metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C34.5D4, C34.1Q8, Dic17⋊1C4, C2.1Dic34, C22.4D34, C17⋊2(C4⋊C4), (C2×C68).1C2, (C2×C4).1D17, C2.4(C4×D17), C34.11(C2×C4), C2.1(C17⋊D4), (C2×C34).4C22, (C2×Dic17).1C2, SmallGroup(272,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C34.D4
G = < a,b,c | a34=b4=1, c2=a17, bab-1=cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C34.D4
►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)
(1 198 215 100)(2 197 216 99)(3 196 217 98)(4 195 218 97)(5 194 219 96)(6 193 220 95)(7 192 221 94)(8 191 222 93)(9 190 223 92)(10 189 224 91)(11 188 225 90)(12 187 226 89)(13 186 227 88)(14 185 228 87)(15 184 229 86)(16 183 230 85)(17 182 231 84)(18 181 232 83)(19 180 233 82)(20 179 234 81)(21 178 235 80)(22 177 236 79)(23 176 237 78)(24 175 238 77)(25 174 205 76)(26 173 206 75)(27 172 207 74)(28 171 208 73)(29 204 209 72)(30 203 210 71)(31 202 211 70)(32 201 212 69)(33 200 213 102)(34 199 214 101)(35 120 250 142)(36 119 251 141)(37 118 252 140)(38 117 253 139)(39 116 254 138)(40 115 255 137)(41 114 256 170)(42 113 257 169)(43 112 258 168)(44 111 259 167)(45 110 260 166)(46 109 261 165)(47 108 262 164)(48 107 263 163)(49 106 264 162)(50 105 265 161)(51 104 266 160)(52 103 267 159)(53 136 268 158)(54 135 269 157)(55 134 270 156)(56 133 271 155)(57 132 272 154)(58 131 239 153)(59 130 240 152)(60 129 241 151)(61 128 242 150)(62 127 243 149)(63 126 244 148)(64 125 245 147)(65 124 246 146)(66 123 247 145)(67 122 248 144)(68 121 249 143)
(1 120 18 103)(2 119 19 136)(3 118 20 135)(4 117 21 134)(5 116 22 133)(6 115 23 132)(7 114 24 131)(8 113 25 130)(9 112 26 129)(10 111 27 128)(11 110 28 127)(12 109 29 126)(13 108 30 125)(14 107 31 124)(15 106 32 123)(16 105 33 122)(17 104 34 121)(35 181 52 198)(36 180 53 197)(37 179 54 196)(38 178 55 195)(39 177 56 194)(40 176 57 193)(41 175 58 192)(42 174 59 191)(43 173 60 190)(44 172 61 189)(45 171 62 188)(46 204 63 187)(47 203 64 186)(48 202 65 185)(49 201 66 184)(50 200 67 183)(51 199 68 182)(69 247 86 264)(70 246 87 263)(71 245 88 262)(72 244 89 261)(73 243 90 260)(74 242 91 259)(75 241 92 258)(76 240 93 257)(77 239 94 256)(78 272 95 255)(79 271 96 254)(80 270 97 253)(81 269 98 252)(82 268 99 251)(83 267 100 250)(84 266 101 249)(85 265 102 248)(137 237 154 220)(138 236 155 219)(139 235 156 218)(140 234 157 217)(141 233 158 216)(142 232 159 215)(143 231 160 214)(144 230 161 213)(145 229 162 212)(146 228 163 211)(147 227 164 210)(148 226 165 209)(149 225 166 208)(150 224 167 207)(151 223 168 206)(152 222 169 205)(153 221 170 238)
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), (1,198,215,100)(2,197,216,99)(3,196,217,98)(4,195,218,97)(5,194,219,96)(6,193,220,95)(7,192,221,94)(8,191,222,93)(9,190,223,92)(10,189,224,91)(11,188,225,90)(12,187,226,89)(13,186,227,88)(14,185,228,87)(15,184,229,86)(16,183,230,85)(17,182,231,84)(18,181,232,83)(19,180,233,82)(20,179,234,81)(21,178,235,80)(22,177,236,79)(23,176,237,78)(24,175,238,77)(25,174,205,76)(26,173,206,75)(27,172,207,74)(28,171,208,73)(29,204,209,72)(30,203,210,71)(31,202,211,70)(32,201,212,69)(33,200,213,102)(34,199,214,101)(35,120,250,142)(36,119,251,141)(37,118,252,140)(38,117,253,139)(39,116,254,138)(40,115,255,137)(41,114,256,170)(42,113,257,169)(43,112,258,168)(44,111,259,167)(45,110,260,166)(46,109,261,165)(47,108,262,164)(48,107,263,163)(49,106,264,162)(50,105,265,161)(51,104,266,160)(52,103,267,159)(53,136,268,158)(54,135,269,157)(55,134,270,156)(56,133,271,155)(57,132,272,154)(58,131,239,153)(59,130,240,152)(60,129,241,151)(61,128,242,150)(62,127,243,149)(63,126,244,148)(64,125,245,147)(65,124,246,146)(66,123,247,145)(67,122,248,144)(68,121,249,143), (1,120,18,103)(2,119,19,136)(3,118,20,135)(4,117,21,134)(5,116,22,133)(6,115,23,132)(7,114,24,131)(8,113,25,130)(9,112,26,129)(10,111,27,128)(11,110,28,127)(12,109,29,126)(13,108,30,125)(14,107,31,124)(15,106,32,123)(16,105,33,122)(17,104,34,121)(35,181,52,198)(36,180,53,197)(37,179,54,196)(38,178,55,195)(39,177,56,194)(40,176,57,193)(41,175,58,192)(42,174,59,191)(43,173,60,190)(44,172,61,189)(45,171,62,188)(46,204,63,187)(47,203,64,186)(48,202,65,185)(49,201,66,184)(50,200,67,183)(51,199,68,182)(69,247,86,264)(70,246,87,263)(71,245,88,262)(72,244,89,261)(73,243,90,260)(74,242,91,259)(75,241,92,258)(76,240,93,257)(77,239,94,256)(78,272,95,255)(79,271,96,254)(80,270,97,253)(81,269,98,252)(82,268,99,251)(83,267,100,250)(84,266,101,249)(85,265,102,248)(137,237,154,220)(138,236,155,219)(139,235,156,218)(140,234,157,217)(141,233,158,216)(142,232,159,215)(143,231,160,214)(144,230,161,213)(145,229,162,212)(146,228,163,211)(147,227,164,210)(148,226,165,209)(149,225,166,208)(150,224,167,207)(151,223,168,206)(152,222,169,205)(153,221,170,238)>;
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), (1,198,215,100)(2,197,216,99)(3,196,217,98)(4,195,218,97)(5,194,219,96)(6,193,220,95)(7,192,221,94)(8,191,222,93)(9,190,223,92)(10,189,224,91)(11,188,225,90)(12,187,226,89)(13,186,227,88)(14,185,228,87)(15,184,229,86)(16,183,230,85)(17,182,231,84)(18,181,232,83)(19,180,233,82)(20,179,234,81)(21,178,235,80)(22,177,236,79)(23,176,237,78)(24,175,238,77)(25,174,205,76)(26,173,206,75)(27,172,207,74)(28,171,208,73)(29,204,209,72)(30,203,210,71)(31,202,211,70)(32,201,212,69)(33,200,213,102)(34,199,214,101)(35,120,250,142)(36,119,251,141)(37,118,252,140)(38,117,253,139)(39,116,254,138)(40,115,255,137)(41,114,256,170)(42,113,257,169)(43,112,258,168)(44,111,259,167)(45,110,260,166)(46,109,261,165)(47,108,262,164)(48,107,263,163)(49,106,264,162)(50,105,265,161)(51,104,266,160)(52,103,267,159)(53,136,268,158)(54,135,269,157)(55,134,270,156)(56,133,271,155)(57,132,272,154)(58,131,239,153)(59,130,240,152)(60,129,241,151)(61,128,242,150)(62,127,243,149)(63,126,244,148)(64,125,245,147)(65,124,246,146)(66,123,247,145)(67,122,248,144)(68,121,249,143), (1,120,18,103)(2,119,19,136)(3,118,20,135)(4,117,21,134)(5,116,22,133)(6,115,23,132)(7,114,24,131)(8,113,25,130)(9,112,26,129)(10,111,27,128)(11,110,28,127)(12,109,29,126)(13,108,30,125)(14,107,31,124)(15,106,32,123)(16,105,33,122)(17,104,34,121)(35,181,52,198)(36,180,53,197)(37,179,54,196)(38,178,55,195)(39,177,56,194)(40,176,57,193)(41,175,58,192)(42,174,59,191)(43,173,60,190)(44,172,61,189)(45,171,62,188)(46,204,63,187)(47,203,64,186)(48,202,65,185)(49,201,66,184)(50,200,67,183)(51,199,68,182)(69,247,86,264)(70,246,87,263)(71,245,88,262)(72,244,89,261)(73,243,90,260)(74,242,91,259)(75,241,92,258)(76,240,93,257)(77,239,94,256)(78,272,95,255)(79,271,96,254)(80,270,97,253)(81,269,98,252)(82,268,99,251)(83,267,100,250)(84,266,101,249)(85,265,102,248)(137,237,154,220)(138,236,155,219)(139,235,156,218)(140,234,157,217)(141,233,158,216)(142,232,159,215)(143,231,160,214)(144,230,161,213)(145,229,162,212)(146,228,163,211)(147,227,164,210)(148,226,165,209)(149,225,166,208)(150,224,167,207)(151,223,168,206)(152,222,169,205)(153,221,170,238) );
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)], [(1,198,215,100),(2,197,216,99),(3,196,217,98),(4,195,218,97),(5,194,219,96),(6,193,220,95),(7,192,221,94),(8,191,222,93),(9,190,223,92),(10,189,224,91),(11,188,225,90),(12,187,226,89),(13,186,227,88),(14,185,228,87),(15,184,229,86),(16,183,230,85),(17,182,231,84),(18,181,232,83),(19,180,233,82),(20,179,234,81),(21,178,235,80),(22,177,236,79),(23,176,237,78),(24,175,238,77),(25,174,205,76),(26,173,206,75),(27,172,207,74),(28,171,208,73),(29,204,209,72),(30,203,210,71),(31,202,211,70),(32,201,212,69),(33,200,213,102),(34,199,214,101),(35,120,250,142),(36,119,251,141),(37,118,252,140),(38,117,253,139),(39,116,254,138),(40,115,255,137),(41,114,256,170),(42,113,257,169),(43,112,258,168),(44,111,259,167),(45,110,260,166),(46,109,261,165),(47,108,262,164),(48,107,263,163),(49,106,264,162),(50,105,265,161),(51,104,266,160),(52,103,267,159),(53,136,268,158),(54,135,269,157),(55,134,270,156),(56,133,271,155),(57,132,272,154),(58,131,239,153),(59,130,240,152),(60,129,241,151),(61,128,242,150),(62,127,243,149),(63,126,244,148),(64,125,245,147),(65,124,246,146),(66,123,247,145),(67,122,248,144),(68,121,249,143)], [(1,120,18,103),(2,119,19,136),(3,118,20,135),(4,117,21,134),(5,116,22,133),(6,115,23,132),(7,114,24,131),(8,113,25,130),(9,112,26,129),(10,111,27,128),(11,110,28,127),(12,109,29,126),(13,108,30,125),(14,107,31,124),(15,106,32,123),(16,105,33,122),(17,104,34,121),(35,181,52,198),(36,180,53,197),(37,179,54,196),(38,178,55,195),(39,177,56,194),(40,176,57,193),(41,175,58,192),(42,174,59,191),(43,173,60,190),(44,172,61,189),(45,171,62,188),(46,204,63,187),(47,203,64,186),(48,202,65,185),(49,201,66,184),(50,200,67,183),(51,199,68,182),(69,247,86,264),(70,246,87,263),(71,245,88,262),(72,244,89,261),(73,243,90,260),(74,242,91,259),(75,241,92,258),(76,240,93,257),(77,239,94,256),(78,272,95,255),(79,271,96,254),(80,270,97,253),(81,269,98,252),(82,268,99,251),(83,267,100,250),(84,266,101,249),(85,265,102,248),(137,237,154,220),(138,236,155,219),(139,235,156,218),(140,234,157,217),(141,233,158,216),(142,232,159,215),(143,231,160,214),(144,230,161,213),(145,229,162,212),(146,228,163,211),(147,227,164,210),(148,226,165,209),(149,225,166,208),(150,224,167,207),(151,223,168,206),(152,222,169,205),(153,221,170,238)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D17 | D34 | Dic34 | C4×D17 | C17⋊D4 |
| kernel | C34.D4 | C2×Dic17 | C2×C68 | Dic17 | C34 | C34 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 8 | 8 | 16 | 16 | 16 |
Matrix representation of C34.D4 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 0 | 105 |
| 0 | 30 | 107 |
| 1 | 0 | 0 |
| 0 | 88 | 136 |
| 0 | 73 | 49 |
| 100 | 0 | 0 |
| 0 | 111 | 11 |
| 0 | 63 | 26 |
G:=sub<GL(3,GF(137))| [136,0,0,0,0,30,0,105,107],[1,0,0,0,88,73,0,136,49],[100,0,0,0,111,63,0,11,26] >;
C34.D4 in GAP, Magma, Sage, TeX
C_{34}.D_4 % in TeX
G:=Group("C34.D4"); // GroupNames label
G:=SmallGroup(272,12);
// by ID
G=gap.SmallGroup(272,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,40,101,26,6404]);
// Polycyclic
G:=Group<a,b,c|a^34=b^4=1,c^2=a^17,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export