,153),(85,142,95,152),(86,141,96,151),(87,160,97,150),(88,159,98,149),(89,158,99,148),(90,157,100,147),(101,134,111,124),(102,133,112,123),(103,132,113,122),(104,131,114,121),(105,130,115,140),(106,129,116,139),(107,128,117,138),(108,127,118,137),(109,126,119,136),(110,125,120,135)], [(1,82,36,116),(2,83,37,117),(3,84,38,118),(4,85,39,119),(5,86,40,120),(6,87,21,101),(7,88,22,102),(8,89,23,103),(9,90,24,104),(10,91,25,105),(11,92,26,106),(12,93,27,107),(13,94,28,108),(14,95,29,109),(15,96,30,110),(16,97,31,111),(17,98,32,112),(18,99,33,113),(19,100,34,114),(20,81,35,115),(41,156,63,130),(42,157,64,131),(43,158,65,132),(44,159,66,133),(45,160,67,134),(46,141,68,135),(47,142,69,136),(48,143,70,137),(49,144,71,138),(50,145,72,139),(51,146,73,140),(52,147,74,121),(53,148,75,122),(54,149,76,123),(55,150,77,124),(56,151,78,125),(57,152,79,126),(58,153,80,127),(59,154,61,128),(60,155,62,129)], [(1,139),(2,130),(3,121),(4,132),(5,123),(6,134),(7,125),(8,136),(9,127),(10,138),(11,129),(12,140),(13,131),(14,122),(15,133),(16,124),(17,135),(18,126),(19,137),(20,128),(21,160),(22,151),(23,142),(24,153),(25,144),(26,155),(27,146),(28,157),(29,148),(30,159),(31,150),(32,141),(33,152),(34,143),(35,154),(36,145),(37,156),(38,147),(39,158),(40,149),(41,117),(42,108),(43,119),(44,110),(45,101),(46,112),(47,103),(48,114),(49,105),(50,116),(51,107),(52,118),(53,109),(54,120),(55,111),(56,102),(57,113),(58,104),(59,115),(60,106),(61,81),(62,92),(63,83),(64,94),(65,85),(66,96),(67,87),(68,98),(69,89),(70,100),(71,91),(72,82),(73,93),(74,84),(75,95),(76,86),(77,97),(78,88),(79,99),(80,90)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 40 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 5 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,36,0,0,0,0,40,6,0,0,0,0,0,0,32,32,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,16,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,36,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | 2+ (1+4) | D4×D5 | D4⋊2D5 | D4⋊6D10 |
| kernel | Dic10⋊11D4 | C4×Dic10 | C4×D20 | D4×Dic5 | C20⋊2D4 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊1D4 | C2×D4⋊2D5 | Dic10 | C4⋊1D4 | C20 | C42 | C2×D4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{11}D_4 % in TeX
G:=Group("Dic10:11D4"); // GroupNames label
G:=SmallGroup(320,1390);
// by ID
G=gap.SmallGroup(320,1390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀