direct product, abelian, monomial, 2-elementary
Aliases: C4×C56, SmallGroup(224,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C56 |
| C1 — C4×C56 |
| C1 — C4×C56 |
Generators and relations for C4×C56
G = < a,b | a4=b56=1, ab=ba >
Smallest permutation representation of C4×C56
►Regular action on 224 points
Generators in S224
(1 196 74 135)(2 197 75 136)(3 198 76 137)(4 199 77 138)(5 200 78 139)(6 201 79 140)(7 202 80 141)(8 203 81 142)(9 204 82 143)(10 205 83 144)(11 206 84 145)(12 207 85 146)(13 208 86 147)(14 209 87 148)(15 210 88 149)(16 211 89 150)(17 212 90 151)(18 213 91 152)(19 214 92 153)(20 215 93 154)(21 216 94 155)(22 217 95 156)(23 218 96 157)(24 219 97 158)(25 220 98 159)(26 221 99 160)(27 222 100 161)(28 223 101 162)(29 224 102 163)(30 169 103 164)(31 170 104 165)(32 171 105 166)(33 172 106 167)(34 173 107 168)(35 174 108 113)(36 175 109 114)(37 176 110 115)(38 177 111 116)(39 178 112 117)(40 179 57 118)(41 180 58 119)(42 181 59 120)(43 182 60 121)(44 183 61 122)(45 184 62 123)(46 185 63 124)(47 186 64 125)(48 187 65 126)(49 188 66 127)(50 189 67 128)(51 190 68 129)(52 191 69 130)(53 192 70 131)(54 193 71 132)(55 194 72 133)(56 195 73 134)
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
G:=sub<Sym(224)| (1,196,74,135)(2,197,75,136)(3,198,76,137)(4,199,77,138)(5,200,78,139)(6,201,79,140)(7,202,80,141)(8,203,81,142)(9,204,82,143)(10,205,83,144)(11,206,84,145)(12,207,85,146)(13,208,86,147)(14,209,87,148)(15,210,88,149)(16,211,89,150)(17,212,90,151)(18,213,91,152)(19,214,92,153)(20,215,93,154)(21,216,94,155)(22,217,95,156)(23,218,96,157)(24,219,97,158)(25,220,98,159)(26,221,99,160)(27,222,100,161)(28,223,101,162)(29,224,102,163)(30,169,103,164)(31,170,104,165)(32,171,105,166)(33,172,106,167)(34,173,107,168)(35,174,108,113)(36,175,109,114)(37,176,110,115)(38,177,111,116)(39,178,112,117)(40,179,57,118)(41,180,58,119)(42,181,59,120)(43,182,60,121)(44,183,61,122)(45,184,62,123)(46,185,63,124)(47,186,64,125)(48,187,65,126)(49,188,66,127)(50,189,67,128)(51,190,68,129)(52,191,69,130)(53,192,70,131)(54,193,71,132)(55,194,72,133)(56,195,73,134), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)>;
G:=Group( (1,196,74,135)(2,197,75,136)(3,198,76,137)(4,199,77,138)(5,200,78,139)(6,201,79,140)(7,202,80,141)(8,203,81,142)(9,204,82,143)(10,205,83,144)(11,206,84,145)(12,207,85,146)(13,208,86,147)(14,209,87,148)(15,210,88,149)(16,211,89,150)(17,212,90,151)(18,213,91,152)(19,214,92,153)(20,215,93,154)(21,216,94,155)(22,217,95,156)(23,218,96,157)(24,219,97,158)(25,220,98,159)(26,221,99,160)(27,222,100,161)(28,223,101,162)(29,224,102,163)(30,169,103,164)(31,170,104,165)(32,171,105,166)(33,172,106,167)(34,173,107,168)(35,174,108,113)(36,175,109,114)(37,176,110,115)(38,177,111,116)(39,178,112,117)(40,179,57,118)(41,180,58,119)(42,181,59,120)(43,182,60,121)(44,183,61,122)(45,184,62,123)(46,185,63,124)(47,186,64,125)(48,187,65,126)(49,188,66,127)(50,189,67,128)(51,190,68,129)(52,191,69,130)(53,192,70,131)(54,193,71,132)(55,194,72,133)(56,195,73,134), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224) );
G=PermutationGroup([[(1,196,74,135),(2,197,75,136),(3,198,76,137),(4,199,77,138),(5,200,78,139),(6,201,79,140),(7,202,80,141),(8,203,81,142),(9,204,82,143),(10,205,83,144),(11,206,84,145),(12,207,85,146),(13,208,86,147),(14,209,87,148),(15,210,88,149),(16,211,89,150),(17,212,90,151),(18,213,91,152),(19,214,92,153),(20,215,93,154),(21,216,94,155),(22,217,95,156),(23,218,96,157),(24,219,97,158),(25,220,98,159),(26,221,99,160),(27,222,100,161),(28,223,101,162),(29,224,102,163),(30,169,103,164),(31,170,104,165),(32,171,105,166),(33,172,106,167),(34,173,107,168),(35,174,108,113),(36,175,109,114),(37,176,110,115),(38,177,111,116),(39,178,112,117),(40,179,57,118),(41,180,58,119),(42,181,59,120),(43,182,60,121),(44,183,61,122),(45,184,62,123),(46,185,63,124),(47,186,64,125),(48,187,65,126),(49,188,66,127),(50,189,67,128),(51,190,68,129),(52,191,69,130),(53,192,70,131),(54,193,71,132),(55,194,72,133),(56,195,73,134)], [(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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)]])
C4×C56 is a maximal subgroup of
C42.279D14 C56⋊C8 C4.8Dic28 C56⋊2C8 C56⋊1C8 C4.17D56 C56.C8 C28⋊C16 C56.16Q8 C56⋊11Q8 C56⋊9Q8 C28.14Q16 C56⋊8Q8 C56.13Q8 C42.282D14 C8⋊6D28 D14.C42 C42.243D14 C8⋊5D28 C4.5D56 C28⋊4D8 C8.8D28 C42.264D14 C28⋊4Q16 D56⋊11C4
224 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8P | 14A | ··· | 14R | 28A | ··· | 28BT | 56A | ··· | 56CR |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
224 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C28 | C28 | C56 |
| kernel | C4×C56 | C4×C28 | C2×C56 | C56 | C2×C28 | C4×C8 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 4 | 6 | 16 | 6 | 12 | 48 | 24 | 96 |
Matrix representation of C4×C56 ►in GL2(𝔽113) generated by
| 15 | 0 |
| 0 | 15 |
| 85 | 0 |
| 0 | 25 |
G:=sub<GL(2,GF(113))| [15,0,0,15],[85,0,0,25] >;
C4×C56 in GAP, Magma, Sage, TeX
C_4\times C_{56} % in TeX
G:=Group("C4xC56"); // GroupNames label
G:=SmallGroup(224,45);
// by ID
G=gap.SmallGroup(224,45);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,168,343,117]);
// Polycyclic
G:=Group<a,b|a^4=b^56=1,a*b=b*a>;
// generators/relations
Export