direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C37, C4.C74, C148.3C2, C74.7C22, C2.2(C2×C74), SmallGroup(296,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C37
G = < a,b,c | a37=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C37
►Regular action on 296 points
Generators in S296
(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)(260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296)
(1 71 104 161)(2 72 105 162)(3 73 106 163)(4 74 107 164)(5 38 108 165)(6 39 109 166)(7 40 110 167)(8 41 111 168)(9 42 75 169)(10 43 76 170)(11 44 77 171)(12 45 78 172)(13 46 79 173)(14 47 80 174)(15 48 81 175)(16 49 82 176)(17 50 83 177)(18 51 84 178)(19 52 85 179)(20 53 86 180)(21 54 87 181)(22 55 88 182)(23 56 89 183)(24 57 90 184)(25 58 91 185)(26 59 92 149)(27 60 93 150)(28 61 94 151)(29 62 95 152)(30 63 96 153)(31 64 97 154)(32 65 98 155)(33 66 99 156)(34 67 100 157)(35 68 101 158)(36 69 102 159)(37 70 103 160)(112 267 221 235)(113 268 222 236)(114 269 186 237)(115 270 187 238)(116 271 188 239)(117 272 189 240)(118 273 190 241)(119 274 191 242)(120 275 192 243)(121 276 193 244)(122 277 194 245)(123 278 195 246)(124 279 196 247)(125 280 197 248)(126 281 198 249)(127 282 199 250)(128 283 200 251)(129 284 201 252)(130 285 202 253)(131 286 203 254)(132 287 204 255)(133 288 205 256)(134 289 206 257)(135 290 207 258)(136 291 208 259)(137 292 209 223)(138 293 210 224)(139 294 211 225)(140 295 212 226)(141 296 213 227)(142 260 214 228)(143 261 215 229)(144 262 216 230)(145 263 217 231)(146 264 218 232)(147 265 219 233)(148 266 220 234)
(1 291 104 259)(2 292 105 223)(3 293 106 224)(4 294 107 225)(5 295 108 226)(6 296 109 227)(7 260 110 228)(8 261 111 229)(9 262 75 230)(10 263 76 231)(11 264 77 232)(12 265 78 233)(13 266 79 234)(14 267 80 235)(15 268 81 236)(16 269 82 237)(17 270 83 238)(18 271 84 239)(19 272 85 240)(20 273 86 241)(21 274 87 242)(22 275 88 243)(23 276 89 244)(24 277 90 245)(25 278 91 246)(26 279 92 247)(27 280 93 248)(28 281 94 249)(29 282 95 250)(30 283 96 251)(31 284 97 252)(32 285 98 253)(33 286 99 254)(34 287 100 255)(35 288 101 256)(36 289 102 257)(37 290 103 258)(38 140 165 212)(39 141 166 213)(40 142 167 214)(41 143 168 215)(42 144 169 216)(43 145 170 217)(44 146 171 218)(45 147 172 219)(46 148 173 220)(47 112 174 221)(48 113 175 222)(49 114 176 186)(50 115 177 187)(51 116 178 188)(52 117 179 189)(53 118 180 190)(54 119 181 191)(55 120 182 192)(56 121 183 193)(57 122 184 194)(58 123 185 195)(59 124 149 196)(60 125 150 197)(61 126 151 198)(62 127 152 199)(63 128 153 200)(64 129 154 201)(65 130 155 202)(66 131 156 203)(67 132 157 204)(68 133 158 205)(69 134 159 206)(70 135 160 207)(71 136 161 208)(72 137 162 209)(73 138 163 210)(74 139 164 211)
G:=sub<Sym(296)| (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)(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296), (1,71,104,161)(2,72,105,162)(3,73,106,163)(4,74,107,164)(5,38,108,165)(6,39,109,166)(7,40,110,167)(8,41,111,168)(9,42,75,169)(10,43,76,170)(11,44,77,171)(12,45,78,172)(13,46,79,173)(14,47,80,174)(15,48,81,175)(16,49,82,176)(17,50,83,177)(18,51,84,178)(19,52,85,179)(20,53,86,180)(21,54,87,181)(22,55,88,182)(23,56,89,183)(24,57,90,184)(25,58,91,185)(26,59,92,149)(27,60,93,150)(28,61,94,151)(29,62,95,152)(30,63,96,153)(31,64,97,154)(32,65,98,155)(33,66,99,156)(34,67,100,157)(35,68,101,158)(36,69,102,159)(37,70,103,160)(112,267,221,235)(113,268,222,236)(114,269,186,237)(115,270,187,238)(116,271,188,239)(117,272,189,240)(118,273,190,241)(119,274,191,242)(120,275,192,243)(121,276,193,244)(122,277,194,245)(123,278,195,246)(124,279,196,247)(125,280,197,248)(126,281,198,249)(127,282,199,250)(128,283,200,251)(129,284,201,252)(130,285,202,253)(131,286,203,254)(132,287,204,255)(133,288,205,256)(134,289,206,257)(135,290,207,258)(136,291,208,259)(137,292,209,223)(138,293,210,224)(139,294,211,225)(140,295,212,226)(141,296,213,227)(142,260,214,228)(143,261,215,229)(144,262,216,230)(145,263,217,231)(146,264,218,232)(147,265,219,233)(148,266,220,234), (1,291,104,259)(2,292,105,223)(3,293,106,224)(4,294,107,225)(5,295,108,226)(6,296,109,227)(7,260,110,228)(8,261,111,229)(9,262,75,230)(10,263,76,231)(11,264,77,232)(12,265,78,233)(13,266,79,234)(14,267,80,235)(15,268,81,236)(16,269,82,237)(17,270,83,238)(18,271,84,239)(19,272,85,240)(20,273,86,241)(21,274,87,242)(22,275,88,243)(23,276,89,244)(24,277,90,245)(25,278,91,246)(26,279,92,247)(27,280,93,248)(28,281,94,249)(29,282,95,250)(30,283,96,251)(31,284,97,252)(32,285,98,253)(33,286,99,254)(34,287,100,255)(35,288,101,256)(36,289,102,257)(37,290,103,258)(38,140,165,212)(39,141,166,213)(40,142,167,214)(41,143,168,215)(42,144,169,216)(43,145,170,217)(44,146,171,218)(45,147,172,219)(46,148,173,220)(47,112,174,221)(48,113,175,222)(49,114,176,186)(50,115,177,187)(51,116,178,188)(52,117,179,189)(53,118,180,190)(54,119,181,191)(55,120,182,192)(56,121,183,193)(57,122,184,194)(58,123,185,195)(59,124,149,196)(60,125,150,197)(61,126,151,198)(62,127,152,199)(63,128,153,200)(64,129,154,201)(65,130,155,202)(66,131,156,203)(67,132,157,204)(68,133,158,205)(69,134,159,206)(70,135,160,207)(71,136,161,208)(72,137,162,209)(73,138,163,210)(74,139,164,211)>;
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)(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296), (1,71,104,161)(2,72,105,162)(3,73,106,163)(4,74,107,164)(5,38,108,165)(6,39,109,166)(7,40,110,167)(8,41,111,168)(9,42,75,169)(10,43,76,170)(11,44,77,171)(12,45,78,172)(13,46,79,173)(14,47,80,174)(15,48,81,175)(16,49,82,176)(17,50,83,177)(18,51,84,178)(19,52,85,179)(20,53,86,180)(21,54,87,181)(22,55,88,182)(23,56,89,183)(24,57,90,184)(25,58,91,185)(26,59,92,149)(27,60,93,150)(28,61,94,151)(29,62,95,152)(30,63,96,153)(31,64,97,154)(32,65,98,155)(33,66,99,156)(34,67,100,157)(35,68,101,158)(36,69,102,159)(37,70,103,160)(112,267,221,235)(113,268,222,236)(114,269,186,237)(115,270,187,238)(116,271,188,239)(117,272,189,240)(118,273,190,241)(119,274,191,242)(120,275,192,243)(121,276,193,244)(122,277,194,245)(123,278,195,246)(124,279,196,247)(125,280,197,248)(126,281,198,249)(127,282,199,250)(128,283,200,251)(129,284,201,252)(130,285,202,253)(131,286,203,254)(132,287,204,255)(133,288,205,256)(134,289,206,257)(135,290,207,258)(136,291,208,259)(137,292,209,223)(138,293,210,224)(139,294,211,225)(140,295,212,226)(141,296,213,227)(142,260,214,228)(143,261,215,229)(144,262,216,230)(145,263,217,231)(146,264,218,232)(147,265,219,233)(148,266,220,234), (1,291,104,259)(2,292,105,223)(3,293,106,224)(4,294,107,225)(5,295,108,226)(6,296,109,227)(7,260,110,228)(8,261,111,229)(9,262,75,230)(10,263,76,231)(11,264,77,232)(12,265,78,233)(13,266,79,234)(14,267,80,235)(15,268,81,236)(16,269,82,237)(17,270,83,238)(18,271,84,239)(19,272,85,240)(20,273,86,241)(21,274,87,242)(22,275,88,243)(23,276,89,244)(24,277,90,245)(25,278,91,246)(26,279,92,247)(27,280,93,248)(28,281,94,249)(29,282,95,250)(30,283,96,251)(31,284,97,252)(32,285,98,253)(33,286,99,254)(34,287,100,255)(35,288,101,256)(36,289,102,257)(37,290,103,258)(38,140,165,212)(39,141,166,213)(40,142,167,214)(41,143,168,215)(42,144,169,216)(43,145,170,217)(44,146,171,218)(45,147,172,219)(46,148,173,220)(47,112,174,221)(48,113,175,222)(49,114,176,186)(50,115,177,187)(51,116,178,188)(52,117,179,189)(53,118,180,190)(54,119,181,191)(55,120,182,192)(56,121,183,193)(57,122,184,194)(58,123,185,195)(59,124,149,196)(60,125,150,197)(61,126,151,198)(62,127,152,199)(63,128,153,200)(64,129,154,201)(65,130,155,202)(66,131,156,203)(67,132,157,204)(68,133,158,205)(69,134,159,206)(70,135,160,207)(71,136,161,208)(72,137,162,209)(73,138,163,210)(74,139,164,211) );
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),(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)], [(1,71,104,161),(2,72,105,162),(3,73,106,163),(4,74,107,164),(5,38,108,165),(6,39,109,166),(7,40,110,167),(8,41,111,168),(9,42,75,169),(10,43,76,170),(11,44,77,171),(12,45,78,172),(13,46,79,173),(14,47,80,174),(15,48,81,175),(16,49,82,176),(17,50,83,177),(18,51,84,178),(19,52,85,179),(20,53,86,180),(21,54,87,181),(22,55,88,182),(23,56,89,183),(24,57,90,184),(25,58,91,185),(26,59,92,149),(27,60,93,150),(28,61,94,151),(29,62,95,152),(30,63,96,153),(31,64,97,154),(32,65,98,155),(33,66,99,156),(34,67,100,157),(35,68,101,158),(36,69,102,159),(37,70,103,160),(112,267,221,235),(113,268,222,236),(114,269,186,237),(115,270,187,238),(116,271,188,239),(117,272,189,240),(118,273,190,241),(119,274,191,242),(120,275,192,243),(121,276,193,244),(122,277,194,245),(123,278,195,246),(124,279,196,247),(125,280,197,248),(126,281,198,249),(127,282,199,250),(128,283,200,251),(129,284,201,252),(130,285,202,253),(131,286,203,254),(132,287,204,255),(133,288,205,256),(134,289,206,257),(135,290,207,258),(136,291,208,259),(137,292,209,223),(138,293,210,224),(139,294,211,225),(140,295,212,226),(141,296,213,227),(142,260,214,228),(143,261,215,229),(144,262,216,230),(145,263,217,231),(146,264,218,232),(147,265,219,233),(148,266,220,234)], [(1,291,104,259),(2,292,105,223),(3,293,106,224),(4,294,107,225),(5,295,108,226),(6,296,109,227),(7,260,110,228),(8,261,111,229),(9,262,75,230),(10,263,76,231),(11,264,77,232),(12,265,78,233),(13,266,79,234),(14,267,80,235),(15,268,81,236),(16,269,82,237),(17,270,83,238),(18,271,84,239),(19,272,85,240),(20,273,86,241),(21,274,87,242),(22,275,88,243),(23,276,89,244),(24,277,90,245),(25,278,91,246),(26,279,92,247),(27,280,93,248),(28,281,94,249),(29,282,95,250),(30,283,96,251),(31,284,97,252),(32,285,98,253),(33,286,99,254),(34,287,100,255),(35,288,101,256),(36,289,102,257),(37,290,103,258),(38,140,165,212),(39,141,166,213),(40,142,167,214),(41,143,168,215),(42,144,169,216),(43,145,170,217),(44,146,171,218),(45,147,172,219),(46,148,173,220),(47,112,174,221),(48,113,175,222),(49,114,176,186),(50,115,177,187),(51,116,178,188),(52,117,179,189),(53,118,180,190),(54,119,181,191),(55,120,182,192),(56,121,183,193),(57,122,184,194),(58,123,185,195),(59,124,149,196),(60,125,150,197),(61,126,151,198),(62,127,152,199),(63,128,153,200),(64,129,154,201),(65,130,155,202),(66,131,156,203),(67,132,157,204),(68,133,158,205),(69,134,159,206),(70,135,160,207),(71,136,161,208),(72,137,162,209),(73,138,163,210),(74,139,164,211)]])
185 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 37A | ··· | 37AJ | 74A | ··· | 74AJ | 148A | ··· | 148DD |
| order | 1 | 2 | 4 | 4 | 4 | 37 | ··· | 37 | 74 | ··· | 74 | 148 | ··· | 148 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
185 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C37 | C74 | Q8 | Q8×C37 |
| kernel | Q8×C37 | C148 | Q8 | C4 | C37 | C1 |
| # reps | 1 | 3 | 36 | 108 | 1 | 36 |
Matrix representation of Q8×C37 ►in GL2(𝔽149) generated by
| 19 | 0 |
| 0 | 19 |
| 14 | 147 |
| 24 | 135 |
| 54 | 68 |
| 82 | 95 |
G:=sub<GL(2,GF(149))| [19,0,0,19],[14,24,147,135],[54,82,68,95] >;
Q8×C37 in GAP, Magma, Sage, TeX
Q_8\times C_{37} % in TeX
G:=Group("Q8xC37"); // GroupNames label
G:=SmallGroup(296,11);
// by ID
G=gap.SmallGroup(296,11);
# by ID
G:=PCGroup([4,-2,-2,-37,-2,592,1201,597]);
// Polycyclic
G:=Group<a,b,c|a^37=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export