direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C34, C68.20C22, C34.12C23, (C2×C4).3C34, C4.4(C2×C34), (C2×C68).9C2, C22.4(C2×C34), C2.2(C22×C34), (C2×C34).15C22, SmallGroup(272,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C34
G = < a,b,c | a34=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C34
►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)
(1 263 209 200)(2 264 210 201)(3 265 211 202)(4 266 212 203)(5 267 213 204)(6 268 214 171)(7 269 215 172)(8 270 216 173)(9 271 217 174)(10 272 218 175)(11 239 219 176)(12 240 220 177)(13 241 221 178)(14 242 222 179)(15 243 223 180)(16 244 224 181)(17 245 225 182)(18 246 226 183)(19 247 227 184)(20 248 228 185)(21 249 229 186)(22 250 230 187)(23 251 231 188)(24 252 232 189)(25 253 233 190)(26 254 234 191)(27 255 235 192)(28 256 236 193)(29 257 237 194)(30 258 238 195)(31 259 205 196)(32 260 206 197)(33 261 207 198)(34 262 208 199)(35 154 113 86)(36 155 114 87)(37 156 115 88)(38 157 116 89)(39 158 117 90)(40 159 118 91)(41 160 119 92)(42 161 120 93)(43 162 121 94)(44 163 122 95)(45 164 123 96)(46 165 124 97)(47 166 125 98)(48 167 126 99)(49 168 127 100)(50 169 128 101)(51 170 129 102)(52 137 130 69)(53 138 131 70)(54 139 132 71)(55 140 133 72)(56 141 134 73)(57 142 135 74)(58 143 136 75)(59 144 103 76)(60 145 104 77)(61 146 105 78)(62 147 106 79)(63 148 107 80)(64 149 108 81)(65 150 109 82)(66 151 110 83)(67 152 111 84)(68 153 112 85)
(1 134 209 56)(2 135 210 57)(3 136 211 58)(4 103 212 59)(5 104 213 60)(6 105 214 61)(7 106 215 62)(8 107 216 63)(9 108 217 64)(10 109 218 65)(11 110 219 66)(12 111 220 67)(13 112 221 68)(14 113 222 35)(15 114 223 36)(16 115 224 37)(17 116 225 38)(18 117 226 39)(19 118 227 40)(20 119 228 41)(21 120 229 42)(22 121 230 43)(23 122 231 44)(24 123 232 45)(25 124 233 46)(26 125 234 47)(27 126 235 48)(28 127 236 49)(29 128 237 50)(30 129 238 51)(31 130 205 52)(32 131 206 53)(33 132 207 54)(34 133 208 55)(69 259 137 196)(70 260 138 197)(71 261 139 198)(72 262 140 199)(73 263 141 200)(74 264 142 201)(75 265 143 202)(76 266 144 203)(77 267 145 204)(78 268 146 171)(79 269 147 172)(80 270 148 173)(81 271 149 174)(82 272 150 175)(83 239 151 176)(84 240 152 177)(85 241 153 178)(86 242 154 179)(87 243 155 180)(88 244 156 181)(89 245 157 182)(90 246 158 183)(91 247 159 184)(92 248 160 185)(93 249 161 186)(94 250 162 187)(95 251 163 188)(96 252 164 189)(97 253 165 190)(98 254 166 191)(99 255 167 192)(100 256 168 193)(101 257 169 194)(102 258 170 195)
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), (1,263,209,200)(2,264,210,201)(3,265,211,202)(4,266,212,203)(5,267,213,204)(6,268,214,171)(7,269,215,172)(8,270,216,173)(9,271,217,174)(10,272,218,175)(11,239,219,176)(12,240,220,177)(13,241,221,178)(14,242,222,179)(15,243,223,180)(16,244,224,181)(17,245,225,182)(18,246,226,183)(19,247,227,184)(20,248,228,185)(21,249,229,186)(22,250,230,187)(23,251,231,188)(24,252,232,189)(25,253,233,190)(26,254,234,191)(27,255,235,192)(28,256,236,193)(29,257,237,194)(30,258,238,195)(31,259,205,196)(32,260,206,197)(33,261,207,198)(34,262,208,199)(35,154,113,86)(36,155,114,87)(37,156,115,88)(38,157,116,89)(39,158,117,90)(40,159,118,91)(41,160,119,92)(42,161,120,93)(43,162,121,94)(44,163,122,95)(45,164,123,96)(46,165,124,97)(47,166,125,98)(48,167,126,99)(49,168,127,100)(50,169,128,101)(51,170,129,102)(52,137,130,69)(53,138,131,70)(54,139,132,71)(55,140,133,72)(56,141,134,73)(57,142,135,74)(58,143,136,75)(59,144,103,76)(60,145,104,77)(61,146,105,78)(62,147,106,79)(63,148,107,80)(64,149,108,81)(65,150,109,82)(66,151,110,83)(67,152,111,84)(68,153,112,85), (1,134,209,56)(2,135,210,57)(3,136,211,58)(4,103,212,59)(5,104,213,60)(6,105,214,61)(7,106,215,62)(8,107,216,63)(9,108,217,64)(10,109,218,65)(11,110,219,66)(12,111,220,67)(13,112,221,68)(14,113,222,35)(15,114,223,36)(16,115,224,37)(17,116,225,38)(18,117,226,39)(19,118,227,40)(20,119,228,41)(21,120,229,42)(22,121,230,43)(23,122,231,44)(24,123,232,45)(25,124,233,46)(26,125,234,47)(27,126,235,48)(28,127,236,49)(29,128,237,50)(30,129,238,51)(31,130,205,52)(32,131,206,53)(33,132,207,54)(34,133,208,55)(69,259,137,196)(70,260,138,197)(71,261,139,198)(72,262,140,199)(73,263,141,200)(74,264,142,201)(75,265,143,202)(76,266,144,203)(77,267,145,204)(78,268,146,171)(79,269,147,172)(80,270,148,173)(81,271,149,174)(82,272,150,175)(83,239,151,176)(84,240,152,177)(85,241,153,178)(86,242,154,179)(87,243,155,180)(88,244,156,181)(89,245,157,182)(90,246,158,183)(91,247,159,184)(92,248,160,185)(93,249,161,186)(94,250,162,187)(95,251,163,188)(96,252,164,189)(97,253,165,190)(98,254,166,191)(99,255,167,192)(100,256,168,193)(101,257,169,194)(102,258,170,195)>;
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), (1,263,209,200)(2,264,210,201)(3,265,211,202)(4,266,212,203)(5,267,213,204)(6,268,214,171)(7,269,215,172)(8,270,216,173)(9,271,217,174)(10,272,218,175)(11,239,219,176)(12,240,220,177)(13,241,221,178)(14,242,222,179)(15,243,223,180)(16,244,224,181)(17,245,225,182)(18,246,226,183)(19,247,227,184)(20,248,228,185)(21,249,229,186)(22,250,230,187)(23,251,231,188)(24,252,232,189)(25,253,233,190)(26,254,234,191)(27,255,235,192)(28,256,236,193)(29,257,237,194)(30,258,238,195)(31,259,205,196)(32,260,206,197)(33,261,207,198)(34,262,208,199)(35,154,113,86)(36,155,114,87)(37,156,115,88)(38,157,116,89)(39,158,117,90)(40,159,118,91)(41,160,119,92)(42,161,120,93)(43,162,121,94)(44,163,122,95)(45,164,123,96)(46,165,124,97)(47,166,125,98)(48,167,126,99)(49,168,127,100)(50,169,128,101)(51,170,129,102)(52,137,130,69)(53,138,131,70)(54,139,132,71)(55,140,133,72)(56,141,134,73)(57,142,135,74)(58,143,136,75)(59,144,103,76)(60,145,104,77)(61,146,105,78)(62,147,106,79)(63,148,107,80)(64,149,108,81)(65,150,109,82)(66,151,110,83)(67,152,111,84)(68,153,112,85), (1,134,209,56)(2,135,210,57)(3,136,211,58)(4,103,212,59)(5,104,213,60)(6,105,214,61)(7,106,215,62)(8,107,216,63)(9,108,217,64)(10,109,218,65)(11,110,219,66)(12,111,220,67)(13,112,221,68)(14,113,222,35)(15,114,223,36)(16,115,224,37)(17,116,225,38)(18,117,226,39)(19,118,227,40)(20,119,228,41)(21,120,229,42)(22,121,230,43)(23,122,231,44)(24,123,232,45)(25,124,233,46)(26,125,234,47)(27,126,235,48)(28,127,236,49)(29,128,237,50)(30,129,238,51)(31,130,205,52)(32,131,206,53)(33,132,207,54)(34,133,208,55)(69,259,137,196)(70,260,138,197)(71,261,139,198)(72,262,140,199)(73,263,141,200)(74,264,142,201)(75,265,143,202)(76,266,144,203)(77,267,145,204)(78,268,146,171)(79,269,147,172)(80,270,148,173)(81,271,149,174)(82,272,150,175)(83,239,151,176)(84,240,152,177)(85,241,153,178)(86,242,154,179)(87,243,155,180)(88,244,156,181)(89,245,157,182)(90,246,158,183)(91,247,159,184)(92,248,160,185)(93,249,161,186)(94,250,162,187)(95,251,163,188)(96,252,164,189)(97,253,165,190)(98,254,166,191)(99,255,167,192)(100,256,168,193)(101,257,169,194)(102,258,170,195) );
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)], [(1,263,209,200),(2,264,210,201),(3,265,211,202),(4,266,212,203),(5,267,213,204),(6,268,214,171),(7,269,215,172),(8,270,216,173),(9,271,217,174),(10,272,218,175),(11,239,219,176),(12,240,220,177),(13,241,221,178),(14,242,222,179),(15,243,223,180),(16,244,224,181),(17,245,225,182),(18,246,226,183),(19,247,227,184),(20,248,228,185),(21,249,229,186),(22,250,230,187),(23,251,231,188),(24,252,232,189),(25,253,233,190),(26,254,234,191),(27,255,235,192),(28,256,236,193),(29,257,237,194),(30,258,238,195),(31,259,205,196),(32,260,206,197),(33,261,207,198),(34,262,208,199),(35,154,113,86),(36,155,114,87),(37,156,115,88),(38,157,116,89),(39,158,117,90),(40,159,118,91),(41,160,119,92),(42,161,120,93),(43,162,121,94),(44,163,122,95),(45,164,123,96),(46,165,124,97),(47,166,125,98),(48,167,126,99),(49,168,127,100),(50,169,128,101),(51,170,129,102),(52,137,130,69),(53,138,131,70),(54,139,132,71),(55,140,133,72),(56,141,134,73),(57,142,135,74),(58,143,136,75),(59,144,103,76),(60,145,104,77),(61,146,105,78),(62,147,106,79),(63,148,107,80),(64,149,108,81),(65,150,109,82),(66,151,110,83),(67,152,111,84),(68,153,112,85)], [(1,134,209,56),(2,135,210,57),(3,136,211,58),(4,103,212,59),(5,104,213,60),(6,105,214,61),(7,106,215,62),(8,107,216,63),(9,108,217,64),(10,109,218,65),(11,110,219,66),(12,111,220,67),(13,112,221,68),(14,113,222,35),(15,114,223,36),(16,115,224,37),(17,116,225,38),(18,117,226,39),(19,118,227,40),(20,119,228,41),(21,120,229,42),(22,121,230,43),(23,122,231,44),(24,123,232,45),(25,124,233,46),(26,125,234,47),(27,126,235,48),(28,127,236,49),(29,128,237,50),(30,129,238,51),(31,130,205,52),(32,131,206,53),(33,132,207,54),(34,133,208,55),(69,259,137,196),(70,260,138,197),(71,261,139,198),(72,262,140,199),(73,263,141,200),(74,264,142,201),(75,265,143,202),(76,266,144,203),(77,267,145,204),(78,268,146,171),(79,269,147,172),(80,270,148,173),(81,271,149,174),(82,272,150,175),(83,239,151,176),(84,240,152,177),(85,241,153,178),(86,242,154,179),(87,243,155,180),(88,244,156,181),(89,245,157,182),(90,246,158,183),(91,247,159,184),(92,248,160,185),(93,249,161,186),(94,250,162,187),(95,251,163,188),(96,252,164,189),(97,253,165,190),(98,254,166,191),(99,255,167,192),(100,256,168,193),(101,257,169,194),(102,258,170,195)]])
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 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | Q8 | Q8×C17 |
| kernel | Q8×C34 | C2×C68 | Q8×C17 | C2×Q8 | C2×C4 | Q8 | C34 | C2 |
| # reps | 1 | 3 | 4 | 16 | 48 | 64 | 2 | 32 |
Matrix representation of Q8×C34 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 1 | 0 | 0 |
| 0 | 136 | 135 |
| 0 | 1 | 1 |
| 136 | 0 | 0 |
| 0 | 51 | 51 |
| 0 | 43 | 86 |
G:=sub<GL(3,GF(137))| [136,0,0,0,22,0,0,0,22],[1,0,0,0,136,1,0,135,1],[136,0,0,0,51,43,0,51,86] >;
Q8×C34 in GAP, Magma, Sage, TeX
Q_8\times C_{34} % in TeX
G:=Group("Q8xC34"); // GroupNames label
G:=SmallGroup(272,48);
// by ID
G=gap.SmallGroup(272,48);
# by ID
G:=PCGroup([5,-2,-2,-2,-17,-2,680,1381,686]);
// Polycyclic
G:=Group<a,b,c|a^34=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export