metabelian, supersoluble, monomial, A-group
Aliases: C7⋊D35, C35⋊1D7, C72⋊2D5, C5⋊(C7⋊D7), (C7×C35)⋊1C2, SmallGroup(490,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×C35 — C7⋊D35 |
Generators and relations for C7⋊D35
G = < a,b,c | a7=b35=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C7⋊D35
►On 245 points
Generators in S245
(1 187 67 119 144 73 222)(2 188 68 120 145 74 223)(3 189 69 121 146 75 224)(4 190 70 122 147 76 225)(5 191 36 123 148 77 226)(6 192 37 124 149 78 227)(7 193 38 125 150 79 228)(8 194 39 126 151 80 229)(9 195 40 127 152 81 230)(10 196 41 128 153 82 231)(11 197 42 129 154 83 232)(12 198 43 130 155 84 233)(13 199 44 131 156 85 234)(14 200 45 132 157 86 235)(15 201 46 133 158 87 236)(16 202 47 134 159 88 237)(17 203 48 135 160 89 238)(18 204 49 136 161 90 239)(19 205 50 137 162 91 240)(20 206 51 138 163 92 241)(21 207 52 139 164 93 242)(22 208 53 140 165 94 243)(23 209 54 106 166 95 244)(24 210 55 107 167 96 245)(25 176 56 108 168 97 211)(26 177 57 109 169 98 212)(27 178 58 110 170 99 213)(28 179 59 111 171 100 214)(29 180 60 112 172 101 215)(30 181 61 113 173 102 216)(31 182 62 114 174 103 217)(32 183 63 115 175 104 218)(33 184 64 116 141 105 219)(34 185 65 117 142 71 220)(35 186 66 118 143 72 221)
(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)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 105)(70 104)(106 156)(107 155)(108 154)(109 153)(110 152)(111 151)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 144)(119 143)(120 142)(121 141)(122 175)(123 174)(124 173)(125 172)(126 171)(127 170)(128 169)(129 168)(130 167)(131 166)(132 165)(133 164)(134 163)(135 162)(136 161)(137 160)(138 159)(139 158)(140 157)(176 232)(177 231)(178 230)(179 229)(180 228)(181 227)(182 226)(183 225)(184 224)(185 223)(186 222)(187 221)(188 220)(189 219)(190 218)(191 217)(192 216)(193 215)(194 214)(195 213)(196 212)(197 211)(198 245)(199 244)(200 243)(201 242)(202 241)(203 240)(204 239)(205 238)(206 237)(207 236)(208 235)(209 234)(210 233)
G:=sub<Sym(245)| (1,187,67,119,144,73,222)(2,188,68,120,145,74,223)(3,189,69,121,146,75,224)(4,190,70,122,147,76,225)(5,191,36,123,148,77,226)(6,192,37,124,149,78,227)(7,193,38,125,150,79,228)(8,194,39,126,151,80,229)(9,195,40,127,152,81,230)(10,196,41,128,153,82,231)(11,197,42,129,154,83,232)(12,198,43,130,155,84,233)(13,199,44,131,156,85,234)(14,200,45,132,157,86,235)(15,201,46,133,158,87,236)(16,202,47,134,159,88,237)(17,203,48,135,160,89,238)(18,204,49,136,161,90,239)(19,205,50,137,162,91,240)(20,206,51,138,163,92,241)(21,207,52,139,164,93,242)(22,208,53,140,165,94,243)(23,209,54,106,166,95,244)(24,210,55,107,167,96,245)(25,176,56,108,168,97,211)(26,177,57,109,169,98,212)(27,178,58,110,170,99,213)(28,179,59,111,171,100,214)(29,180,60,112,172,101,215)(30,181,61,113,173,102,216)(31,182,62,114,174,103,217)(32,183,63,115,175,104,218)(33,184,64,116,141,105,219)(34,185,65,117,142,71,220)(35,186,66,118,143,72,221), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,105)(70,104)(106,156)(107,155)(108,154)(109,153)(110,152)(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,175)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,161)(137,160)(138,159)(139,158)(140,157)(176,232)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,225)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219)(190,218)(191,217)(192,216)(193,215)(194,214)(195,213)(196,212)(197,211)(198,245)(199,244)(200,243)(201,242)(202,241)(203,240)(204,239)(205,238)(206,237)(207,236)(208,235)(209,234)(210,233)>;
G:=Group( (1,187,67,119,144,73,222)(2,188,68,120,145,74,223)(3,189,69,121,146,75,224)(4,190,70,122,147,76,225)(5,191,36,123,148,77,226)(6,192,37,124,149,78,227)(7,193,38,125,150,79,228)(8,194,39,126,151,80,229)(9,195,40,127,152,81,230)(10,196,41,128,153,82,231)(11,197,42,129,154,83,232)(12,198,43,130,155,84,233)(13,199,44,131,156,85,234)(14,200,45,132,157,86,235)(15,201,46,133,158,87,236)(16,202,47,134,159,88,237)(17,203,48,135,160,89,238)(18,204,49,136,161,90,239)(19,205,50,137,162,91,240)(20,206,51,138,163,92,241)(21,207,52,139,164,93,242)(22,208,53,140,165,94,243)(23,209,54,106,166,95,244)(24,210,55,107,167,96,245)(25,176,56,108,168,97,211)(26,177,57,109,169,98,212)(27,178,58,110,170,99,213)(28,179,59,111,171,100,214)(29,180,60,112,172,101,215)(30,181,61,113,173,102,216)(31,182,62,114,174,103,217)(32,183,63,115,175,104,218)(33,184,64,116,141,105,219)(34,185,65,117,142,71,220)(35,186,66,118,143,72,221), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,105)(70,104)(106,156)(107,155)(108,154)(109,153)(110,152)(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,175)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,161)(137,160)(138,159)(139,158)(140,157)(176,232)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,225)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219)(190,218)(191,217)(192,216)(193,215)(194,214)(195,213)(196,212)(197,211)(198,245)(199,244)(200,243)(201,242)(202,241)(203,240)(204,239)(205,238)(206,237)(207,236)(208,235)(209,234)(210,233) );
G=PermutationGroup([[(1,187,67,119,144,73,222),(2,188,68,120,145,74,223),(3,189,69,121,146,75,224),(4,190,70,122,147,76,225),(5,191,36,123,148,77,226),(6,192,37,124,149,78,227),(7,193,38,125,150,79,228),(8,194,39,126,151,80,229),(9,195,40,127,152,81,230),(10,196,41,128,153,82,231),(11,197,42,129,154,83,232),(12,198,43,130,155,84,233),(13,199,44,131,156,85,234),(14,200,45,132,157,86,235),(15,201,46,133,158,87,236),(16,202,47,134,159,88,237),(17,203,48,135,160,89,238),(18,204,49,136,161,90,239),(19,205,50,137,162,91,240),(20,206,51,138,163,92,241),(21,207,52,139,164,93,242),(22,208,53,140,165,94,243),(23,209,54,106,166,95,244),(24,210,55,107,167,96,245),(25,176,56,108,168,97,211),(26,177,57,109,169,98,212),(27,178,58,110,170,99,213),(28,179,59,111,171,100,214),(29,180,60,112,172,101,215),(30,181,61,113,173,102,216),(31,182,62,114,174,103,217),(32,183,63,115,175,104,218),(33,184,64,116,141,105,219),(34,185,65,117,142,71,220),(35,186,66,118,143,72,221)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,90),(50,89),(51,88),(52,87),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,105),(70,104),(106,156),(107,155),(108,154),(109,153),(110,152),(111,151),(112,150),(113,149),(114,148),(115,147),(116,146),(117,145),(118,144),(119,143),(120,142),(121,141),(122,175),(123,174),(124,173),(125,172),(126,171),(127,170),(128,169),(129,168),(130,167),(131,166),(132,165),(133,164),(134,163),(135,162),(136,161),(137,160),(138,159),(139,158),(140,157),(176,232),(177,231),(178,230),(179,229),(180,228),(181,227),(182,226),(183,225),(184,224),(185,223),(186,222),(187,221),(188,220),(189,219),(190,218),(191,217),(192,216),(193,215),(194,214),(195,213),(196,212),(197,211),(198,245),(199,244),(200,243),(201,242),(202,241),(203,240),(204,239),(205,238),(206,237),(207,236),(208,235),(209,234),(210,233)]])
124 conjugacy classes
| class | 1 | 2 | 5A | 5B | 7A | ··· | 7X | 35A | ··· | 35CR |
| order | 1 | 2 | 5 | 5 | 7 | ··· | 7 | 35 | ··· | 35 |
| size | 1 | 245 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D5 | D7 | D35 |
| kernel | C7⋊D35 | C7×C35 | C72 | C35 | C7 |
| # reps | 1 | 1 | 2 | 24 | 96 |
Matrix representation of C7⋊D35 ►in GL4(𝔽71) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 1 |
| 0 | 0 | 17 | 53 |
| 47 | 53 | 0 | 0 |
| 18 | 46 | 0 | 0 |
| 0 | 0 | 45 | 2 |
| 0 | 0 | 34 | 11 |
| 47 | 53 | 0 | 0 |
| 28 | 24 | 0 | 0 |
| 0 | 0 | 63 | 47 |
| 0 | 0 | 47 | 8 |
G:=sub<GL(4,GF(71))| [1,0,0,0,0,1,0,0,0,0,70,17,0,0,1,53],[47,18,0,0,53,46,0,0,0,0,45,34,0,0,2,11],[47,28,0,0,53,24,0,0,0,0,63,47,0,0,47,8] >;
C7⋊D35 in GAP, Magma, Sage, TeX
C_7\rtimes D_{35} % in TeX
G:=Group("C7:D35"); // GroupNames label
G:=SmallGroup(490,9);
// by ID
G=gap.SmallGroup(490,9);
# by ID
G:=PCGroup([4,-2,-5,-7,-7,65,722,6723]);
// Polycyclic
G:=Group<a,b,c|a^7=b^35=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export