direct product, abelian, monomial, 11-elementary
Aliases: C11×C22, SmallGroup(242,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C11×C22 |
| C1 — C11×C22 |
| C1 — C11×C22 |
Generators and relations for C11×C22
G = < a,b | a11=b22=1, ab=ba >
Smallest permutation representation of C11×C22
►Regular action on 242 points
Generators in S242
(1 65 127 36 209 179 73 164 91 233 134)(2 66 128 37 210 180 74 165 92 234 135)(3 45 129 38 211 181 75 166 93 235 136)(4 46 130 39 212 182 76 167 94 236 137)(5 47 131 40 213 183 77 168 95 237 138)(6 48 132 41 214 184 78 169 96 238 139)(7 49 111 42 215 185 79 170 97 239 140)(8 50 112 43 216 186 80 171 98 240 141)(9 51 113 44 217 187 81 172 99 241 142)(10 52 114 23 218 188 82 173 100 242 143)(11 53 115 24 219 189 83 174 101 221 144)(12 54 116 25 220 190 84 175 102 222 145)(13 55 117 26 199 191 85 176 103 223 146)(14 56 118 27 200 192 86 155 104 224 147)(15 57 119 28 201 193 87 156 105 225 148)(16 58 120 29 202 194 88 157 106 226 149)(17 59 121 30 203 195 67 158 107 227 150)(18 60 122 31 204 196 68 159 108 228 151)(19 61 123 32 205 197 69 160 109 229 152)(20 62 124 33 206 198 70 161 110 230 153)(21 63 125 34 207 177 71 162 89 231 154)(22 64 126 35 208 178 72 163 90 232 133)
(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)(155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)(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)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242)
G:=sub<Sym(242)| (1,65,127,36,209,179,73,164,91,233,134)(2,66,128,37,210,180,74,165,92,234,135)(3,45,129,38,211,181,75,166,93,235,136)(4,46,130,39,212,182,76,167,94,236,137)(5,47,131,40,213,183,77,168,95,237,138)(6,48,132,41,214,184,78,169,96,238,139)(7,49,111,42,215,185,79,170,97,239,140)(8,50,112,43,216,186,80,171,98,240,141)(9,51,113,44,217,187,81,172,99,241,142)(10,52,114,23,218,188,82,173,100,242,143)(11,53,115,24,219,189,83,174,101,221,144)(12,54,116,25,220,190,84,175,102,222,145)(13,55,117,26,199,191,85,176,103,223,146)(14,56,118,27,200,192,86,155,104,224,147)(15,57,119,28,201,193,87,156,105,225,148)(16,58,120,29,202,194,88,157,106,226,149)(17,59,121,30,203,195,67,158,107,227,150)(18,60,122,31,204,196,68,159,108,228,151)(19,61,123,32,205,197,69,160,109,229,152)(20,62,124,33,206,198,70,161,110,230,153)(21,63,125,34,207,177,71,162,89,231,154)(22,64,126,35,208,178,72,163,90,232,133), (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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)(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)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242)>;
G:=Group( (1,65,127,36,209,179,73,164,91,233,134)(2,66,128,37,210,180,74,165,92,234,135)(3,45,129,38,211,181,75,166,93,235,136)(4,46,130,39,212,182,76,167,94,236,137)(5,47,131,40,213,183,77,168,95,237,138)(6,48,132,41,214,184,78,169,96,238,139)(7,49,111,42,215,185,79,170,97,239,140)(8,50,112,43,216,186,80,171,98,240,141)(9,51,113,44,217,187,81,172,99,241,142)(10,52,114,23,218,188,82,173,100,242,143)(11,53,115,24,219,189,83,174,101,221,144)(12,54,116,25,220,190,84,175,102,222,145)(13,55,117,26,199,191,85,176,103,223,146)(14,56,118,27,200,192,86,155,104,224,147)(15,57,119,28,201,193,87,156,105,225,148)(16,58,120,29,202,194,88,157,106,226,149)(17,59,121,30,203,195,67,158,107,227,150)(18,60,122,31,204,196,68,159,108,228,151)(19,61,123,32,205,197,69,160,109,229,152)(20,62,124,33,206,198,70,161,110,230,153)(21,63,125,34,207,177,71,162,89,231,154)(22,64,126,35,208,178,72,163,90,232,133), (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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)(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)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242) );
G=PermutationGroup([[(1,65,127,36,209,179,73,164,91,233,134),(2,66,128,37,210,180,74,165,92,234,135),(3,45,129,38,211,181,75,166,93,235,136),(4,46,130,39,212,182,76,167,94,236,137),(5,47,131,40,213,183,77,168,95,237,138),(6,48,132,41,214,184,78,169,96,238,139),(7,49,111,42,215,185,79,170,97,239,140),(8,50,112,43,216,186,80,171,98,240,141),(9,51,113,44,217,187,81,172,99,241,142),(10,52,114,23,218,188,82,173,100,242,143),(11,53,115,24,219,189,83,174,101,221,144),(12,54,116,25,220,190,84,175,102,222,145),(13,55,117,26,199,191,85,176,103,223,146),(14,56,118,27,200,192,86,155,104,224,147),(15,57,119,28,201,193,87,156,105,225,148),(16,58,120,29,202,194,88,157,106,226,149),(17,59,121,30,203,195,67,158,107,227,150),(18,60,122,31,204,196,68,159,108,228,151),(19,61,123,32,205,197,69,160,109,229,152),(20,62,124,33,206,198,70,161,110,230,153),(21,63,125,34,207,177,71,162,89,231,154),(22,64,126,35,208,178,72,163,90,232,133)], [(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),(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176),(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),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242)]])
C11×C22 is a maximal subgroup of
C11⋊Dic11
242 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11DP | 22A | ··· | 22DP |
| order | 1 | 2 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
242 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C11 | C22 |
| kernel | C11×C22 | C112 | C22 | C11 |
| # reps | 1 | 1 | 120 | 120 |
Matrix representation of C11×C22 ►in GL2(𝔽23) generated by
| 18 | 0 |
| 0 | 2 |
| 6 | 0 |
| 0 | 11 |
G:=sub<GL(2,GF(23))| [18,0,0,2],[6,0,0,11] >;
C11×C22 in GAP, Magma, Sage, TeX
C_{11}\times C_{22} % in TeX
G:=Group("C11xC22"); // GroupNames label
G:=SmallGroup(242,5);
// by ID
G=gap.SmallGroup(242,5);
# by ID
G:=PCGroup([3,-2,-11,-11]);
// Polycyclic
G:=Group<a,b|a^11=b^22=1,a*b=b*a>;
// generators/relations
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