direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20.2C4, C20.69C24, C40.46C23, M4(2)⋊25D10, C10⋊4(C8○D4), C4○D20.8C4, (C2×D20).28C4, D20.42(C2×C4), (C2×C8).279D10, (C8×D5)⋊22C22, C8.43(C22×D5), C23.30(C4×D5), C4.68(C23×D5), C8⋊D5⋊18C22, (C10×M4(2))⋊9C2, (C2×M4(2))⋊17D5, C10.53(C23×C4), C5⋊2C8.42C23, (C4×D5).71C23, (C2×C40).238C22, C20.151(C22×C4), (C2×C20).882C23, Dic10.44(C2×C4), (C2×Dic10).29C4, C4○D20.49C22, D10.22(C22×C4), (C22×C4).374D10, (C5×M4(2))⋊25C22, Dic5.21(C22×C4), (C22×C20).264C22, C5⋊5(C2×C8○D4), (D5×C2×C8)⋊29C2, C4.123(C2×C4×D5), C22.8(C2×C4×D5), (C2×C4).87(C4×D5), C5⋊D4.7(C2×C4), (C2×C8⋊D5)⋊27C2, C2.33(D5×C22×C4), (C4×D5).60(C2×C4), (C2×C5⋊D4).23C4, (C2×C20).304(C2×C4), (C2×C5⋊2C8)⋊33C22, (C22×C5⋊2C8)⋊10C2, (C2×C4○D20).21C2, (C2×C4×D5).386C22, (C22×D5).83(C2×C4), (C2×C4).825(C22×D5), (C2×C10).126(C22×C4), (C22×C10).146(C2×C4), (C2×Dic5).117(C2×C4), SmallGroup(320,1416)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D20.2C4
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >
Subgroups: 718 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×14], M4(2) [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8 [×3], C2×M4(2), C2×M4(2) [×2], C8○D4 [×8], C2×C4○D4, C5⋊2C8 [×4], C40 [×4], Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C2×C8○D4, C8×D5 [×8], C8⋊D5 [×8], C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, D5×C2×C8 [×2], C2×C8⋊D5 [×2], D20.2C4 [×8], C22×C5⋊2C8, C10×M4(2), C2×C4○D20, C2×D20.2C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C8○D4 [×2], C23×C4, C4×D5 [×4], C22×D5 [×7], C2×C8○D4, C2×C4×D5 [×6], C23×D5, D20.2C4 [×2], D5×C22×C4, C2×D20.2C4
►On 160 points
Generators in S160
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 61)(39 62)(40 63)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(101 139)(102 140)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)
(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 98)(2 97)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 100)(20 99)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 145)(42 144)(43 143)(44 142)(45 141)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(101 123)(102 122)(103 121)(104 140)(105 139)(106 138)(107 137)(108 136)(109 135)(110 134)(111 133)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)
(1 116 58 66 11 106 48 76)(2 107 59 77 12 117 49 67)(3 118 60 68 13 108 50 78)(4 109 41 79 14 119 51 69)(5 120 42 70 15 110 52 80)(6 111 43 61 16 101 53 71)(7 102 44 72 17 112 54 62)(8 113 45 63 18 103 55 73)(9 104 46 74 19 114 56 64)(10 115 47 65 20 105 57 75)(21 82 122 152 31 92 132 142)(22 93 123 143 32 83 133 153)(23 84 124 154 33 94 134 144)(24 95 125 145 34 85 135 155)(25 86 126 156 35 96 136 146)(26 97 127 147 36 87 137 157)(27 88 128 158 37 98 138 148)(28 99 129 149 38 89 139 159)(29 90 130 160 39 100 140 150)(30 81 131 151 40 91 121 141)
G:=sub<Sym(160)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,61)(39,62)(40,63)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(101,139)(102,140)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,100)(20,99)(21,62)(22,61)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,145)(42,144)(43,143)(44,142)(45,141)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(101,123)(102,122)(103,121)(104,140)(105,139)(106,138)(107,137)(108,136)(109,135)(110,134)(111,133)(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124), (1,116,58,66,11,106,48,76)(2,107,59,77,12,117,49,67)(3,118,60,68,13,108,50,78)(4,109,41,79,14,119,51,69)(5,120,42,70,15,110,52,80)(6,111,43,61,16,101,53,71)(7,102,44,72,17,112,54,62)(8,113,45,63,18,103,55,73)(9,104,46,74,19,114,56,64)(10,115,47,65,20,105,57,75)(21,82,122,152,31,92,132,142)(22,93,123,143,32,83,133,153)(23,84,124,154,33,94,134,144)(24,95,125,145,34,85,135,155)(25,86,126,156,35,96,136,146)(26,97,127,147,36,87,137,157)(27,88,128,158,37,98,138,148)(28,99,129,149,38,89,139,159)(29,90,130,160,39,100,140,150)(30,81,131,151,40,91,121,141)>;
G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,61)(39,62)(40,63)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(101,139)(102,140)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,100)(20,99)(21,62)(22,61)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,145)(42,144)(43,143)(44,142)(45,141)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(101,123)(102,122)(103,121)(104,140)(105,139)(106,138)(107,137)(108,136)(109,135)(110,134)(111,133)(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124), (1,116,58,66,11,106,48,76)(2,107,59,77,12,117,49,67)(3,118,60,68,13,108,50,78)(4,109,41,79,14,119,51,69)(5,120,42,70,15,110,52,80)(6,111,43,61,16,101,53,71)(7,102,44,72,17,112,54,62)(8,113,45,63,18,103,55,73)(9,104,46,74,19,114,56,64)(10,115,47,65,20,105,57,75)(21,82,122,152,31,92,132,142)(22,93,123,143,32,83,133,153)(23,84,124,154,33,94,134,144)(24,95,125,145,34,85,135,155)(25,86,126,156,35,96,136,146)(26,97,127,147,36,87,137,157)(27,88,128,158,37,98,138,148)(28,99,129,149,38,89,139,159)(29,90,130,160,39,100,140,150)(30,81,131,151,40,91,121,141) );
G=PermutationGroup([(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,81),(9,82),(10,83),(11,84),(12,85),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,61),(39,62),(40,63),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,155),(50,156),(51,157),(52,158),(53,159),(54,160),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(101,139),(102,140),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138)], [(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)], [(1,98),(2,97),(3,96),(4,95),(5,94),(6,93),(7,92),(8,91),(9,90),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,81),(19,100),(20,99),(21,62),(22,61),(23,80),(24,79),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(41,145),(42,144),(43,143),(44,142),(45,141),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,154),(53,153),(54,152),(55,151),(56,150),(57,149),(58,148),(59,147),(60,146),(101,123),(102,122),(103,121),(104,140),(105,139),(106,138),(107,137),(108,136),(109,135),(110,134),(111,133),(112,132),(113,131),(114,130),(115,129),(116,128),(117,127),(118,126),(119,125),(120,124)], [(1,116,58,66,11,106,48,76),(2,107,59,77,12,117,49,67),(3,118,60,68,13,108,50,78),(4,109,41,79,14,119,51,69),(5,120,42,70,15,110,52,80),(6,111,43,61,16,101,53,71),(7,102,44,72,17,112,54,62),(8,113,45,63,18,103,55,73),(9,104,46,74,19,114,56,64),(10,115,47,65,20,105,57,75),(21,82,122,152,31,92,132,142),(22,93,123,143,32,83,133,153),(23,84,124,154,33,94,134,144),(24,95,125,145,34,85,135,155),(25,86,126,156,35,96,136,146),(26,97,127,147,36,87,137,157),(27,88,128,158,37,98,138,148),(28,99,129,149,38,89,139,159),(29,90,130,160,39,100,140,150),(30,81,131,151,40,91,121,141)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 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 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.2C4 |
| kernel | C2×D20.2C4 | D5×C2×C8 | C2×C8⋊D5 | D20.2C4 | C22×C5⋊2C8 | C10×M4(2) | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C5⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 8 | 2 | 8 | 12 | 4 | 8 |
Matrix representation of C2×D20.2C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 34 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 23 | 9 |
| 7 | 1 | 0 | 0 |
| 34 | 34 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 14 | 27 |
| 0 | 0 | 28 | 27 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[34,1,0,0,40,0,0,0,0,0,32,23,0,0,0,9],[7,34,0,0,1,34,0,0,0,0,40,0,0,0,1,1],[32,0,0,0,0,32,0,0,0,0,14,28,0,0,27,27] >;
C2×D20.2C4 in GAP, Magma, Sage, TeX
C_2\times D_{20}._2C_4 % in TeX
G:=Group("C2xD20.2C4"); // GroupNames label
G:=SmallGroup(320,1416);
// by ID
G=gap.SmallGroup(320,1416);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,297,80,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations