metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.82D4, C22⋊1Dic20, C23.26D20, (C2×C10)⋊5Q16, C40⋊5C4⋊15C2, (C2×C4).69D20, (C22×C8).9D5, (C2×Dic20)⋊9C2, (C2×C20).357D4, (C2×C8).309D10, C20.414(C2×D4), C8.39(C5⋊D4), C5⋊4(C8.18D4), C10.11(C2×Q16), C10.19(C4○D8), C20.44D4⋊3C2, (C22×C40).15C2, C2.11(C2×Dic20), C20.229(C4○D4), C4.113(C4○D20), C2.20(C20⋊7D4), C10.72(C4⋊D4), (C2×C20).770C23, (C2×C40).381C22, C20.48D4.5C2, (C22×C4).432D10, (C22×C10).142D4, C22.133(C2×D20), C4⋊Dic5.25C22, C2.19(D40⋊7C2), (C22×C20).520C22, (C2×Dic10).20C22, C4.107(C2×C5⋊D4), (C2×C10).160(C2×D4), (C2×C4).718(C22×D5), SmallGroup(320,743)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.82D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=cac-1=a-1, cbc-1=a20b-1 >
Subgroups: 406 in 114 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C40, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C8.18D4, Dic20, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×C40, C2×Dic10, C22×C20, C20.44D4, C40⋊5C4, C2×Dic20, C20.48D4, C22×C40, C40.82D4
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C4○D8, D20, C5⋊D4, C22×D5, C8.18D4, Dic20, C2×D20, C4○D20, C2×C5⋊D4, D40⋊7C2, C2×Dic20, C20⋊7D4, C40.82D4
►On 160 points
Generators in S160
(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 47 100 145)(2 46 101 144)(3 45 102 143)(4 44 103 142)(5 43 104 141)(6 42 105 140)(7 41 106 139)(8 80 107 138)(9 79 108 137)(10 78 109 136)(11 77 110 135)(12 76 111 134)(13 75 112 133)(14 74 113 132)(15 73 114 131)(16 72 115 130)(17 71 116 129)(18 70 117 128)(19 69 118 127)(20 68 119 126)(21 67 120 125)(22 66 81 124)(23 65 82 123)(24 64 83 122)(25 63 84 121)(26 62 85 160)(27 61 86 159)(28 60 87 158)(29 59 88 157)(30 58 89 156)(31 57 90 155)(32 56 91 154)(33 55 92 153)(34 54 93 152)(35 53 94 151)(36 52 95 150)(37 51 96 149)(38 50 97 148)(39 49 98 147)(40 48 99 146)
(1 125 21 145)(2 124 22 144)(3 123 23 143)(4 122 24 142)(5 121 25 141)(6 160 26 140)(7 159 27 139)(8 158 28 138)(9 157 29 137)(10 156 30 136)(11 155 31 135)(12 154 32 134)(13 153 33 133)(14 152 34 132)(15 151 35 131)(16 150 36 130)(17 149 37 129)(18 148 38 128)(19 147 39 127)(20 146 40 126)(41 106 61 86)(42 105 62 85)(43 104 63 84)(44 103 64 83)(45 102 65 82)(46 101 66 81)(47 100 67 120)(48 99 68 119)(49 98 69 118)(50 97 70 117)(51 96 71 116)(52 95 72 115)(53 94 73 114)(54 93 74 113)(55 92 75 112)(56 91 76 111)(57 90 77 110)(58 89 78 109)(59 88 79 108)(60 87 80 107)
G:=sub<Sym(160)| (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), (1,47,100,145)(2,46,101,144)(3,45,102,143)(4,44,103,142)(5,43,104,141)(6,42,105,140)(7,41,106,139)(8,80,107,138)(9,79,108,137)(10,78,109,136)(11,77,110,135)(12,76,111,134)(13,75,112,133)(14,74,113,132)(15,73,114,131)(16,72,115,130)(17,71,116,129)(18,70,117,128)(19,69,118,127)(20,68,119,126)(21,67,120,125)(22,66,81,124)(23,65,82,123)(24,64,83,122)(25,63,84,121)(26,62,85,160)(27,61,86,159)(28,60,87,158)(29,59,88,157)(30,58,89,156)(31,57,90,155)(32,56,91,154)(33,55,92,153)(34,54,93,152)(35,53,94,151)(36,52,95,150)(37,51,96,149)(38,50,97,148)(39,49,98,147)(40,48,99,146), (1,125,21,145)(2,124,22,144)(3,123,23,143)(4,122,24,142)(5,121,25,141)(6,160,26,140)(7,159,27,139)(8,158,28,138)(9,157,29,137)(10,156,30,136)(11,155,31,135)(12,154,32,134)(13,153,33,133)(14,152,34,132)(15,151,35,131)(16,150,36,130)(17,149,37,129)(18,148,38,128)(19,147,39,127)(20,146,40,126)(41,106,61,86)(42,105,62,85)(43,104,63,84)(44,103,64,83)(45,102,65,82)(46,101,66,81)(47,100,67,120)(48,99,68,119)(49,98,69,118)(50,97,70,117)(51,96,71,116)(52,95,72,115)(53,94,73,114)(54,93,74,113)(55,92,75,112)(56,91,76,111)(57,90,77,110)(58,89,78,109)(59,88,79,108)(60,87,80,107)>;
G:=Group( (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), (1,47,100,145)(2,46,101,144)(3,45,102,143)(4,44,103,142)(5,43,104,141)(6,42,105,140)(7,41,106,139)(8,80,107,138)(9,79,108,137)(10,78,109,136)(11,77,110,135)(12,76,111,134)(13,75,112,133)(14,74,113,132)(15,73,114,131)(16,72,115,130)(17,71,116,129)(18,70,117,128)(19,69,118,127)(20,68,119,126)(21,67,120,125)(22,66,81,124)(23,65,82,123)(24,64,83,122)(25,63,84,121)(26,62,85,160)(27,61,86,159)(28,60,87,158)(29,59,88,157)(30,58,89,156)(31,57,90,155)(32,56,91,154)(33,55,92,153)(34,54,93,152)(35,53,94,151)(36,52,95,150)(37,51,96,149)(38,50,97,148)(39,49,98,147)(40,48,99,146), (1,125,21,145)(2,124,22,144)(3,123,23,143)(4,122,24,142)(5,121,25,141)(6,160,26,140)(7,159,27,139)(8,158,28,138)(9,157,29,137)(10,156,30,136)(11,155,31,135)(12,154,32,134)(13,153,33,133)(14,152,34,132)(15,151,35,131)(16,150,36,130)(17,149,37,129)(18,148,38,128)(19,147,39,127)(20,146,40,126)(41,106,61,86)(42,105,62,85)(43,104,63,84)(44,103,64,83)(45,102,65,82)(46,101,66,81)(47,100,67,120)(48,99,68,119)(49,98,69,118)(50,97,70,117)(51,96,71,116)(52,95,72,115)(53,94,73,114)(54,93,74,113)(55,92,75,112)(56,91,76,111)(57,90,77,110)(58,89,78,109)(59,88,79,108)(60,87,80,107) );
G=PermutationGroup([[(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)], [(1,47,100,145),(2,46,101,144),(3,45,102,143),(4,44,103,142),(5,43,104,141),(6,42,105,140),(7,41,106,139),(8,80,107,138),(9,79,108,137),(10,78,109,136),(11,77,110,135),(12,76,111,134),(13,75,112,133),(14,74,113,132),(15,73,114,131),(16,72,115,130),(17,71,116,129),(18,70,117,128),(19,69,118,127),(20,68,119,126),(21,67,120,125),(22,66,81,124),(23,65,82,123),(24,64,83,122),(25,63,84,121),(26,62,85,160),(27,61,86,159),(28,60,87,158),(29,59,88,157),(30,58,89,156),(31,57,90,155),(32,56,91,154),(33,55,92,153),(34,54,93,152),(35,53,94,151),(36,52,95,150),(37,51,96,149),(38,50,97,148),(39,49,98,147),(40,48,99,146)], [(1,125,21,145),(2,124,22,144),(3,123,23,143),(4,122,24,142),(5,121,25,141),(6,160,26,140),(7,159,27,139),(8,158,28,138),(9,157,29,137),(10,156,30,136),(11,155,31,135),(12,154,32,134),(13,153,33,133),(14,152,34,132),(15,151,35,131),(16,150,36,130),(17,149,37,129),(18,148,38,128),(19,147,39,127),(20,146,40,126),(41,106,61,86),(42,105,62,85),(43,104,63,84),(44,103,64,83),(45,102,65,82),(46,101,66,81),(47,100,67,120),(48,99,68,119),(49,98,69,118),(50,97,70,117),(51,96,71,116),(52,95,72,115),(53,94,73,114),(54,93,74,113),(55,92,75,112),(56,91,76,111),(57,90,77,110),(58,89,78,109),(59,88,79,108),(60,87,80,107)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | Q16 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | D20 | C4○D20 | Dic20 | D40⋊7C2 |
| kernel | C40.82D4 | C20.44D4 | C40⋊5C4 | C2×Dic20 | C20.48D4 | C22×C40 | C40 | C2×C20 | C22×C10 | C22×C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C40.82D4 ►in GL4(𝔽41) generated by
| 7 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 23 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 0 | 18 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [7,0,0,0,0,6,0,0,0,0,18,23,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,40,18,0,0,9,1],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,9,1] >;
C40.82D4 in GAP, Magma, Sage, TeX
C_{40}._{82}D_4 % in TeX
G:=Group("C40.82D4"); // GroupNames label
G:=SmallGroup(320,743);
// by ID
G=gap.SmallGroup(320,743);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,254,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^20*b^-1>;
// generators/relations