direct product, abelian, monomial, 5-elementary
Aliases: C52×C10, SmallGroup(250,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C52×C10 |
| C1 — C52×C10 |
| C1 — C52×C10 |
Generators and relations for C52×C10
G = < a,b,c | a5=b5=c10=1, ab=ba, ac=ca, bc=cb >
Subgroups: 128, all normal (4 characteristic)
C1, C2, C5, C10, C52, C5×C10, C53, C52×C10
Quotients: C1, C2, C5, C10, C52, C5×C10, C53, C52×C10
►Regular action on 250 points
Generators in S250
(1 233 167 95 138)(2 234 168 96 139)(3 235 169 97 140)(4 236 170 98 131)(5 237 161 99 132)(6 238 162 100 133)(7 239 163 91 134)(8 240 164 92 135)(9 231 165 93 136)(10 232 166 94 137)(11 188 116 159 204)(12 189 117 160 205)(13 190 118 151 206)(14 181 119 152 207)(15 182 120 153 208)(16 183 111 154 209)(17 184 112 155 210)(18 185 113 156 201)(19 186 114 157 202)(20 187 115 158 203)(21 174 229 36 81)(22 175 230 37 82)(23 176 221 38 83)(24 177 222 39 84)(25 178 223 40 85)(26 179 224 31 86)(27 180 225 32 87)(28 171 226 33 88)(29 172 227 34 89)(30 173 228 35 90)(41 199 127 56 101)(42 200 128 57 102)(43 191 129 58 103)(44 192 130 59 104)(45 193 121 60 105)(46 194 122 51 106)(47 195 123 52 107)(48 196 124 53 108)(49 197 125 54 109)(50 198 126 55 110)(61 219 143 71 249)(62 220 144 72 250)(63 211 145 73 241)(64 212 146 74 242)(65 213 147 75 243)(66 214 148 76 244)(67 215 149 77 245)(68 216 150 78 246)(69 217 141 79 247)(70 218 142 80 248)
(1 193 20 75 25)(2 194 11 76 26)(3 195 12 77 27)(4 196 13 78 28)(5 197 14 79 29)(6 198 15 80 30)(7 199 16 71 21)(8 200 17 72 22)(9 191 18 73 23)(10 192 19 74 24)(31 96 106 159 214)(32 97 107 160 215)(33 98 108 151 216)(34 99 109 152 217)(35 100 110 153 218)(36 91 101 154 219)(37 92 102 155 220)(38 93 103 156 211)(39 94 104 157 212)(40 95 105 158 213)(41 209 143 81 134)(42 210 144 82 135)(43 201 145 83 136)(44 202 146 84 137)(45 203 147 85 138)(46 204 148 86 139)(47 205 149 87 140)(48 206 150 88 131)(49 207 141 89 132)(50 208 142 90 133)(51 116 66 224 168)(52 117 67 225 169)(53 118 68 226 170)(54 119 69 227 161)(55 120 70 228 162)(56 111 61 229 163)(57 112 62 230 164)(58 113 63 221 165)(59 114 64 222 166)(60 115 65 223 167)(121 187 243 178 233)(122 188 244 179 234)(123 189 245 180 235)(124 190 246 171 236)(125 181 247 172 237)(126 182 248 173 238)(127 183 249 174 239)(128 184 250 175 240)(129 185 241 176 231)(130 186 242 177 232)
(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)(241 242 243 244 245 246 247 248 249 250)
G:=sub<Sym(250)| (1,233,167,95,138)(2,234,168,96,139)(3,235,169,97,140)(4,236,170,98,131)(5,237,161,99,132)(6,238,162,100,133)(7,239,163,91,134)(8,240,164,92,135)(9,231,165,93,136)(10,232,166,94,137)(11,188,116,159,204)(12,189,117,160,205)(13,190,118,151,206)(14,181,119,152,207)(15,182,120,153,208)(16,183,111,154,209)(17,184,112,155,210)(18,185,113,156,201)(19,186,114,157,202)(20,187,115,158,203)(21,174,229,36,81)(22,175,230,37,82)(23,176,221,38,83)(24,177,222,39,84)(25,178,223,40,85)(26,179,224,31,86)(27,180,225,32,87)(28,171,226,33,88)(29,172,227,34,89)(30,173,228,35,90)(41,199,127,56,101)(42,200,128,57,102)(43,191,129,58,103)(44,192,130,59,104)(45,193,121,60,105)(46,194,122,51,106)(47,195,123,52,107)(48,196,124,53,108)(49,197,125,54,109)(50,198,126,55,110)(61,219,143,71,249)(62,220,144,72,250)(63,211,145,73,241)(64,212,146,74,242)(65,213,147,75,243)(66,214,148,76,244)(67,215,149,77,245)(68,216,150,78,246)(69,217,141,79,247)(70,218,142,80,248), (1,193,20,75,25)(2,194,11,76,26)(3,195,12,77,27)(4,196,13,78,28)(5,197,14,79,29)(6,198,15,80,30)(7,199,16,71,21)(8,200,17,72,22)(9,191,18,73,23)(10,192,19,74,24)(31,96,106,159,214)(32,97,107,160,215)(33,98,108,151,216)(34,99,109,152,217)(35,100,110,153,218)(36,91,101,154,219)(37,92,102,155,220)(38,93,103,156,211)(39,94,104,157,212)(40,95,105,158,213)(41,209,143,81,134)(42,210,144,82,135)(43,201,145,83,136)(44,202,146,84,137)(45,203,147,85,138)(46,204,148,86,139)(47,205,149,87,140)(48,206,150,88,131)(49,207,141,89,132)(50,208,142,90,133)(51,116,66,224,168)(52,117,67,225,169)(53,118,68,226,170)(54,119,69,227,161)(55,120,70,228,162)(56,111,61,229,163)(57,112,62,230,164)(58,113,63,221,165)(59,114,64,222,166)(60,115,65,223,167)(121,187,243,178,233)(122,188,244,179,234)(123,189,245,180,235)(124,190,246,171,236)(125,181,247,172,237)(126,182,248,173,238)(127,183,249,174,239)(128,184,250,175,240)(129,185,241,176,231)(130,186,242,177,232), (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)(241,242,243,244,245,246,247,248,249,250)>;
G:=Group( (1,233,167,95,138)(2,234,168,96,139)(3,235,169,97,140)(4,236,170,98,131)(5,237,161,99,132)(6,238,162,100,133)(7,239,163,91,134)(8,240,164,92,135)(9,231,165,93,136)(10,232,166,94,137)(11,188,116,159,204)(12,189,117,160,205)(13,190,118,151,206)(14,181,119,152,207)(15,182,120,153,208)(16,183,111,154,209)(17,184,112,155,210)(18,185,113,156,201)(19,186,114,157,202)(20,187,115,158,203)(21,174,229,36,81)(22,175,230,37,82)(23,176,221,38,83)(24,177,222,39,84)(25,178,223,40,85)(26,179,224,31,86)(27,180,225,32,87)(28,171,226,33,88)(29,172,227,34,89)(30,173,228,35,90)(41,199,127,56,101)(42,200,128,57,102)(43,191,129,58,103)(44,192,130,59,104)(45,193,121,60,105)(46,194,122,51,106)(47,195,123,52,107)(48,196,124,53,108)(49,197,125,54,109)(50,198,126,55,110)(61,219,143,71,249)(62,220,144,72,250)(63,211,145,73,241)(64,212,146,74,242)(65,213,147,75,243)(66,214,148,76,244)(67,215,149,77,245)(68,216,150,78,246)(69,217,141,79,247)(70,218,142,80,248), (1,193,20,75,25)(2,194,11,76,26)(3,195,12,77,27)(4,196,13,78,28)(5,197,14,79,29)(6,198,15,80,30)(7,199,16,71,21)(8,200,17,72,22)(9,191,18,73,23)(10,192,19,74,24)(31,96,106,159,214)(32,97,107,160,215)(33,98,108,151,216)(34,99,109,152,217)(35,100,110,153,218)(36,91,101,154,219)(37,92,102,155,220)(38,93,103,156,211)(39,94,104,157,212)(40,95,105,158,213)(41,209,143,81,134)(42,210,144,82,135)(43,201,145,83,136)(44,202,146,84,137)(45,203,147,85,138)(46,204,148,86,139)(47,205,149,87,140)(48,206,150,88,131)(49,207,141,89,132)(50,208,142,90,133)(51,116,66,224,168)(52,117,67,225,169)(53,118,68,226,170)(54,119,69,227,161)(55,120,70,228,162)(56,111,61,229,163)(57,112,62,230,164)(58,113,63,221,165)(59,114,64,222,166)(60,115,65,223,167)(121,187,243,178,233)(122,188,244,179,234)(123,189,245,180,235)(124,190,246,171,236)(125,181,247,172,237)(126,182,248,173,238)(127,183,249,174,239)(128,184,250,175,240)(129,185,241,176,231)(130,186,242,177,232), (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)(241,242,243,244,245,246,247,248,249,250) );
G=PermutationGroup([[(1,233,167,95,138),(2,234,168,96,139),(3,235,169,97,140),(4,236,170,98,131),(5,237,161,99,132),(6,238,162,100,133),(7,239,163,91,134),(8,240,164,92,135),(9,231,165,93,136),(10,232,166,94,137),(11,188,116,159,204),(12,189,117,160,205),(13,190,118,151,206),(14,181,119,152,207),(15,182,120,153,208),(16,183,111,154,209),(17,184,112,155,210),(18,185,113,156,201),(19,186,114,157,202),(20,187,115,158,203),(21,174,229,36,81),(22,175,230,37,82),(23,176,221,38,83),(24,177,222,39,84),(25,178,223,40,85),(26,179,224,31,86),(27,180,225,32,87),(28,171,226,33,88),(29,172,227,34,89),(30,173,228,35,90),(41,199,127,56,101),(42,200,128,57,102),(43,191,129,58,103),(44,192,130,59,104),(45,193,121,60,105),(46,194,122,51,106),(47,195,123,52,107),(48,196,124,53,108),(49,197,125,54,109),(50,198,126,55,110),(61,219,143,71,249),(62,220,144,72,250),(63,211,145,73,241),(64,212,146,74,242),(65,213,147,75,243),(66,214,148,76,244),(67,215,149,77,245),(68,216,150,78,246),(69,217,141,79,247),(70,218,142,80,248)], [(1,193,20,75,25),(2,194,11,76,26),(3,195,12,77,27),(4,196,13,78,28),(5,197,14,79,29),(6,198,15,80,30),(7,199,16,71,21),(8,200,17,72,22),(9,191,18,73,23),(10,192,19,74,24),(31,96,106,159,214),(32,97,107,160,215),(33,98,108,151,216),(34,99,109,152,217),(35,100,110,153,218),(36,91,101,154,219),(37,92,102,155,220),(38,93,103,156,211),(39,94,104,157,212),(40,95,105,158,213),(41,209,143,81,134),(42,210,144,82,135),(43,201,145,83,136),(44,202,146,84,137),(45,203,147,85,138),(46,204,148,86,139),(47,205,149,87,140),(48,206,150,88,131),(49,207,141,89,132),(50,208,142,90,133),(51,116,66,224,168),(52,117,67,225,169),(53,118,68,226,170),(54,119,69,227,161),(55,120,70,228,162),(56,111,61,229,163),(57,112,62,230,164),(58,113,63,221,165),(59,114,64,222,166),(60,115,65,223,167),(121,187,243,178,233),(122,188,244,179,234),(123,189,245,180,235),(124,190,246,171,236),(125,181,247,172,237),(126,182,248,173,238),(127,183,249,174,239),(128,184,250,175,240),(129,185,241,176,231),(130,186,242,177,232)], [(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),(241,242,243,244,245,246,247,248,249,250)]])
C52×C10 is a maximal subgroup of
C53⋊12C4
250 conjugacy classes
| class | 1 | 2 | 5A | ··· | 5DT | 10A | ··· | 10DT |
| order | 1 | 2 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
250 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C5 | C10 |
| kernel | C52×C10 | C53 | C5×C10 | C52 |
| # reps | 1 | 1 | 124 | 124 |
Matrix representation of C52×C10 ►in GL3(𝔽11) generated by
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 7 |
G:=sub<GL(3,GF(11))| [9,0,0,0,1,0,0,0,1],[1,0,0,0,3,0,0,0,5],[1,0,0,0,9,0,0,0,7] >;
C52×C10 in GAP, Magma, Sage, TeX
C_5^2\times C_{10} % in TeX
G:=Group("C5^2xC10"); // GroupNames label
G:=SmallGroup(250,15);
// by ID
G=gap.SmallGroup(250,15);
# by ID
G:=PCGroup([4,-2,-5,-5,-5]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^10=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations