direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C17, C4⋊C68, C68⋊5C4, C34.3Q8, C34.13D4, C2.(Q8×C17), (C2×C4).1C34, (C2×C68).2C2, C2.2(C2×C68), C2.2(D4×C17), C34.18(C2×C4), C22.3(C2×C34), (C2×C34).14C22, SmallGroup(272,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C17
G = < a,b,c | a17=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4⋊C4×C17
►Regular action on 272 points
Generators in S272
(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)
(1 140 211 84)(2 141 212 85)(3 142 213 69)(4 143 214 70)(5 144 215 71)(6 145 216 72)(7 146 217 73)(8 147 218 74)(9 148 219 75)(10 149 220 76)(11 150 221 77)(12 151 205 78)(13 152 206 79)(14 153 207 80)(15 137 208 81)(16 138 209 82)(17 139 210 83)(18 98 195 133)(19 99 196 134)(20 100 197 135)(21 101 198 136)(22 102 199 120)(23 86 200 121)(24 87 201 122)(25 88 202 123)(26 89 203 124)(27 90 204 125)(28 91 188 126)(29 92 189 127)(30 93 190 128)(31 94 191 129)(32 95 192 130)(33 96 193 131)(34 97 194 132)(35 173 234 112)(36 174 235 113)(37 175 236 114)(38 176 237 115)(39 177 238 116)(40 178 222 117)(41 179 223 118)(42 180 224 119)(43 181 225 103)(44 182 226 104)(45 183 227 105)(46 184 228 106)(47 185 229 107)(48 186 230 108)(49 187 231 109)(50 171 232 110)(51 172 233 111)(52 255 271 165)(53 239 272 166)(54 240 256 167)(55 241 257 168)(56 242 258 169)(57 243 259 170)(58 244 260 154)(59 245 261 155)(60 246 262 156)(61 247 263 157)(62 248 264 158)(63 249 265 159)(64 250 266 160)(65 251 267 161)(66 252 268 162)(67 253 269 163)(68 254 270 164)
(1 24 108 52)(2 25 109 53)(3 26 110 54)(4 27 111 55)(5 28 112 56)(6 29 113 57)(7 30 114 58)(8 31 115 59)(9 32 116 60)(10 33 117 61)(11 34 118 62)(12 18 119 63)(13 19 103 64)(14 20 104 65)(15 21 105 66)(16 22 106 67)(17 23 107 68)(35 169 144 126)(36 170 145 127)(37 154 146 128)(38 155 147 129)(39 156 148 130)(40 157 149 131)(41 158 150 132)(42 159 151 133)(43 160 152 134)(44 161 153 135)(45 162 137 136)(46 163 138 120)(47 164 139 121)(48 165 140 122)(49 166 141 123)(50 167 142 124)(51 168 143 125)(69 89 232 240)(70 90 233 241)(71 91 234 242)(72 92 235 243)(73 93 236 244)(74 94 237 245)(75 95 238 246)(76 96 222 247)(77 97 223 248)(78 98 224 249)(79 99 225 250)(80 100 226 251)(81 101 227 252)(82 102 228 253)(83 86 229 254)(84 87 230 255)(85 88 231 239)(171 256 213 203)(172 257 214 204)(173 258 215 188)(174 259 216 189)(175 260 217 190)(176 261 218 191)(177 262 219 192)(178 263 220 193)(179 264 221 194)(180 265 205 195)(181 266 206 196)(182 267 207 197)(183 268 208 198)(184 269 209 199)(185 270 210 200)(186 271 211 201)(187 272 212 202)
G:=sub<Sym(272)| (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), (1,140,211,84)(2,141,212,85)(3,142,213,69)(4,143,214,70)(5,144,215,71)(6,145,216,72)(7,146,217,73)(8,147,218,74)(9,148,219,75)(10,149,220,76)(11,150,221,77)(12,151,205,78)(13,152,206,79)(14,153,207,80)(15,137,208,81)(16,138,209,82)(17,139,210,83)(18,98,195,133)(19,99,196,134)(20,100,197,135)(21,101,198,136)(22,102,199,120)(23,86,200,121)(24,87,201,122)(25,88,202,123)(26,89,203,124)(27,90,204,125)(28,91,188,126)(29,92,189,127)(30,93,190,128)(31,94,191,129)(32,95,192,130)(33,96,193,131)(34,97,194,132)(35,173,234,112)(36,174,235,113)(37,175,236,114)(38,176,237,115)(39,177,238,116)(40,178,222,117)(41,179,223,118)(42,180,224,119)(43,181,225,103)(44,182,226,104)(45,183,227,105)(46,184,228,106)(47,185,229,107)(48,186,230,108)(49,187,231,109)(50,171,232,110)(51,172,233,111)(52,255,271,165)(53,239,272,166)(54,240,256,167)(55,241,257,168)(56,242,258,169)(57,243,259,170)(58,244,260,154)(59,245,261,155)(60,246,262,156)(61,247,263,157)(62,248,264,158)(63,249,265,159)(64,250,266,160)(65,251,267,161)(66,252,268,162)(67,253,269,163)(68,254,270,164), (1,24,108,52)(2,25,109,53)(3,26,110,54)(4,27,111,55)(5,28,112,56)(6,29,113,57)(7,30,114,58)(8,31,115,59)(9,32,116,60)(10,33,117,61)(11,34,118,62)(12,18,119,63)(13,19,103,64)(14,20,104,65)(15,21,105,66)(16,22,106,67)(17,23,107,68)(35,169,144,126)(36,170,145,127)(37,154,146,128)(38,155,147,129)(39,156,148,130)(40,157,149,131)(41,158,150,132)(42,159,151,133)(43,160,152,134)(44,161,153,135)(45,162,137,136)(46,163,138,120)(47,164,139,121)(48,165,140,122)(49,166,141,123)(50,167,142,124)(51,168,143,125)(69,89,232,240)(70,90,233,241)(71,91,234,242)(72,92,235,243)(73,93,236,244)(74,94,237,245)(75,95,238,246)(76,96,222,247)(77,97,223,248)(78,98,224,249)(79,99,225,250)(80,100,226,251)(81,101,227,252)(82,102,228,253)(83,86,229,254)(84,87,230,255)(85,88,231,239)(171,256,213,203)(172,257,214,204)(173,258,215,188)(174,259,216,189)(175,260,217,190)(176,261,218,191)(177,262,219,192)(178,263,220,193)(179,264,221,194)(180,265,205,195)(181,266,206,196)(182,267,207,197)(183,268,208,198)(184,269,209,199)(185,270,210,200)(186,271,211,201)(187,272,212,202)>;
G:=Group( (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), (1,140,211,84)(2,141,212,85)(3,142,213,69)(4,143,214,70)(5,144,215,71)(6,145,216,72)(7,146,217,73)(8,147,218,74)(9,148,219,75)(10,149,220,76)(11,150,221,77)(12,151,205,78)(13,152,206,79)(14,153,207,80)(15,137,208,81)(16,138,209,82)(17,139,210,83)(18,98,195,133)(19,99,196,134)(20,100,197,135)(21,101,198,136)(22,102,199,120)(23,86,200,121)(24,87,201,122)(25,88,202,123)(26,89,203,124)(27,90,204,125)(28,91,188,126)(29,92,189,127)(30,93,190,128)(31,94,191,129)(32,95,192,130)(33,96,193,131)(34,97,194,132)(35,173,234,112)(36,174,235,113)(37,175,236,114)(38,176,237,115)(39,177,238,116)(40,178,222,117)(41,179,223,118)(42,180,224,119)(43,181,225,103)(44,182,226,104)(45,183,227,105)(46,184,228,106)(47,185,229,107)(48,186,230,108)(49,187,231,109)(50,171,232,110)(51,172,233,111)(52,255,271,165)(53,239,272,166)(54,240,256,167)(55,241,257,168)(56,242,258,169)(57,243,259,170)(58,244,260,154)(59,245,261,155)(60,246,262,156)(61,247,263,157)(62,248,264,158)(63,249,265,159)(64,250,266,160)(65,251,267,161)(66,252,268,162)(67,253,269,163)(68,254,270,164), (1,24,108,52)(2,25,109,53)(3,26,110,54)(4,27,111,55)(5,28,112,56)(6,29,113,57)(7,30,114,58)(8,31,115,59)(9,32,116,60)(10,33,117,61)(11,34,118,62)(12,18,119,63)(13,19,103,64)(14,20,104,65)(15,21,105,66)(16,22,106,67)(17,23,107,68)(35,169,144,126)(36,170,145,127)(37,154,146,128)(38,155,147,129)(39,156,148,130)(40,157,149,131)(41,158,150,132)(42,159,151,133)(43,160,152,134)(44,161,153,135)(45,162,137,136)(46,163,138,120)(47,164,139,121)(48,165,140,122)(49,166,141,123)(50,167,142,124)(51,168,143,125)(69,89,232,240)(70,90,233,241)(71,91,234,242)(72,92,235,243)(73,93,236,244)(74,94,237,245)(75,95,238,246)(76,96,222,247)(77,97,223,248)(78,98,224,249)(79,99,225,250)(80,100,226,251)(81,101,227,252)(82,102,228,253)(83,86,229,254)(84,87,230,255)(85,88,231,239)(171,256,213,203)(172,257,214,204)(173,258,215,188)(174,259,216,189)(175,260,217,190)(176,261,218,191)(177,262,219,192)(178,263,220,193)(179,264,221,194)(180,265,205,195)(181,266,206,196)(182,267,207,197)(183,268,208,198)(184,269,209,199)(185,270,210,200)(186,271,211,201)(187,272,212,202) );
G=PermutationGroup([[(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)], [(1,140,211,84),(2,141,212,85),(3,142,213,69),(4,143,214,70),(5,144,215,71),(6,145,216,72),(7,146,217,73),(8,147,218,74),(9,148,219,75),(10,149,220,76),(11,150,221,77),(12,151,205,78),(13,152,206,79),(14,153,207,80),(15,137,208,81),(16,138,209,82),(17,139,210,83),(18,98,195,133),(19,99,196,134),(20,100,197,135),(21,101,198,136),(22,102,199,120),(23,86,200,121),(24,87,201,122),(25,88,202,123),(26,89,203,124),(27,90,204,125),(28,91,188,126),(29,92,189,127),(30,93,190,128),(31,94,191,129),(32,95,192,130),(33,96,193,131),(34,97,194,132),(35,173,234,112),(36,174,235,113),(37,175,236,114),(38,176,237,115),(39,177,238,116),(40,178,222,117),(41,179,223,118),(42,180,224,119),(43,181,225,103),(44,182,226,104),(45,183,227,105),(46,184,228,106),(47,185,229,107),(48,186,230,108),(49,187,231,109),(50,171,232,110),(51,172,233,111),(52,255,271,165),(53,239,272,166),(54,240,256,167),(55,241,257,168),(56,242,258,169),(57,243,259,170),(58,244,260,154),(59,245,261,155),(60,246,262,156),(61,247,263,157),(62,248,264,158),(63,249,265,159),(64,250,266,160),(65,251,267,161),(66,252,268,162),(67,253,269,163),(68,254,270,164)], [(1,24,108,52),(2,25,109,53),(3,26,110,54),(4,27,111,55),(5,28,112,56),(6,29,113,57),(7,30,114,58),(8,31,115,59),(9,32,116,60),(10,33,117,61),(11,34,118,62),(12,18,119,63),(13,19,103,64),(14,20,104,65),(15,21,105,66),(16,22,106,67),(17,23,107,68),(35,169,144,126),(36,170,145,127),(37,154,146,128),(38,155,147,129),(39,156,148,130),(40,157,149,131),(41,158,150,132),(42,159,151,133),(43,160,152,134),(44,161,153,135),(45,162,137,136),(46,163,138,120),(47,164,139,121),(48,165,140,122),(49,166,141,123),(50,167,142,124),(51,168,143,125),(69,89,232,240),(70,90,233,241),(71,91,234,242),(72,92,235,243),(73,93,236,244),(74,94,237,245),(75,95,238,246),(76,96,222,247),(77,97,223,248),(78,98,224,249),(79,99,225,250),(80,100,226,251),(81,101,227,252),(82,102,228,253),(83,86,229,254),(84,87,230,255),(85,88,231,239),(171,256,213,203),(172,257,214,204),(173,258,215,188),(174,259,216,189),(175,260,217,190),(176,261,218,191),(177,262,219,192),(178,263,220,193),(179,264,221,194),(180,265,205,195),(181,266,206,196),(182,267,207,197),(183,268,208,198),(184,269,209,199),(185,270,210,200),(186,271,211,201),(187,272,212,202)]])
170 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 17A | ··· | 17P | 34A | ··· | 34AV | 68A | ··· | 68CR |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
170 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C17 | C34 | C68 | D4 | Q8 | D4×C17 | Q8×C17 |
| kernel | C4⋊C4×C17 | C2×C68 | C68 | C4⋊C4 | C2×C4 | C4 | C34 | C34 | C2 | C2 |
| # reps | 1 | 3 | 4 | 16 | 48 | 64 | 1 | 1 | 16 | 16 |
Matrix representation of C4⋊C4×C17 ►in GL3(𝔽137) generated by
| 1 | 0 | 0 |
| 0 | 50 | 0 |
| 0 | 0 | 50 |
| 136 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 136 | 0 |
| 37 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(137))| [1,0,0,0,50,0,0,0,50],[136,0,0,0,0,136,0,1,0],[37,0,0,0,0,1,0,1,0] >;
C4⋊C4×C17 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{17} % in TeX
G:=Group("C4:C4xC17"); // GroupNames label
G:=SmallGroup(272,22);
// by ID
G=gap.SmallGroup(272,22);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,680,701,346]);
// Polycyclic
G:=Group<a,b,c|a^17=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export