direct product, abelian, monomial, 2-elementary
Aliases: C4×C40, SmallGroup(160,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C40 |
| C1 — C4×C40 |
| C1 — C4×C40 |
Generators and relations for C4×C40
G = < a,b | a4=b40=1, ab=ba >
Smallest permutation representation of C4×C40
►Regular action on 160 points
Generators in S160
(1 54 107 141)(2 55 108 142)(3 56 109 143)(4 57 110 144)(5 58 111 145)(6 59 112 146)(7 60 113 147)(8 61 114 148)(9 62 115 149)(10 63 116 150)(11 64 117 151)(12 65 118 152)(13 66 119 153)(14 67 120 154)(15 68 81 155)(16 69 82 156)(17 70 83 157)(18 71 84 158)(19 72 85 159)(20 73 86 160)(21 74 87 121)(22 75 88 122)(23 76 89 123)(24 77 90 124)(25 78 91 125)(26 79 92 126)(27 80 93 127)(28 41 94 128)(29 42 95 129)(30 43 96 130)(31 44 97 131)(32 45 98 132)(33 46 99 133)(34 47 100 134)(35 48 101 135)(36 49 102 136)(37 50 103 137)(38 51 104 138)(39 52 105 139)(40 53 106 140)
(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)
G:=sub<Sym(160)| (1,54,107,141)(2,55,108,142)(3,56,109,143)(4,57,110,144)(5,58,111,145)(6,59,112,146)(7,60,113,147)(8,61,114,148)(9,62,115,149)(10,63,116,150)(11,64,117,151)(12,65,118,152)(13,66,119,153)(14,67,120,154)(15,68,81,155)(16,69,82,156)(17,70,83,157)(18,71,84,158)(19,72,85,159)(20,73,86,160)(21,74,87,121)(22,75,88,122)(23,76,89,123)(24,77,90,124)(25,78,91,125)(26,79,92,126)(27,80,93,127)(28,41,94,128)(29,42,95,129)(30,43,96,130)(31,44,97,131)(32,45,98,132)(33,46,99,133)(34,47,100,134)(35,48,101,135)(36,49,102,136)(37,50,103,137)(38,51,104,138)(39,52,105,139)(40,53,106,140), (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)>;
G:=Group( (1,54,107,141)(2,55,108,142)(3,56,109,143)(4,57,110,144)(5,58,111,145)(6,59,112,146)(7,60,113,147)(8,61,114,148)(9,62,115,149)(10,63,116,150)(11,64,117,151)(12,65,118,152)(13,66,119,153)(14,67,120,154)(15,68,81,155)(16,69,82,156)(17,70,83,157)(18,71,84,158)(19,72,85,159)(20,73,86,160)(21,74,87,121)(22,75,88,122)(23,76,89,123)(24,77,90,124)(25,78,91,125)(26,79,92,126)(27,80,93,127)(28,41,94,128)(29,42,95,129)(30,43,96,130)(31,44,97,131)(32,45,98,132)(33,46,99,133)(34,47,100,134)(35,48,101,135)(36,49,102,136)(37,50,103,137)(38,51,104,138)(39,52,105,139)(40,53,106,140), (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) );
G=PermutationGroup([[(1,54,107,141),(2,55,108,142),(3,56,109,143),(4,57,110,144),(5,58,111,145),(6,59,112,146),(7,60,113,147),(8,61,114,148),(9,62,115,149),(10,63,116,150),(11,64,117,151),(12,65,118,152),(13,66,119,153),(14,67,120,154),(15,68,81,155),(16,69,82,156),(17,70,83,157),(18,71,84,158),(19,72,85,159),(20,73,86,160),(21,74,87,121),(22,75,88,122),(23,76,89,123),(24,77,90,124),(25,78,91,125),(26,79,92,126),(27,80,93,127),(28,41,94,128),(29,42,95,129),(30,43,96,130),(31,44,97,131),(32,45,98,132),(33,46,99,133),(34,47,100,134),(35,48,101,135),(36,49,102,136),(37,50,103,137),(38,51,104,138),(39,52,105,139),(40,53,106,140)], [(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)]])
C4×C40 is a maximal subgroup of
C42.279D10 C40⋊8C8 Dic10⋊3C8 C40⋊6C8 C40⋊5C8 D20⋊3C8 C40.10C8 C20⋊3C16 C40.7C8 C40⋊11Q8 C40⋊9Q8 C20.14Q16 C40⋊8Q8 C40.13Q8 C42.282D10 C8⋊6D20 D10.5C42 C42.243D10 C8⋊5D20 C4.5D40 C20⋊4D8 C8.8D20 C42.264D10 C20⋊4Q16 D40⋊17C4
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10L | 20A | ··· | 20AV | 40A | ··· | 40BL |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 |
| kernel | C4×C40 | C4×C20 | C2×C40 | C40 | C2×C20 | C4×C8 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 4 | 4 | 16 | 4 | 8 | 32 | 16 | 64 |
Matrix representation of C4×C40 ►in GL2(𝔽41) generated by
| 32 | 0 |
| 0 | 40 |
| 32 | 0 |
| 0 | 24 |
G:=sub<GL(2,GF(41))| [32,0,0,40],[32,0,0,24] >;
C4×C40 in GAP, Magma, Sage, TeX
C_4\times C_{40} % in TeX
G:=Group("C4xC40"); // GroupNames label
G:=SmallGroup(160,46);
// by ID
G=gap.SmallGroup(160,46);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,120,247,117]);
// Polycyclic
G:=Group<a,b|a^4=b^40=1,a*b=b*a>;
// generators/relations
Export