metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊2Dic20, C23.41D20, Dic10.32D4, (C2×C10)⋊1Q16, (C2×C8).5D10, C4.124(D4×D5), (C2×C4).36D20, (C2×C20).47D4, C22⋊C8.4D5, C10.6(C2×Q16), (C2×Dic20)⋊3C2, C20.336(C2×D4), (C2×C40).5C22, C5⋊1(C22⋊Q16), C2.8(C2×Dic20), C10.12C22≀C2, C20.44D4⋊7C2, (C22×C4).89D10, (C22×C10).59D4, (C2×C20).749C23, C20.48D4.4C2, C22.112(C2×D20), C4⋊Dic5.16C22, C2.15(C22⋊D20), C2.15(C8.D10), C10.12(C8.C22), (C22×C20).55C22, (C22×Dic10).3C2, (C2×Dic10).15C22, (C5×C22⋊C8).6C2, (C2×C10).132(C2×D4), (C2×C4).694(C22×D5), SmallGroup(320,366)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊Dic20
G = < a,b,c,d | a2=b2=c40=1, d2=c20, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 590 in 148 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×12], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8 [×8], Dic5 [×6], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C40 [×2], Dic10 [×4], Dic10 [×8], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C22×C10, C22⋊Q16, Dic20 [×4], C10.D4, C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10, C2×Dic10 [×2], C2×Dic10 [×5], C22×Dic5, C22×C20, C20.44D4 [×2], C5×C22⋊C8, C2×Dic20 [×2], C20.48D4, C22×Dic10, C22⋊Dic20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, D20 [×2], C22×D5, C22⋊Q16, Dic20 [×2], C2×D20, D4×D5 [×2], C22⋊D20, C2×Dic20, C8.D10, C22⋊Dic20
►On 160 points
Generators in S160
(1 21)(2 72)(3 23)(4 74)(5 25)(6 76)(7 27)(8 78)(9 29)(10 80)(11 31)(12 42)(13 33)(14 44)(15 35)(16 46)(17 37)(18 48)(19 39)(20 50)(22 52)(24 54)(26 56)(28 58)(30 60)(32 62)(34 64)(36 66)(38 68)(40 70)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)(81 101)(82 123)(83 103)(84 125)(85 105)(86 127)(87 107)(88 129)(89 109)(90 131)(91 111)(92 133)(93 113)(94 135)(95 115)(96 137)(97 117)(98 139)(99 119)(100 141)(102 143)(104 145)(106 147)(108 149)(110 151)(112 153)(114 155)(116 157)(118 159)(120 121)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(81 142)(82 143)(83 144)(84 145)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 141)
(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)
(1 146 21 126)(2 145 22 125)(3 144 23 124)(4 143 24 123)(5 142 25 122)(6 141 26 121)(7 140 27 160)(8 139 28 159)(9 138 29 158)(10 137 30 157)(11 136 31 156)(12 135 32 155)(13 134 33 154)(14 133 34 153)(15 132 35 152)(16 131 36 151)(17 130 37 150)(18 129 38 149)(19 128 39 148)(20 127 40 147)(41 95 61 115)(42 94 62 114)(43 93 63 113)(44 92 64 112)(45 91 65 111)(46 90 66 110)(47 89 67 109)(48 88 68 108)(49 87 69 107)(50 86 70 106)(51 85 71 105)(52 84 72 104)(53 83 73 103)(54 82 74 102)(55 81 75 101)(56 120 76 100)(57 119 77 99)(58 118 78 98)(59 117 79 97)(60 116 80 96)
G:=sub<Sym(160)| (1,21)(2,72)(3,23)(4,74)(5,25)(6,76)(7,27)(8,78)(9,29)(10,80)(11,31)(12,42)(13,33)(14,44)(15,35)(16,46)(17,37)(18,48)(19,39)(20,50)(22,52)(24,54)(26,56)(28,58)(30,60)(32,62)(34,64)(36,66)(38,68)(40,70)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79)(81,101)(82,123)(83,103)(84,125)(85,105)(86,127)(87,107)(88,129)(89,109)(90,131)(91,111)(92,133)(93,113)(94,135)(95,115)(96,137)(97,117)(98,139)(99,119)(100,141)(102,143)(104,145)(106,147)(108,149)(110,151)(112,153)(114,155)(116,157)(118,159)(120,121)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(81,142)(82,143)(83,144)(84,145)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157)(97,158)(98,159)(99,160)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141), (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), (1,146,21,126)(2,145,22,125)(3,144,23,124)(4,143,24,123)(5,142,25,122)(6,141,26,121)(7,140,27,160)(8,139,28,159)(9,138,29,158)(10,137,30,157)(11,136,31,156)(12,135,32,155)(13,134,33,154)(14,133,34,153)(15,132,35,152)(16,131,36,151)(17,130,37,150)(18,129,38,149)(19,128,39,148)(20,127,40,147)(41,95,61,115)(42,94,62,114)(43,93,63,113)(44,92,64,112)(45,91,65,111)(46,90,66,110)(47,89,67,109)(48,88,68,108)(49,87,69,107)(50,86,70,106)(51,85,71,105)(52,84,72,104)(53,83,73,103)(54,82,74,102)(55,81,75,101)(56,120,76,100)(57,119,77,99)(58,118,78,98)(59,117,79,97)(60,116,80,96)>;
G:=Group( (1,21)(2,72)(3,23)(4,74)(5,25)(6,76)(7,27)(8,78)(9,29)(10,80)(11,31)(12,42)(13,33)(14,44)(15,35)(16,46)(17,37)(18,48)(19,39)(20,50)(22,52)(24,54)(26,56)(28,58)(30,60)(32,62)(34,64)(36,66)(38,68)(40,70)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79)(81,101)(82,123)(83,103)(84,125)(85,105)(86,127)(87,107)(88,129)(89,109)(90,131)(91,111)(92,133)(93,113)(94,135)(95,115)(96,137)(97,117)(98,139)(99,119)(100,141)(102,143)(104,145)(106,147)(108,149)(110,151)(112,153)(114,155)(116,157)(118,159)(120,121)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(81,142)(82,143)(83,144)(84,145)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157)(97,158)(98,159)(99,160)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141), (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), (1,146,21,126)(2,145,22,125)(3,144,23,124)(4,143,24,123)(5,142,25,122)(6,141,26,121)(7,140,27,160)(8,139,28,159)(9,138,29,158)(10,137,30,157)(11,136,31,156)(12,135,32,155)(13,134,33,154)(14,133,34,153)(15,132,35,152)(16,131,36,151)(17,130,37,150)(18,129,38,149)(19,128,39,148)(20,127,40,147)(41,95,61,115)(42,94,62,114)(43,93,63,113)(44,92,64,112)(45,91,65,111)(46,90,66,110)(47,89,67,109)(48,88,68,108)(49,87,69,107)(50,86,70,106)(51,85,71,105)(52,84,72,104)(53,83,73,103)(54,82,74,102)(55,81,75,101)(56,120,76,100)(57,119,77,99)(58,118,78,98)(59,117,79,97)(60,116,80,96) );
G=PermutationGroup([(1,21),(2,72),(3,23),(4,74),(5,25),(6,76),(7,27),(8,78),(9,29),(10,80),(11,31),(12,42),(13,33),(14,44),(15,35),(16,46),(17,37),(18,48),(19,39),(20,50),(22,52),(24,54),(26,56),(28,58),(30,60),(32,62),(34,64),(36,66),(38,68),(40,70),(41,61),(43,63),(45,65),(47,67),(49,69),(51,71),(53,73),(55,75),(57,77),(59,79),(81,101),(82,123),(83,103),(84,125),(85,105),(86,127),(87,107),(88,129),(89,109),(90,131),(91,111),(92,133),(93,113),(94,135),(95,115),(96,137),(97,117),(98,139),(99,119),(100,141),(102,143),(104,145),(106,147),(108,149),(110,151),(112,153),(114,155),(116,157),(118,159),(120,121),(122,142),(124,144),(126,146),(128,148),(130,150),(132,152),(134,154),(136,156),(138,158),(140,160)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(81,142),(82,143),(83,144),(84,145),(85,146),(86,147),(87,148),(88,149),(89,150),(90,151),(91,152),(92,153),(93,154),(94,155),(95,156),(96,157),(97,158),(98,159),(99,160),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,141)], [(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)], [(1,146,21,126),(2,145,22,125),(3,144,23,124),(4,143,24,123),(5,142,25,122),(6,141,26,121),(7,140,27,160),(8,139,28,159),(9,138,29,158),(10,137,30,157),(11,136,31,156),(12,135,32,155),(13,134,33,154),(14,133,34,153),(15,132,35,152),(16,131,36,151),(17,130,37,150),(18,129,38,149),(19,128,39,148),(20,127,40,147),(41,95,61,115),(42,94,62,114),(43,93,63,113),(44,92,64,112),(45,91,65,111),(46,90,66,110),(47,89,67,109),(48,88,68,108),(49,87,69,107),(50,86,70,106),(51,85,71,105),(52,84,72,104),(53,83,73,103),(54,82,74,102),(55,81,75,101),(56,120,76,100),(57,119,77,99),(58,118,78,98),(59,117,79,97),(60,116,80,96)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | Q16 | D10 | D10 | D20 | D20 | Dic20 | C8.C22 | D4×D5 | C8.D10 |
| kernel | C22⋊Dic20 | C20.44D4 | C5×C22⋊C8 | C2×Dic20 | C20.48D4 | C22×Dic10 | Dic10 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C22⋊Dic20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 39 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 24 |
| 0 | 0 | 0 | 0 | 29 | 24 |
| 3 | 5 | 0 | 0 | 0 | 0 |
| 23 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 35 |
| 0 | 0 | 0 | 0 | 21 | 17 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,0,0,29,0,0,0,0,24,24],[3,23,0,0,0,0,5,38,0,0,0,0,0,0,40,2,0,0,0,0,0,1,0,0,0,0,0,0,24,21,0,0,0,0,35,17] >;
C22⋊Dic20 in GAP, Magma, Sage, TeX
C_2^2\rtimes {\rm Dic}_{20} % in TeX
G:=Group("C2^2:Dic20"); // GroupNames label
G:=SmallGroup(320,366);
// by ID
G=gap.SmallGroup(320,366);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,226,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=c^20,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations