direct product, abelian, monomial, 2-elementary
Aliases: C22×C68, SmallGroup(272,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C68 |
| C1 — C22×C68 |
| C1 — C22×C68 |
Generators and relations for C22×C68
G = < a,b,c | a2=b2=c68=1, ab=ba, ac=ca, bc=cb >
Subgroups: 54, all normal (8 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C17, C34, C34, C68, C2×C34, C2×C68, C22×C34, C22×C68
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C17, C34, C68, C2×C34, C2×C68, C22×C34, C22×C68
►Regular action on 272 points
Generators in S272
(1 262)(2 263)(3 264)(4 265)(5 266)(6 267)(7 268)(8 269)(9 270)(10 271)(11 272)(12 205)(13 206)(14 207)(15 208)(16 209)(17 210)(18 211)(19 212)(20 213)(21 214)(22 215)(23 216)(24 217)(25 218)(26 219)(27 220)(28 221)(29 222)(30 223)(31 224)(32 225)(33 226)(34 227)(35 228)(36 229)(37 230)(38 231)(39 232)(40 233)(41 234)(42 235)(43 236)(44 237)(45 238)(46 239)(47 240)(48 241)(49 242)(50 243)(51 244)(52 245)(53 246)(54 247)(55 248)(56 249)(57 250)(58 251)(59 252)(60 253)(61 254)(62 255)(63 256)(64 257)(65 258)(66 259)(67 260)(68 261)(69 137)(70 138)(71 139)(72 140)(73 141)(74 142)(75 143)(76 144)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 159)(92 160)(93 161)(94 162)(95 163)(96 164)(97 165)(98 166)(99 167)(100 168)(101 169)(102 170)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 181)(114 182)(115 183)(116 184)(117 185)(118 186)(119 187)(120 188)(121 189)(122 190)(123 191)(124 192)(125 193)(126 194)(127 195)(128 196)(129 197)(130 198)(131 199)(132 200)(133 201)(134 202)(135 203)(136 204)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 181)(7 182)(8 183)(9 184)(10 185)(11 186)(12 187)(13 188)(14 189)(15 190)(16 191)(17 192)(18 193)(19 194)(20 195)(21 196)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 204)(30 137)(31 138)(32 139)(33 140)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 156)(50 157)(51 158)(52 159)(53 160)(54 161)(55 162)(56 163)(57 164)(58 165)(59 166)(60 167)(61 168)(62 169)(63 170)(64 171)(65 172)(66 173)(67 174)(68 175)(69 223)(70 224)(71 225)(72 226)(73 227)(74 228)(75 229)(76 230)(77 231)(78 232)(79 233)(80 234)(81 235)(82 236)(83 237)(84 238)(85 239)(86 240)(87 241)(88 242)(89 243)(90 244)(91 245)(92 246)(93 247)(94 248)(95 249)(96 250)(97 251)(98 252)(99 253)(100 254)(101 255)(102 256)(103 257)(104 258)(105 259)(106 260)(107 261)(108 262)(109 263)(110 264)(111 265)(112 266)(113 267)(114 268)(115 269)(116 270)(117 271)(118 272)(119 205)(120 206)(121 207)(122 208)(123 209)(124 210)(125 211)(126 212)(127 213)(128 214)(129 215)(130 216)(131 217)(132 218)(133 219)(134 220)(135 221)(136 222)
(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)(205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272)
G:=sub<Sym(272)| (1,262)(2,263)(3,264)(4,265)(5,266)(6,267)(7,268)(8,269)(9,270)(10,271)(11,272)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,222)(30,223)(31,224)(32,225)(33,226)(34,227)(35,228)(36,229)(37,230)(38,231)(39,232)(40,233)(41,234)(42,235)(43,236)(44,237)(45,238)(46,239)(47,240)(48,241)(49,242)(50,243)(51,244)(52,245)(53,246)(54,247)(55,248)(56,249)(57,250)(58,251)(59,252)(60,253)(61,254)(62,255)(63,256)(64,257)(65,258)(66,259)(67,260)(68,261)(69,137)(70,138)(71,139)(72,140)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,161)(94,162)(95,163)(96,164)(97,165)(98,166)(99,167)(100,168)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)(120,188)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204), (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,190)(16,191)(17,192)(18,193)(19,194)(20,195)(21,196)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,204)(30,137)(31,138)(32,139)(33,140)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,223)(70,224)(71,225)(72,226)(73,227)(74,228)(75,229)(76,230)(77,231)(78,232)(79,233)(80,234)(81,235)(82,236)(83,237)(84,238)(85,239)(86,240)(87,241)(88,242)(89,243)(90,244)(91,245)(92,246)(93,247)(94,248)(95,249)(96,250)(97,251)(98,252)(99,253)(100,254)(101,255)(102,256)(103,257)(104,258)(105,259)(106,260)(107,261)(108,262)(109,263)(110,264)(111,265)(112,266)(113,267)(114,268)(115,269)(116,270)(117,271)(118,272)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(131,217)(132,218)(133,219)(134,220)(135,221)(136,222), (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)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)>;
G:=Group( (1,262)(2,263)(3,264)(4,265)(5,266)(6,267)(7,268)(8,269)(9,270)(10,271)(11,272)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,222)(30,223)(31,224)(32,225)(33,226)(34,227)(35,228)(36,229)(37,230)(38,231)(39,232)(40,233)(41,234)(42,235)(43,236)(44,237)(45,238)(46,239)(47,240)(48,241)(49,242)(50,243)(51,244)(52,245)(53,246)(54,247)(55,248)(56,249)(57,250)(58,251)(59,252)(60,253)(61,254)(62,255)(63,256)(64,257)(65,258)(66,259)(67,260)(68,261)(69,137)(70,138)(71,139)(72,140)(73,141)(74,142)(75,143)(76,144)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,161)(94,162)(95,163)(96,164)(97,165)(98,166)(99,167)(100,168)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)(120,188)(121,189)(122,190)(123,191)(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204), (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,190)(16,191)(17,192)(18,193)(19,194)(20,195)(21,196)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,204)(30,137)(31,138)(32,139)(33,140)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,223)(70,224)(71,225)(72,226)(73,227)(74,228)(75,229)(76,230)(77,231)(78,232)(79,233)(80,234)(81,235)(82,236)(83,237)(84,238)(85,239)(86,240)(87,241)(88,242)(89,243)(90,244)(91,245)(92,246)(93,247)(94,248)(95,249)(96,250)(97,251)(98,252)(99,253)(100,254)(101,255)(102,256)(103,257)(104,258)(105,259)(106,260)(107,261)(108,262)(109,263)(110,264)(111,265)(112,266)(113,267)(114,268)(115,269)(116,270)(117,271)(118,272)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(131,217)(132,218)(133,219)(134,220)(135,221)(136,222), (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)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272) );
G=PermutationGroup([[(1,262),(2,263),(3,264),(4,265),(5,266),(6,267),(7,268),(8,269),(9,270),(10,271),(11,272),(12,205),(13,206),(14,207),(15,208),(16,209),(17,210),(18,211),(19,212),(20,213),(21,214),(22,215),(23,216),(24,217),(25,218),(26,219),(27,220),(28,221),(29,222),(30,223),(31,224),(32,225),(33,226),(34,227),(35,228),(36,229),(37,230),(38,231),(39,232),(40,233),(41,234),(42,235),(43,236),(44,237),(45,238),(46,239),(47,240),(48,241),(49,242),(50,243),(51,244),(52,245),(53,246),(54,247),(55,248),(56,249),(57,250),(58,251),(59,252),(60,253),(61,254),(62,255),(63,256),(64,257),(65,258),(66,259),(67,260),(68,261),(69,137),(70,138),(71,139),(72,140),(73,141),(74,142),(75,143),(76,144),(77,145),(78,146),(79,147),(80,148),(81,149),(82,150),(83,151),(84,152),(85,153),(86,154),(87,155),(88,156),(89,157),(90,158),(91,159),(92,160),(93,161),(94,162),(95,163),(96,164),(97,165),(98,166),(99,167),(100,168),(101,169),(102,170),(103,171),(104,172),(105,173),(106,174),(107,175),(108,176),(109,177),(110,178),(111,179),(112,180),(113,181),(114,182),(115,183),(116,184),(117,185),(118,186),(119,187),(120,188),(121,189),(122,190),(123,191),(124,192),(125,193),(126,194),(127,195),(128,196),(129,197),(130,198),(131,199),(132,200),(133,201),(134,202),(135,203),(136,204)], [(1,176),(2,177),(3,178),(4,179),(5,180),(6,181),(7,182),(8,183),(9,184),(10,185),(11,186),(12,187),(13,188),(14,189),(15,190),(16,191),(17,192),(18,193),(19,194),(20,195),(21,196),(22,197),(23,198),(24,199),(25,200),(26,201),(27,202),(28,203),(29,204),(30,137),(31,138),(32,139),(33,140),(34,141),(35,142),(36,143),(37,144),(38,145),(39,146),(40,147),(41,148),(42,149),(43,150),(44,151),(45,152),(46,153),(47,154),(48,155),(49,156),(50,157),(51,158),(52,159),(53,160),(54,161),(55,162),(56,163),(57,164),(58,165),(59,166),(60,167),(61,168),(62,169),(63,170),(64,171),(65,172),(66,173),(67,174),(68,175),(69,223),(70,224),(71,225),(72,226),(73,227),(74,228),(75,229),(76,230),(77,231),(78,232),(79,233),(80,234),(81,235),(82,236),(83,237),(84,238),(85,239),(86,240),(87,241),(88,242),(89,243),(90,244),(91,245),(92,246),(93,247),(94,248),(95,249),(96,250),(97,251),(98,252),(99,253),(100,254),(101,255),(102,256),(103,257),(104,258),(105,259),(106,260),(107,261),(108,262),(109,263),(110,264),(111,265),(112,266),(113,267),(114,268),(115,269),(116,270),(117,271),(118,272),(119,205),(120,206),(121,207),(122,208),(123,209),(124,210),(125,211),(126,212),(127,213),(128,214),(129,215),(130,216),(131,217),(132,218),(133,219),(134,220),(135,221),(136,222)], [(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),(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)]])
272 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 17A | ··· | 17P | 34A | ··· | 34DH | 68A | ··· | 68DX |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
272 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C17 | C34 | C34 | C68 |
| kernel | C22×C68 | C2×C68 | C22×C34 | C2×C34 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 8 | 16 | 96 | 16 | 128 |
Matrix representation of C22×C68 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 1 |
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 2 | 0 | 0 |
| 0 | 19 | 0 |
| 0 | 0 | 68 |
G:=sub<GL(3,GF(137))| [136,0,0,0,136,0,0,0,1],[136,0,0,0,1,0,0,0,1],[2,0,0,0,19,0,0,0,68] >;
C22×C68 in GAP, Magma, Sage, TeX
C_2^2\times C_{68} % in TeX
G:=Group("C2^2xC68"); // GroupNames label
G:=SmallGroup(272,46);
// by ID
G=gap.SmallGroup(272,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-17,-2,680]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^68=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations