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, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C22⋊Q16, Dic20, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C20.44D4, C5×C22⋊C8, C2×Dic20, C20.48D4, C22×Dic10, C22⋊Dic20
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C22≀C2, C2×Q16, C8.C22, D20, C22×D5, C22⋊Q16, Dic20, C2×D20, D4×D5, C22⋊D20, C2×Dic20, C8.D10, C22⋊Dic20
►On 160 points
Generators in S160
(1 21)(2 87)(3 23)(4 89)(5 25)(6 91)(7 27)(8 93)(9 29)(10 95)(11 31)(12 97)(13 33)(14 99)(15 35)(16 101)(17 37)(18 103)(19 39)(20 105)(22 107)(24 109)(26 111)(28 113)(30 115)(32 117)(34 119)(36 81)(38 83)(40 85)(41 143)(42 62)(43 145)(44 64)(45 147)(46 66)(47 149)(48 68)(49 151)(50 70)(51 153)(52 72)(53 155)(54 74)(55 157)(56 76)(57 159)(58 78)(59 121)(60 80)(61 123)(63 125)(65 127)(67 129)(69 131)(71 133)(73 135)(75 137)(77 139)(79 141)(82 102)(84 104)(86 106)(88 108)(90 110)(92 112)(94 114)(96 116)(98 118)(100 120)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 121)(80 122)
(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)
(1 156 21 136)(2 155 22 135)(3 154 23 134)(4 153 24 133)(5 152 25 132)(6 151 26 131)(7 150 27 130)(8 149 28 129)(9 148 29 128)(10 147 30 127)(11 146 31 126)(12 145 32 125)(13 144 33 124)(14 143 34 123)(15 142 35 122)(16 141 36 121)(17 140 37 160)(18 139 38 159)(19 138 39 158)(20 137 40 157)(41 119 61 99)(42 118 62 98)(43 117 63 97)(44 116 64 96)(45 115 65 95)(46 114 66 94)(47 113 67 93)(48 112 68 92)(49 111 69 91)(50 110 70 90)(51 109 71 89)(52 108 72 88)(53 107 73 87)(54 106 74 86)(55 105 75 85)(56 104 76 84)(57 103 77 83)(58 102 78 82)(59 101 79 81)(60 100 80 120)
G:=sub<Sym(160)| (1,21)(2,87)(3,23)(4,89)(5,25)(6,91)(7,27)(8,93)(9,29)(10,95)(11,31)(12,97)(13,33)(14,99)(15,35)(16,101)(17,37)(18,103)(19,39)(20,105)(22,107)(24,109)(26,111)(28,113)(30,115)(32,117)(34,119)(36,81)(38,83)(40,85)(41,143)(42,62)(43,145)(44,64)(45,147)(46,66)(47,149)(48,68)(49,151)(50,70)(51,153)(52,72)(53,155)(54,74)(55,157)(56,76)(57,159)(58,78)(59,121)(60,80)(61,123)(63,125)(65,127)(67,129)(69,131)(71,133)(73,135)(75,137)(77,139)(79,141)(82,102)(84,104)(86,106)(88,108)(90,110)(92,112)(94,114)(96,116)(98,118)(100,120)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,121)(80,122), (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), (1,156,21,136)(2,155,22,135)(3,154,23,134)(4,153,24,133)(5,152,25,132)(6,151,26,131)(7,150,27,130)(8,149,28,129)(9,148,29,128)(10,147,30,127)(11,146,31,126)(12,145,32,125)(13,144,33,124)(14,143,34,123)(15,142,35,122)(16,141,36,121)(17,140,37,160)(18,139,38,159)(19,138,39,158)(20,137,40,157)(41,119,61,99)(42,118,62,98)(43,117,63,97)(44,116,64,96)(45,115,65,95)(46,114,66,94)(47,113,67,93)(48,112,68,92)(49,111,69,91)(50,110,70,90)(51,109,71,89)(52,108,72,88)(53,107,73,87)(54,106,74,86)(55,105,75,85)(56,104,76,84)(57,103,77,83)(58,102,78,82)(59,101,79,81)(60,100,80,120)>;
G:=Group( (1,21)(2,87)(3,23)(4,89)(5,25)(6,91)(7,27)(8,93)(9,29)(10,95)(11,31)(12,97)(13,33)(14,99)(15,35)(16,101)(17,37)(18,103)(19,39)(20,105)(22,107)(24,109)(26,111)(28,113)(30,115)(32,117)(34,119)(36,81)(38,83)(40,85)(41,143)(42,62)(43,145)(44,64)(45,147)(46,66)(47,149)(48,68)(49,151)(50,70)(51,153)(52,72)(53,155)(54,74)(55,157)(56,76)(57,159)(58,78)(59,121)(60,80)(61,123)(63,125)(65,127)(67,129)(69,131)(71,133)(73,135)(75,137)(77,139)(79,141)(82,102)(84,104)(86,106)(88,108)(90,110)(92,112)(94,114)(96,116)(98,118)(100,120)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,121)(80,122), (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), (1,156,21,136)(2,155,22,135)(3,154,23,134)(4,153,24,133)(5,152,25,132)(6,151,26,131)(7,150,27,130)(8,149,28,129)(9,148,29,128)(10,147,30,127)(11,146,31,126)(12,145,32,125)(13,144,33,124)(14,143,34,123)(15,142,35,122)(16,141,36,121)(17,140,37,160)(18,139,38,159)(19,138,39,158)(20,137,40,157)(41,119,61,99)(42,118,62,98)(43,117,63,97)(44,116,64,96)(45,115,65,95)(46,114,66,94)(47,113,67,93)(48,112,68,92)(49,111,69,91)(50,110,70,90)(51,109,71,89)(52,108,72,88)(53,107,73,87)(54,106,74,86)(55,105,75,85)(56,104,76,84)(57,103,77,83)(58,102,78,82)(59,101,79,81)(60,100,80,120) );
G=PermutationGroup([[(1,21),(2,87),(3,23),(4,89),(5,25),(6,91),(7,27),(8,93),(9,29),(10,95),(11,31),(12,97),(13,33),(14,99),(15,35),(16,101),(17,37),(18,103),(19,39),(20,105),(22,107),(24,109),(26,111),(28,113),(30,115),(32,117),(34,119),(36,81),(38,83),(40,85),(41,143),(42,62),(43,145),(44,64),(45,147),(46,66),(47,149),(48,68),(49,151),(50,70),(51,153),(52,72),(53,155),(54,74),(55,157),(56,76),(57,159),(58,78),(59,121),(60,80),(61,123),(63,125),(65,127),(67,129),(69,131),(71,133),(73,135),(75,137),(77,139),(79,141),(82,102),(84,104),(86,106),(88,108),(90,110),(92,112),(94,114),(96,116),(98,118),(100,120),(122,142),(124,144),(126,146),(128,148),(130,150),(132,152),(134,154),(136,156),(138,158),(140,160)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,121),(80,122)], [(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)], [(1,156,21,136),(2,155,22,135),(3,154,23,134),(4,153,24,133),(5,152,25,132),(6,151,26,131),(7,150,27,130),(8,149,28,129),(9,148,29,128),(10,147,30,127),(11,146,31,126),(12,145,32,125),(13,144,33,124),(14,143,34,123),(15,142,35,122),(16,141,36,121),(17,140,37,160),(18,139,38,159),(19,138,39,158),(20,137,40,157),(41,119,61,99),(42,118,62,98),(43,117,63,97),(44,116,64,96),(45,115,65,95),(46,114,66,94),(47,113,67,93),(48,112,68,92),(49,111,69,91),(50,110,70,90),(51,109,71,89),(52,108,72,88),(53,107,73,87),(54,106,74,86),(55,105,75,85),(56,104,76,84),(57,103,77,83),(58,102,78,82),(59,101,79,81),(60,100,80,120)]])
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