direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20.3C4, C40.71C23, C20.67C24, (C2×C8)⋊37D10, C10⋊3(C8○D4), (C22×C8)⋊13D5, (C22×C40)⋊19C2, (C2×C40)⋊49C22, C4○D20.11C4, (C2×D20).31C4, D20.44(C2×C4), (C8×D5)⋊19C22, C4.66(C23×D5), C8.65(C22×D5), C23.39(C4×D5), C8⋊D5⋊21C22, C10.51(C23×C4), C5⋊2C8.31C23, (C4×D5).70C23, (C2×C20).880C23, C20.180(C22×C4), (C2×Dic10).32C4, Dic10.47(C2×C4), C4○D20.58C22, D10.21(C22×C4), (C22×C4).441D10, C4.Dic5⋊39C22, Dic5.20(C22×C4), (C22×C20).544C22, C5⋊4(C2×C8○D4), (D5×C2×C8)⋊25C2, C4.121(C2×C4×D5), C5⋊D4.9(C2×C4), (C2×C8⋊D5)⋊29C2, C22.11(C2×C4×D5), C2.31(D5×C22×C4), (C4×D5).59(C2×C4), (C2×C4).119(C4×D5), (C2×C5⋊D4).28C4, (C2×C20).409(C2×C4), (C2×C4○D20).29C2, (C2×C4.Dic5)⋊33C2, (C2×C4×D5).385C22, (C22×D5).82(C2×C4), (C2×C4).824(C22×D5), (C2×C10).257(C22×C4), (C22×C10).171(C2×C4), (C2×C5⋊2C8).335C22, (C2×Dic5).116(C2×C4), SmallGroup(320,1410)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 718 in 266 conjugacy classes, 151 normal (29 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 [×4], C2×C8 [×10], M4(2) [×12], 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, C22×C8 [×2], C2×M4(2) [×3], 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], C4.Dic5 [×4], C2×C40 [×2], C2×C40 [×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], D20.3C4 [×8], C2×C4.Dic5, C22×C40, C2×C4○D20, C2×D20.3C4
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, D20.3C4 [×2], D5×C22×C4, C2×D20.3C4
Generators and relations
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
►On 160 points
Generators in S160
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 121)(20 122)(21 146)(22 147)(23 148)(24 149)(25 150)(26 151)(27 152)(28 153)(29 154)(30 155)(31 156)(32 157)(33 158)(34 159)(35 160)(36 141)(37 142)(38 143)(39 144)(40 145)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 61)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 101)(99 102)(100 103)
(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)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 126)(8 125)(9 124)(10 123)(11 122)(12 121)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 155)(22 154)(23 153)(24 152)(25 151)(26 150)(27 149)(28 148)(29 147)(30 146)(31 145)(32 144)(33 143)(34 142)(35 141)(36 160)(37 159)(38 158)(39 157)(40 156)(41 63)(42 62)(43 61)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(81 117)(82 116)(83 115)(84 114)(85 113)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 120)(99 119)(100 118)
(1 57 31 116 11 47 21 106)(2 58 32 117 12 48 22 107)(3 59 33 118 13 49 23 108)(4 60 34 119 14 50 24 109)(5 41 35 120 15 51 25 110)(6 42 36 101 16 52 26 111)(7 43 37 102 17 53 27 112)(8 44 38 103 18 54 28 113)(9 45 39 104 19 55 29 114)(10 46 40 105 20 56 30 115)(61 159 96 136 71 149 86 126)(62 160 97 137 72 150 87 127)(63 141 98 138 73 151 88 128)(64 142 99 139 74 152 89 129)(65 143 100 140 75 153 90 130)(66 144 81 121 76 154 91 131)(67 145 82 122 77 155 92 132)(68 146 83 123 78 156 93 133)(69 147 84 124 79 157 94 134)(70 148 85 125 80 158 95 135)
G:=sub<Sym(160)| (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,121)(20,122)(21,146)(22,147)(23,148)(24,149)(25,150)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,141)(37,142)(38,143)(39,144)(40,145)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,61)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,101)(99,102)(100,103), (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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,155)(22,154)(23,153)(24,152)(25,151)(26,150)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,143)(34,142)(35,141)(36,160)(37,159)(38,158)(39,157)(40,156)(41,63)(42,62)(43,61)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(81,117)(82,116)(83,115)(84,114)(85,113)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,120)(99,119)(100,118), (1,57,31,116,11,47,21,106)(2,58,32,117,12,48,22,107)(3,59,33,118,13,49,23,108)(4,60,34,119,14,50,24,109)(5,41,35,120,15,51,25,110)(6,42,36,101,16,52,26,111)(7,43,37,102,17,53,27,112)(8,44,38,103,18,54,28,113)(9,45,39,104,19,55,29,114)(10,46,40,105,20,56,30,115)(61,159,96,136,71,149,86,126)(62,160,97,137,72,150,87,127)(63,141,98,138,73,151,88,128)(64,142,99,139,74,152,89,129)(65,143,100,140,75,153,90,130)(66,144,81,121,76,154,91,131)(67,145,82,122,77,155,92,132)(68,146,83,123,78,156,93,133)(69,147,84,124,79,157,94,134)(70,148,85,125,80,158,95,135)>;
G:=Group( (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,121)(20,122)(21,146)(22,147)(23,148)(24,149)(25,150)(26,151)(27,152)(28,153)(29,154)(30,155)(31,156)(32,157)(33,158)(34,159)(35,160)(36,141)(37,142)(38,143)(39,144)(40,145)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,61)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,101)(99,102)(100,103), (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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,155)(22,154)(23,153)(24,152)(25,151)(26,150)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,143)(34,142)(35,141)(36,160)(37,159)(38,158)(39,157)(40,156)(41,63)(42,62)(43,61)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(81,117)(82,116)(83,115)(84,114)(85,113)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,120)(99,119)(100,118), (1,57,31,116,11,47,21,106)(2,58,32,117,12,48,22,107)(3,59,33,118,13,49,23,108)(4,60,34,119,14,50,24,109)(5,41,35,120,15,51,25,110)(6,42,36,101,16,52,26,111)(7,43,37,102,17,53,27,112)(8,44,38,103,18,54,28,113)(9,45,39,104,19,55,29,114)(10,46,40,105,20,56,30,115)(61,159,96,136,71,149,86,126)(62,160,97,137,72,150,87,127)(63,141,98,138,73,151,88,128)(64,142,99,139,74,152,89,129)(65,143,100,140,75,153,90,130)(66,144,81,121,76,154,91,131)(67,145,82,122,77,155,92,132)(68,146,83,123,78,156,93,133)(69,147,84,124,79,157,94,134)(70,148,85,125,80,158,95,135) );
G=PermutationGroup([(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,121),(20,122),(21,146),(22,147),(23,148),(24,149),(25,150),(26,151),(27,152),(28,153),(29,154),(30,155),(31,156),(32,157),(33,158),(34,159),(35,160),(36,141),(37,142),(38,143),(39,144),(40,145),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,61),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,101),(99,102),(100,103)], [(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)], [(1,132),(2,131),(3,130),(4,129),(5,128),(6,127),(7,126),(8,125),(9,124),(10,123),(11,122),(12,121),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,155),(22,154),(23,153),(24,152),(25,151),(26,150),(27,149),(28,148),(29,147),(30,146),(31,145),(32,144),(33,143),(34,142),(35,141),(36,160),(37,159),(38,158),(39,157),(40,156),(41,63),(42,62),(43,61),(44,80),(45,79),(46,78),(47,77),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(81,117),(82,116),(83,115),(84,114),(85,113),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,120),(99,119),(100,118)], [(1,57,31,116,11,47,21,106),(2,58,32,117,12,48,22,107),(3,59,33,118,13,49,23,108),(4,60,34,119,14,50,24,109),(5,41,35,120,15,51,25,110),(6,42,36,101,16,52,26,111),(7,43,37,102,17,53,27,112),(8,44,38,103,18,54,28,113),(9,45,39,104,19,55,29,114),(10,46,40,105,20,56,30,115),(61,159,96,136,71,149,86,126),(62,160,97,137,72,150,87,127),(63,141,98,138,73,151,88,128),(64,142,99,139,74,152,89,129),(65,143,100,140,75,153,90,130),(66,144,81,121,76,154,91,131),(67,145,82,122,77,155,92,132),(68,146,83,123,78,156,93,133),(69,147,84,124,79,157,94,134),(70,148,85,125,80,158,95,135)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 36 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 6 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 36 | 0 | 0 |
| 0 | 16 | 32 | 0 | 0 |
| 0 | 0 | 0 | 6 | 40 |
| 0 | 0 | 0 | 35 | 35 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,36,32,0,0,0,0,0,6,1,0,0,0,40,0],[1,0,0,0,0,0,9,16,0,0,0,36,32,0,0,0,0,0,6,35,0,0,0,40,35],[32,0,0,0,0,0,14,0,0,0,0,0,14,0,0,0,0,0,32,0,0,0,0,0,32] >;
104 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 | 8J | 8K | 8L | 8M | ··· | 8T | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| 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 | 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 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 |
| kernel | C2×D20.3C4 | D5×C2×C8 | C2×C8⋊D5 | D20.3C4 | C2×C4.Dic5 | C22×C40 | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C5⋊D4 | C22×C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 12 | 2 | 8 | 12 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2\times D_{20}._3C_4 % in TeX
G:=Group("C2xD20.3C4"); // GroupNames label
G:=SmallGroup(320,1410);
// by ID
G=gap.SmallGroup(320,1410);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,80,102,12550]);
// 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,b*d=d*b,c*d=d*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀