metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊10D4, C42.142D10, C10.912- (1+4), C4.71(D4×D5), (C4×D20)⋊44C2, C5⋊5(Q8⋊5D4), C20.64(C2×D4), C20⋊2D4⋊34C2, C4.4D4⋊11D5, D10⋊15(C4○D4), D10⋊D4⋊41C2, D10⋊3Q8⋊29C2, (C4×Dic10)⋊45C2, (C2×D4).174D10, (C2×C20).81C23, (C2×Q8).137D10, C22⋊C4.73D10, Dic5.52(C2×D4), C10.91(C22×D4), Dic5⋊4D4⋊30C2, (C4×C20).186C22, (C2×C10).221C24, C23.43(C22×D5), Dic5.5D4⋊40C2, (C2×D20).231C22, (D4×C10).156C22, C4⋊Dic5.377C22, (C22×C10).51C23, (Q8×C10).127C22, C22.242(C23×D5), Dic5.14D4⋊41C2, C23.D5.55C22, (C2×Dic5).263C23, (C4×Dic5).234C22, (C22×D5).226C23, C2.52(D4.10D10), D10⋊C4.135C22, (C2×Dic10).305C22, C10.D4.121C22, (C22×Dic5).143C22, (C2×Q8×D5)⋊11C2, C2.64(C2×D4×D5), C2.77(D5×C4○D4), (C2×D4⋊2D5)⋊19C2, C10.188(C2×C4○D4), (C5×C4.4D4)⋊13C2, (C2×C4×D5).130C22, (C2×C4).196(C22×D5), (C2×C5⋊D4).60C22, (C5×C22⋊C4).65C22, SmallGroup(320,1349)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1046 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10 [×6], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4, C4.4D4 [×2], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×4], C4×D5 [×8], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], Q8⋊5D4, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, D4⋊2D5 [×4], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic5.14D4 [×2], Dic5⋊4D4 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], C20⋊2D4, D10⋊3Q8, C5×C4.4D4, C2×D4⋊2D5, C2×Q8×D5, Dic10⋊10D4
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⋊5D4, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4, D4.10D10, Dic10⋊10D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a9, cbc-1=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)
(1 149 11 159)(2 148 12 158)(3 147 13 157)(4 146 14 156)(5 145 15 155)(6 144 16 154)(7 143 17 153)(8 142 18 152)(9 141 19 151)(10 160 20 150)(21 121 31 131)(22 140 32 130)(23 139 33 129)(24 138 34 128)(25 137 35 127)(26 136 36 126)(27 135 37 125)(28 134 38 124)(29 133 39 123)(30 132 40 122)(41 85 51 95)(42 84 52 94)(43 83 53 93)(44 82 54 92)(45 81 55 91)(46 100 56 90)(47 99 57 89)(48 98 58 88)(49 97 59 87)(50 96 60 86)(61 115 71 105)(62 114 72 104)(63 113 73 103)(64 112 74 102)(65 111 75 101)(66 110 76 120)(67 109 77 119)(68 108 78 118)(69 107 79 117)(70 106 80 116)
(1 48 112 139)(2 49 113 140)(3 50 114 121)(4 51 115 122)(5 52 116 123)(6 53 117 124)(7 54 118 125)(8 55 119 126)(9 56 120 127)(10 57 101 128)(11 58 102 129)(12 59 103 130)(13 60 104 131)(14 41 105 132)(15 42 106 133)(16 43 107 134)(17 44 108 135)(18 45 109 136)(19 46 110 137)(20 47 111 138)(21 147 86 72)(22 148 87 73)(23 149 88 74)(24 150 89 75)(25 151 90 76)(26 152 91 77)(27 153 92 78)(28 154 93 79)(29 155 94 80)(30 156 95 61)(31 157 96 62)(32 158 97 63)(33 159 98 64)(34 160 99 65)(35 141 100 66)(36 142 81 67)(37 143 82 68)(38 144 83 69)(39 145 84 70)(40 146 85 71)
(1 112)(2 101)(3 110)(4 119)(5 108)(6 117)(7 106)(8 115)(9 104)(10 113)(11 102)(12 111)(13 120)(14 109)(15 118)(16 107)(17 116)(18 105)(19 114)(20 103)(21 35)(22 24)(23 33)(25 31)(26 40)(27 29)(28 38)(30 36)(32 34)(37 39)(41 45)(42 54)(44 52)(46 50)(47 59)(49 57)(51 55)(56 60)(61 142)(62 151)(63 160)(64 149)(65 158)(66 147)(67 156)(68 145)(69 154)(70 143)(71 152)(72 141)(73 150)(74 159)(75 148)(76 157)(77 146)(78 155)(79 144)(80 153)(81 95)(82 84)(83 93)(85 91)(86 100)(87 89)(88 98)(90 96)(92 94)(97 99)(121 137)(122 126)(123 135)(125 133)(127 131)(128 140)(130 138)(132 136)
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), (1,149,11,159)(2,148,12,158)(3,147,13,157)(4,146,14,156)(5,145,15,155)(6,144,16,154)(7,143,17,153)(8,142,18,152)(9,141,19,151)(10,160,20,150)(21,121,31,131)(22,140,32,130)(23,139,33,129)(24,138,34,128)(25,137,35,127)(26,136,36,126)(27,135,37,125)(28,134,38,124)(29,133,39,123)(30,132,40,122)(41,85,51,95)(42,84,52,94)(43,83,53,93)(44,82,54,92)(45,81,55,91)(46,100,56,90)(47,99,57,89)(48,98,58,88)(49,97,59,87)(50,96,60,86)(61,115,71,105)(62,114,72,104)(63,113,73,103)(64,112,74,102)(65,111,75,101)(66,110,76,120)(67,109,77,119)(68,108,78,118)(69,107,79,117)(70,106,80,116), (1,48,112,139)(2,49,113,140)(3,50,114,121)(4,51,115,122)(5,52,116,123)(6,53,117,124)(7,54,118,125)(8,55,119,126)(9,56,120,127)(10,57,101,128)(11,58,102,129)(12,59,103,130)(13,60,104,131)(14,41,105,132)(15,42,106,133)(16,43,107,134)(17,44,108,135)(18,45,109,136)(19,46,110,137)(20,47,111,138)(21,147,86,72)(22,148,87,73)(23,149,88,74)(24,150,89,75)(25,151,90,76)(26,152,91,77)(27,153,92,78)(28,154,93,79)(29,155,94,80)(30,156,95,61)(31,157,96,62)(32,158,97,63)(33,159,98,64)(34,160,99,65)(35,141,100,66)(36,142,81,67)(37,143,82,68)(38,144,83,69)(39,145,84,70)(40,146,85,71), (1,112)(2,101)(3,110)(4,119)(5,108)(6,117)(7,106)(8,115)(9,104)(10,113)(11,102)(12,111)(13,120)(14,109)(15,118)(16,107)(17,116)(18,105)(19,114)(20,103)(21,35)(22,24)(23,33)(25,31)(26,40)(27,29)(28,38)(30,36)(32,34)(37,39)(41,45)(42,54)(44,52)(46,50)(47,59)(49,57)(51,55)(56,60)(61,142)(62,151)(63,160)(64,149)(65,158)(66,147)(67,156)(68,145)(69,154)(70,143)(71,152)(72,141)(73,150)(74,159)(75,148)(76,157)(77,146)(78,155)(79,144)(80,153)(81,95)(82,84)(83,93)(85,91)(86,100)(87,89)(88,98)(90,96)(92,94)(97,99)(121,137)(122,126)(123,135)(125,133)(127,131)(128,140)(130,138)(132,136)>;
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), (1,149,11,159)(2,148,12,158)(3,147,13,157)(4,146,14,156)(5,145,15,155)(6,144,16,154)(7,143,17,153)(8,142,18,152)(9,141,19,151)(10,160,20,150)(21,121,31,131)(22,140,32,130)(23,139,33,129)(24,138,34,128)(25,137,35,127)(26,136,36,126)(27,135,37,125)(28,134,38,124)(29,133,39,123)(30,132,40,122)(41,85,51,95)(42,84,52,94)(43,83,53,93)(44,82,54,92)(45,81,55,91)(46,100,56,90)(47,99,57,89)(48,98,58,88)(49,97,59,87)(50,96,60,86)(61,115,71,105)(62,114,72,104)(63,113,73,103)(64,112,74,102)(65,111,75,101)(66,110,76,120)(67,109,77,119)(68,108,78,118)(69,107,79,117)(70,106,80,116), (1,48,112,139)(2,49,113,140)(3,50,114,121)(4,51,115,122)(5,52,116,123)(6,53,117,124)(7,54,118,125)(8,55,119,126)(9,56,120,127)(10,57,101,128)(11,58,102,129)(12,59,103,130)(13,60,104,131)(14,41,105,132)(15,42,106,133)(16,43,107,134)(17,44,108,135)(18,45,109,136)(19,46,110,137)(20,47,111,138)(21,147,86,72)(22,148,87,73)(23,149,88,74)(24,150,89,75)(25,151,90,76)(26,152,91,77)(27,153,92,78)(28,154,93,79)(29,155,94,80)(30,156,95,61)(31,157,96,62)(32,158,97,63)(33,159,98,64)(34,160,99,65)(35,141,100,66)(36,142,81,67)(37,143,82,68)(38,144,83,69)(39,145,84,70)(40,146,85,71), (1,112)(2,101)(3,110)(4,119)(5,108)(6,117)(7,106)(8,115)(9,104)(10,113)(11,102)(12,111)(13,120)(14,109)(15,118)(16,107)(17,116)(18,105)(19,114)(20,103)(21,35)(22,24)(23,33)(25,31)(26,40)(27,29)(28,38)(30,36)(32,34)(37,39)(41,45)(42,54)(44,52)(46,50)(47,59)(49,57)(51,55)(56,60)(61,142)(62,151)(63,160)(64,149)(65,158)(66,147)(67,156)(68,145)(69,154)(70,143)(71,152)(72,141)(73,150)(74,159)(75,148)(76,157)(77,146)(78,155)(79,144)(80,153)(81,95)(82,84)(83,93)(85,91)(86,100)(87,89)(88,98)(90,96)(92,94)(97,99)(121,137)(122,126)(123,135)(125,133)(127,131)(128,140)(130,138)(132,136) );
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)], [(1,149,11,159),(2,148,12,158),(3,147,13,157),(4,146,14,156),(5,145,15,155),(6,144,16,154),(7,143,17,153),(8,142,18,152),(9,141,19,151),(10,160,20,150),(21,121,31,131),(22,140,32,130),(23,139,33,129),(24,138,34,128),(25,137,35,127),(26,136,36,126),(27,135,37,125),(28,134,38,124),(29,133,39,123),(30,132,40,122),(41,85,51,95),(42,84,52,94),(43,83,53,93),(44,82,54,92),(45,81,55,91),(46,100,56,90),(47,99,57,89),(48,98,58,88),(49,97,59,87),(50,96,60,86),(61,115,71,105),(62,114,72,104),(63,113,73,103),(64,112,74,102),(65,111,75,101),(66,110,76,120),(67,109,77,119),(68,108,78,118),(69,107,79,117),(70,106,80,116)], [(1,48,112,139),(2,49,113,140),(3,50,114,121),(4,51,115,122),(5,52,116,123),(6,53,117,124),(7,54,118,125),(8,55,119,126),(9,56,120,127),(10,57,101,128),(11,58,102,129),(12,59,103,130),(13,60,104,131),(14,41,105,132),(15,42,106,133),(16,43,107,134),(17,44,108,135),(18,45,109,136),(19,46,110,137),(20,47,111,138),(21,147,86,72),(22,148,87,73),(23,149,88,74),(24,150,89,75),(25,151,90,76),(26,152,91,77),(27,153,92,78),(28,154,93,79),(29,155,94,80),(30,156,95,61),(31,157,96,62),(32,158,97,63),(33,159,98,64),(34,160,99,65),(35,141,100,66),(36,142,81,67),(37,143,82,68),(38,144,83,69),(39,145,84,70),(40,146,85,71)], [(1,112),(2,101),(3,110),(4,119),(5,108),(6,117),(7,106),(8,115),(9,104),(10,113),(11,102),(12,111),(13,120),(14,109),(15,118),(16,107),(17,116),(18,105),(19,114),(20,103),(21,35),(22,24),(23,33),(25,31),(26,40),(27,29),(28,38),(30,36),(32,34),(37,39),(41,45),(42,54),(44,52),(46,50),(47,59),(49,57),(51,55),(56,60),(61,142),(62,151),(63,160),(64,149),(65,158),(66,147),(67,156),(68,145),(69,154),(70,143),(71,152),(72,141),(73,150),(74,159),(75,148),(76,157),(77,146),(78,155),(79,144),(80,153),(81,95),(82,84),(83,93),(85,91),(86,100),(87,89),(88,98),(90,96),(92,94),(97,99),(121,137),(122,126),(123,135),(125,133),(127,131),(128,140),(130,138),(132,136)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 7 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
G:=sub<GL(6,GF(41))| [7,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | D5×C4○D4 | D4.10D10 |
| kernel | Dic10⋊10D4 | C4×Dic10 | C4×D20 | Dic5.14D4 | Dic5⋊4D4 | D10⋊D4 | Dic5.5D4 | C20⋊2D4 | D10⋊3Q8 | C5×C4.4D4 | C2×D4⋊2D5 | C2×Q8×D5 | Dic10 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{10}D_4 % in TeX
G:=Group("Dic10:10D4"); // GroupNames label
G:=SmallGroup(320,1349);
// by ID
G=gap.SmallGroup(320,1349);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,192,12550]);
// 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^9,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀