direct product, abelian, monomial, 7-elementary
Aliases: C7×C35, SmallGroup(245,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C35 |
| C1 — C7×C35 |
| C1 — C7×C35 |
Generators and relations for C7×C35
G = < a,b | a7=b35=1, ab=ba >
Smallest permutation representation of C7×C35
►Regular action on 245 points
Generators in S245
(1 43 86 113 214 161 178)(2 44 87 114 215 162 179)(3 45 88 115 216 163 180)(4 46 89 116 217 164 181)(5 47 90 117 218 165 182)(6 48 91 118 219 166 183)(7 49 92 119 220 167 184)(8 50 93 120 221 168 185)(9 51 94 121 222 169 186)(10 52 95 122 223 170 187)(11 53 96 123 224 171 188)(12 54 97 124 225 172 189)(13 55 98 125 226 173 190)(14 56 99 126 227 174 191)(15 57 100 127 228 175 192)(16 58 101 128 229 141 193)(17 59 102 129 230 142 194)(18 60 103 130 231 143 195)(19 61 104 131 232 144 196)(20 62 105 132 233 145 197)(21 63 71 133 234 146 198)(22 64 72 134 235 147 199)(23 65 73 135 236 148 200)(24 66 74 136 237 149 201)(25 67 75 137 238 150 202)(26 68 76 138 239 151 203)(27 69 77 139 240 152 204)(28 70 78 140 241 153 205)(29 36 79 106 242 154 206)(30 37 80 107 243 155 207)(31 38 81 108 244 156 208)(32 39 82 109 245 157 209)(33 40 83 110 211 158 210)(34 41 84 111 212 159 176)(35 42 85 112 213 160 177)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245)
G:=sub<Sym(245)| (1,43,86,113,214,161,178)(2,44,87,114,215,162,179)(3,45,88,115,216,163,180)(4,46,89,116,217,164,181)(5,47,90,117,218,165,182)(6,48,91,118,219,166,183)(7,49,92,119,220,167,184)(8,50,93,120,221,168,185)(9,51,94,121,222,169,186)(10,52,95,122,223,170,187)(11,53,96,123,224,171,188)(12,54,97,124,225,172,189)(13,55,98,125,226,173,190)(14,56,99,126,227,174,191)(15,57,100,127,228,175,192)(16,58,101,128,229,141,193)(17,59,102,129,230,142,194)(18,60,103,130,231,143,195)(19,61,104,131,232,144,196)(20,62,105,132,233,145,197)(21,63,71,133,234,146,198)(22,64,72,134,235,147,199)(23,65,73,135,236,148,200)(24,66,74,136,237,149,201)(25,67,75,137,238,150,202)(26,68,76,138,239,151,203)(27,69,77,139,240,152,204)(28,70,78,140,241,153,205)(29,36,79,106,242,154,206)(30,37,80,107,243,155,207)(31,38,81,108,244,156,208)(32,39,82,109,245,157,209)(33,40,83,110,211,158,210)(34,41,84,111,212,159,176)(35,42,85,112,213,160,177), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245)>;
G:=Group( (1,43,86,113,214,161,178)(2,44,87,114,215,162,179)(3,45,88,115,216,163,180)(4,46,89,116,217,164,181)(5,47,90,117,218,165,182)(6,48,91,118,219,166,183)(7,49,92,119,220,167,184)(8,50,93,120,221,168,185)(9,51,94,121,222,169,186)(10,52,95,122,223,170,187)(11,53,96,123,224,171,188)(12,54,97,124,225,172,189)(13,55,98,125,226,173,190)(14,56,99,126,227,174,191)(15,57,100,127,228,175,192)(16,58,101,128,229,141,193)(17,59,102,129,230,142,194)(18,60,103,130,231,143,195)(19,61,104,131,232,144,196)(20,62,105,132,233,145,197)(21,63,71,133,234,146,198)(22,64,72,134,235,147,199)(23,65,73,135,236,148,200)(24,66,74,136,237,149,201)(25,67,75,137,238,150,202)(26,68,76,138,239,151,203)(27,69,77,139,240,152,204)(28,70,78,140,241,153,205)(29,36,79,106,242,154,206)(30,37,80,107,243,155,207)(31,38,81,108,244,156,208)(32,39,82,109,245,157,209)(33,40,83,110,211,158,210)(34,41,84,111,212,159,176)(35,42,85,112,213,160,177), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245) );
G=PermutationGroup([[(1,43,86,113,214,161,178),(2,44,87,114,215,162,179),(3,45,88,115,216,163,180),(4,46,89,116,217,164,181),(5,47,90,117,218,165,182),(6,48,91,118,219,166,183),(7,49,92,119,220,167,184),(8,50,93,120,221,168,185),(9,51,94,121,222,169,186),(10,52,95,122,223,170,187),(11,53,96,123,224,171,188),(12,54,97,124,225,172,189),(13,55,98,125,226,173,190),(14,56,99,126,227,174,191),(15,57,100,127,228,175,192),(16,58,101,128,229,141,193),(17,59,102,129,230,142,194),(18,60,103,130,231,143,195),(19,61,104,131,232,144,196),(20,62,105,132,233,145,197),(21,63,71,133,234,146,198),(22,64,72,134,235,147,199),(23,65,73,135,236,148,200),(24,66,74,136,237,149,201),(25,67,75,137,238,150,202),(26,68,76,138,239,151,203),(27,69,77,139,240,152,204),(28,70,78,140,241,153,205),(29,36,79,106,242,154,206),(30,37,80,107,243,155,207),(31,38,81,108,244,156,208),(32,39,82,109,245,157,209),(33,40,83,110,211,158,210),(34,41,84,111,212,159,176),(35,42,85,112,213,160,177)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245)]])
C7×C35 is a maximal subgroup of
C7⋊D35
245 conjugacy classes
| class | 1 | 5A | 5B | 5C | 5D | 7A | ··· | 7AV | 35A | ··· | 35GJ |
| order | 1 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 35 | ··· | 35 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
245 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C7 | C35 |
| kernel | C7×C35 | C72 | C35 | C7 |
| # reps | 1 | 4 | 48 | 192 |
Matrix representation of C7×C35 ►in GL2(𝔽71) generated by
| 30 | 0 |
| 0 | 48 |
| 60 | 0 |
| 0 | 8 |
G:=sub<GL(2,GF(71))| [30,0,0,48],[60,0,0,8] >;
C7×C35 in GAP, Magma, Sage, TeX
C_7\times C_{35} % in TeX
G:=Group("C7xC35"); // GroupNames label
G:=SmallGroup(245,2);
// by ID
G=gap.SmallGroup(245,2);
# by ID
G:=PCGroup([3,-5,-7,-7]);
// Polycyclic
G:=Group<a,b|a^7=b^35=1,a*b=b*a>;
// generators/relations
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