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, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C8○D4, C8×D5, C8⋊D5, C2×C5⋊2C8, C2×C5⋊2C8, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D5×C2×C8, C2×C8⋊D5, D20.2C4, C22×C5⋊2C8, C10×M4(2), C2×C4○D20, C2×D20.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C8○D4, C23×C4, C4×D5, C22×D5, C2×C8○D4, C2×C4×D5, C23×D5, D20.2C4, D5×C22×C4, C2×D20.2C4
►On 160 points
Generators in S160
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)
(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)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 135)(22 134)(23 133)(24 132)(25 131)(26 130)(27 129)(28 128)(29 127)(30 126)(31 125)(32 124)(33 123)(34 122)(35 121)(36 140)(37 139)(38 138)(39 137)(40 136)(41 85)(42 84)(43 83)(44 82)(45 81)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 144)(62 143)(63 142)(64 141)(65 160)(66 159)(67 158)(68 157)(69 156)(70 155)(71 154)(72 153)(73 152)(74 151)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)
(1 126 84 70 11 136 94 80)(2 137 85 61 12 127 95 71)(3 128 86 72 13 138 96 62)(4 139 87 63 14 129 97 73)(5 130 88 74 15 140 98 64)(6 121 89 65 16 131 99 75)(7 132 90 76 17 122 100 66)(8 123 91 67 18 133 81 77)(9 134 92 78 19 124 82 68)(10 125 93 69 20 135 83 79)(21 53 146 113 31 43 156 103)(22 44 147 104 32 54 157 114)(23 55 148 115 33 45 158 105)(24 46 149 106 34 56 159 116)(25 57 150 117 35 47 160 107)(26 48 151 108 36 58 141 118)(27 59 152 119 37 49 142 109)(28 50 153 110 38 60 143 120)(29 41 154 101 39 51 144 111)(30 52 155 112 40 42 145 102)
G:=sub<Sym(160)| (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,140)(37,139)(38,138)(39,137)(40,136)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,144)(62,143)(63,142)(64,141)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145), (1,126,84,70,11,136,94,80)(2,137,85,61,12,127,95,71)(3,128,86,72,13,138,96,62)(4,139,87,63,14,129,97,73)(5,130,88,74,15,140,98,64)(6,121,89,65,16,131,99,75)(7,132,90,76,17,122,100,66)(8,123,91,67,18,133,81,77)(9,134,92,78,19,124,82,68)(10,125,93,69,20,135,83,79)(21,53,146,113,31,43,156,103)(22,44,147,104,32,54,157,114)(23,55,148,115,33,45,158,105)(24,46,149,106,34,56,159,116)(25,57,150,117,35,47,160,107)(26,48,151,108,36,58,141,118)(27,59,152,119,37,49,142,109)(28,50,153,110,38,60,143,120)(29,41,154,101,39,51,144,111)(30,52,155,112,40,42,145,102)>;
G:=Group( (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,140)(37,139)(38,138)(39,137)(40,136)(41,85)(42,84)(43,83)(44,82)(45,81)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,144)(62,143)(63,142)(64,141)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145), (1,126,84,70,11,136,94,80)(2,137,85,61,12,127,95,71)(3,128,86,72,13,138,96,62)(4,139,87,63,14,129,97,73)(5,130,88,74,15,140,98,64)(6,121,89,65,16,131,99,75)(7,132,90,76,17,122,100,66)(8,123,91,67,18,133,81,77)(9,134,92,78,19,124,82,68)(10,125,93,69,20,135,83,79)(21,53,146,113,31,43,156,103)(22,44,147,104,32,54,157,114)(23,55,148,115,33,45,158,105)(24,46,149,106,34,56,159,116)(25,57,150,117,35,47,160,107)(26,48,151,108,36,58,141,118)(27,59,152,119,37,49,142,109)(28,50,153,110,38,60,143,120)(29,41,154,101,39,51,144,111)(30,52,155,112,40,42,145,102) );
G=PermutationGroup([[(1,108),(2,109),(3,110),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,120),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,127),(38,128),(39,129),(40,130),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151)], [(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)], [(1,112),(2,111),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,102),(12,101),(13,120),(14,119),(15,118),(16,117),(17,116),(18,115),(19,114),(20,113),(21,135),(22,134),(23,133),(24,132),(25,131),(26,130),(27,129),(28,128),(29,127),(30,126),(31,125),(32,124),(33,123),(34,122),(35,121),(36,140),(37,139),(38,138),(39,137),(40,136),(41,85),(42,84),(43,83),(44,82),(45,81),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(61,144),(62,143),(63,142),(64,141),(65,160),(66,159),(67,158),(68,157),(69,156),(70,155),(71,154),(72,153),(73,152),(74,151),(75,150),(76,149),(77,148),(78,147),(79,146),(80,145)], [(1,126,84,70,11,136,94,80),(2,137,85,61,12,127,95,71),(3,128,86,72,13,138,96,62),(4,139,87,63,14,129,97,73),(5,130,88,74,15,140,98,64),(6,121,89,65,16,131,99,75),(7,132,90,76,17,122,100,66),(8,123,91,67,18,133,81,77),(9,134,92,78,19,124,82,68),(10,125,93,69,20,135,83,79),(21,53,146,113,31,43,156,103),(22,44,147,104,32,54,157,114),(23,55,148,115,33,45,158,105),(24,46,149,106,34,56,159,116),(25,57,150,117,35,47,160,107),(26,48,151,108,36,58,141,118),(27,59,152,119,37,49,142,109),(28,50,153,110,38,60,143,120),(29,41,154,101,39,51,144,111),(30,52,155,112,40,42,145,102)]])
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