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, C23.30(C4×D5), C4.68(C23×D5), C8.43(C22×D5), C8⋊D5⋊18C22, (C2×M4(2))⋊17D5, (C10×M4(2))⋊9C2, C10.53(C23×C4), C5⋊2C8.42C23, (C4×D5).71C23, (C2×C20).882C23, C20.151(C22×C4), (C2×C40).238C22, 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), (C22×C10).146(C2×C4), (C2×C10).126(C22×C4), (C2×Dic5).117(C2×C4), SmallGroup(320,1416)
Series: Derived ►Chief ►Lower central ►Upper central
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
Generators and relations
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 >
►On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 81)(17 82)(18 83)(19 84)(20 85)(21 159)(22 160)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 117)(42 118)(43 119)(44 120)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)
(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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 143)(22 142)(23 141)(24 160)(25 159)(26 158)(27 157)(28 156)(29 155)(30 154)(31 153)(32 152)(33 151)(34 150)(35 149)(36 148)(37 147)(38 146)(39 145)(40 144)(41 103)(42 102)(43 101)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 105)(60 104)(61 124)(62 123)(63 122)(64 121)(65 140)(66 139)(67 138)(68 137)(69 136)(70 135)(71 134)(72 133)(73 132)(74 131)(75 130)(76 129)(77 128)(78 127)(79 126)(80 125)
(1 57 137 154 11 47 127 144)(2 48 138 145 12 58 128 155)(3 59 139 156 13 49 129 146)(4 50 140 147 14 60 130 157)(5 41 121 158 15 51 131 148)(6 52 122 149 16 42 132 159)(7 43 123 160 17 53 133 150)(8 54 124 151 18 44 134 141)(9 45 125 142 19 55 135 152)(10 56 126 153 20 46 136 143)(21 91 108 69 31 81 118 79)(22 82 109 80 32 92 119 70)(23 93 110 71 33 83 120 61)(24 84 111 62 34 94 101 72)(25 95 112 73 35 85 102 63)(26 86 113 64 36 96 103 74)(27 97 114 75 37 87 104 65)(28 88 115 66 38 98 105 76)(29 99 116 77 39 89 106 67)(30 90 117 68 40 100 107 78)
G:=sub<Sym(160)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,81)(17,82)(18,83)(19,84)(20,85)(21,159)(22,160)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,117)(42,118)(43,119)(44,120)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133), (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,143)(22,142)(23,141)(24,160)(25,159)(26,158)(27,157)(28,156)(29,155)(30,154)(31,153)(32,152)(33,151)(34,150)(35,149)(36,148)(37,147)(38,146)(39,145)(40,144)(41,103)(42,102)(43,101)(44,120)(45,119)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,108)(57,107)(58,106)(59,105)(60,104)(61,124)(62,123)(63,122)(64,121)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125), (1,57,137,154,11,47,127,144)(2,48,138,145,12,58,128,155)(3,59,139,156,13,49,129,146)(4,50,140,147,14,60,130,157)(5,41,121,158,15,51,131,148)(6,52,122,149,16,42,132,159)(7,43,123,160,17,53,133,150)(8,54,124,151,18,44,134,141)(9,45,125,142,19,55,135,152)(10,56,126,153,20,46,136,143)(21,91,108,69,31,81,118,79)(22,82,109,80,32,92,119,70)(23,93,110,71,33,83,120,61)(24,84,111,62,34,94,101,72)(25,95,112,73,35,85,102,63)(26,86,113,64,36,96,103,74)(27,97,114,75,37,87,104,65)(28,88,115,66,38,98,105,76)(29,99,116,77,39,89,106,67)(30,90,117,68,40,100,107,78)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,81)(17,82)(18,83)(19,84)(20,85)(21,159)(22,160)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,117)(42,118)(43,119)(44,120)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133), (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,143)(22,142)(23,141)(24,160)(25,159)(26,158)(27,157)(28,156)(29,155)(30,154)(31,153)(32,152)(33,151)(34,150)(35,149)(36,148)(37,147)(38,146)(39,145)(40,144)(41,103)(42,102)(43,101)(44,120)(45,119)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,108)(57,107)(58,106)(59,105)(60,104)(61,124)(62,123)(63,122)(64,121)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125), (1,57,137,154,11,47,127,144)(2,48,138,145,12,58,128,155)(3,59,139,156,13,49,129,146)(4,50,140,147,14,60,130,157)(5,41,121,158,15,51,131,148)(6,52,122,149,16,42,132,159)(7,43,123,160,17,53,133,150)(8,54,124,151,18,44,134,141)(9,45,125,142,19,55,135,152)(10,56,126,153,20,46,136,143)(21,91,108,69,31,81,118,79)(22,82,109,80,32,92,119,70)(23,93,110,71,33,83,120,61)(24,84,111,62,34,94,101,72)(25,95,112,73,35,85,102,63)(26,86,113,64,36,96,103,74)(27,97,114,75,37,87,104,65)(28,88,115,66,38,98,105,76)(29,99,116,77,39,89,106,67)(30,90,117,68,40,100,107,78) );
G=PermutationGroup([(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,81),(17,82),(18,83),(19,84),(20,85),(21,159),(22,160),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,117),(42,118),(43,119),(44,120),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,129),(77,130),(78,131),(79,132),(80,133)], [(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)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,84),(8,83),(9,82),(10,81),(11,100),(12,99),(13,98),(14,97),(15,96),(16,95),(17,94),(18,93),(19,92),(20,91),(21,143),(22,142),(23,141),(24,160),(25,159),(26,158),(27,157),(28,156),(29,155),(30,154),(31,153),(32,152),(33,151),(34,150),(35,149),(36,148),(37,147),(38,146),(39,145),(40,144),(41,103),(42,102),(43,101),(44,120),(45,119),(46,118),(47,117),(48,116),(49,115),(50,114),(51,113),(52,112),(53,111),(54,110),(55,109),(56,108),(57,107),(58,106),(59,105),(60,104),(61,124),(62,123),(63,122),(64,121),(65,140),(66,139),(67,138),(68,137),(69,136),(70,135),(71,134),(72,133),(73,132),(74,131),(75,130),(76,129),(77,128),(78,127),(79,126),(80,125)], [(1,57,137,154,11,47,127,144),(2,48,138,145,12,58,128,155),(3,59,139,156,13,49,129,146),(4,50,140,147,14,60,130,157),(5,41,121,158,15,51,131,148),(6,52,122,149,16,42,132,159),(7,43,123,160,17,53,133,150),(8,54,124,151,18,44,134,141),(9,45,125,142,19,55,135,152),(10,56,126,153,20,46,136,143),(21,91,108,69,31,81,118,79),(22,82,109,80,32,92,119,70),(23,93,110,71,33,83,120,61),(24,84,111,62,34,94,101,72),(25,95,112,73,35,85,102,63),(26,86,113,64,36,96,103,74),(27,97,114,75,37,87,104,65),(28,88,115,66,38,98,105,76),(29,99,116,77,39,89,106,67),(30,90,117,68,40,100,107,78)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀