direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C38, C38.12C23, C76.20C22, C4.4(C2×C38), (C2×C76).9C2, (C2×C4).3C38, C22.4(C2×C38), C2.2(C22×C38), (C2×C38).15C22, SmallGroup(304,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C38
G = < a,b,c | a38=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C38
►Regular action on 304 points
Generators in S304
(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)(267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)
(1 110 273 132)(2 111 274 133)(3 112 275 134)(4 113 276 135)(5 114 277 136)(6 77 278 137)(7 78 279 138)(8 79 280 139)(9 80 281 140)(10 81 282 141)(11 82 283 142)(12 83 284 143)(13 84 285 144)(14 85 286 145)(15 86 287 146)(16 87 288 147)(17 88 289 148)(18 89 290 149)(19 90 291 150)(20 91 292 151)(21 92 293 152)(22 93 294 115)(23 94 295 116)(24 95 296 117)(25 96 297 118)(26 97 298 119)(27 98 299 120)(28 99 300 121)(29 100 301 122)(30 101 302 123)(31 102 303 124)(32 103 304 125)(33 104 267 126)(34 105 268 127)(35 106 269 128)(36 107 270 129)(37 108 271 130)(38 109 272 131)(39 227 240 158)(40 228 241 159)(41 191 242 160)(42 192 243 161)(43 193 244 162)(44 194 245 163)(45 195 246 164)(46 196 247 165)(47 197 248 166)(48 198 249 167)(49 199 250 168)(50 200 251 169)(51 201 252 170)(52 202 253 171)(53 203 254 172)(54 204 255 173)(55 205 256 174)(56 206 257 175)(57 207 258 176)(58 208 259 177)(59 209 260 178)(60 210 261 179)(61 211 262 180)(62 212 263 181)(63 213 264 182)(64 214 265 183)(65 215 266 184)(66 216 229 185)(67 217 230 186)(68 218 231 187)(69 219 232 188)(70 220 233 189)(71 221 234 190)(72 222 235 153)(73 223 236 154)(74 224 237 155)(75 225 238 156)(76 226 239 157)
(1 59 273 260)(2 60 274 261)(3 61 275 262)(4 62 276 263)(5 63 277 264)(6 64 278 265)(7 65 279 266)(8 66 280 229)(9 67 281 230)(10 68 282 231)(11 69 283 232)(12 70 284 233)(13 71 285 234)(14 72 286 235)(15 73 287 236)(16 74 288 237)(17 75 289 238)(18 76 290 239)(19 39 291 240)(20 40 292 241)(21 41 293 242)(22 42 294 243)(23 43 295 244)(24 44 296 245)(25 45 297 246)(26 46 298 247)(27 47 299 248)(28 48 300 249)(29 49 301 250)(30 50 302 251)(31 51 303 252)(32 52 304 253)(33 53 267 254)(34 54 268 255)(35 55 269 256)(36 56 270 257)(37 57 271 258)(38 58 272 259)(77 183 137 214)(78 184 138 215)(79 185 139 216)(80 186 140 217)(81 187 141 218)(82 188 142 219)(83 189 143 220)(84 190 144 221)(85 153 145 222)(86 154 146 223)(87 155 147 224)(88 156 148 225)(89 157 149 226)(90 158 150 227)(91 159 151 228)(92 160 152 191)(93 161 115 192)(94 162 116 193)(95 163 117 194)(96 164 118 195)(97 165 119 196)(98 166 120 197)(99 167 121 198)(100 168 122 199)(101 169 123 200)(102 170 124 201)(103 171 125 202)(104 172 126 203)(105 173 127 204)(106 174 128 205)(107 175 129 206)(108 176 130 207)(109 177 131 208)(110 178 132 209)(111 179 133 210)(112 180 134 211)(113 181 135 212)(114 182 136 213)
G:=sub<Sym(304)| (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)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,110,273,132)(2,111,274,133)(3,112,275,134)(4,113,276,135)(5,114,277,136)(6,77,278,137)(7,78,279,138)(8,79,280,139)(9,80,281,140)(10,81,282,141)(11,82,283,142)(12,83,284,143)(13,84,285,144)(14,85,286,145)(15,86,287,146)(16,87,288,147)(17,88,289,148)(18,89,290,149)(19,90,291,150)(20,91,292,151)(21,92,293,152)(22,93,294,115)(23,94,295,116)(24,95,296,117)(25,96,297,118)(26,97,298,119)(27,98,299,120)(28,99,300,121)(29,100,301,122)(30,101,302,123)(31,102,303,124)(32,103,304,125)(33,104,267,126)(34,105,268,127)(35,106,269,128)(36,107,270,129)(37,108,271,130)(38,109,272,131)(39,227,240,158)(40,228,241,159)(41,191,242,160)(42,192,243,161)(43,193,244,162)(44,194,245,163)(45,195,246,164)(46,196,247,165)(47,197,248,166)(48,198,249,167)(49,199,250,168)(50,200,251,169)(51,201,252,170)(52,202,253,171)(53,203,254,172)(54,204,255,173)(55,205,256,174)(56,206,257,175)(57,207,258,176)(58,208,259,177)(59,209,260,178)(60,210,261,179)(61,211,262,180)(62,212,263,181)(63,213,264,182)(64,214,265,183)(65,215,266,184)(66,216,229,185)(67,217,230,186)(68,218,231,187)(69,219,232,188)(70,220,233,189)(71,221,234,190)(72,222,235,153)(73,223,236,154)(74,224,237,155)(75,225,238,156)(76,226,239,157), (1,59,273,260)(2,60,274,261)(3,61,275,262)(4,62,276,263)(5,63,277,264)(6,64,278,265)(7,65,279,266)(8,66,280,229)(9,67,281,230)(10,68,282,231)(11,69,283,232)(12,70,284,233)(13,71,285,234)(14,72,286,235)(15,73,287,236)(16,74,288,237)(17,75,289,238)(18,76,290,239)(19,39,291,240)(20,40,292,241)(21,41,293,242)(22,42,294,243)(23,43,295,244)(24,44,296,245)(25,45,297,246)(26,46,298,247)(27,47,299,248)(28,48,300,249)(29,49,301,250)(30,50,302,251)(31,51,303,252)(32,52,304,253)(33,53,267,254)(34,54,268,255)(35,55,269,256)(36,56,270,257)(37,57,271,258)(38,58,272,259)(77,183,137,214)(78,184,138,215)(79,185,139,216)(80,186,140,217)(81,187,141,218)(82,188,142,219)(83,189,143,220)(84,190,144,221)(85,153,145,222)(86,154,146,223)(87,155,147,224)(88,156,148,225)(89,157,149,226)(90,158,150,227)(91,159,151,228)(92,160,152,191)(93,161,115,192)(94,162,116,193)(95,163,117,194)(96,164,118,195)(97,165,119,196)(98,166,120,197)(99,167,121,198)(100,168,122,199)(101,169,123,200)(102,170,124,201)(103,171,125,202)(104,172,126,203)(105,173,127,204)(106,174,128,205)(107,175,129,206)(108,176,130,207)(109,177,131,208)(110,178,132,209)(111,179,133,210)(112,180,134,211)(113,181,135,212)(114,182,136,213)>;
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)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,110,273,132)(2,111,274,133)(3,112,275,134)(4,113,276,135)(5,114,277,136)(6,77,278,137)(7,78,279,138)(8,79,280,139)(9,80,281,140)(10,81,282,141)(11,82,283,142)(12,83,284,143)(13,84,285,144)(14,85,286,145)(15,86,287,146)(16,87,288,147)(17,88,289,148)(18,89,290,149)(19,90,291,150)(20,91,292,151)(21,92,293,152)(22,93,294,115)(23,94,295,116)(24,95,296,117)(25,96,297,118)(26,97,298,119)(27,98,299,120)(28,99,300,121)(29,100,301,122)(30,101,302,123)(31,102,303,124)(32,103,304,125)(33,104,267,126)(34,105,268,127)(35,106,269,128)(36,107,270,129)(37,108,271,130)(38,109,272,131)(39,227,240,158)(40,228,241,159)(41,191,242,160)(42,192,243,161)(43,193,244,162)(44,194,245,163)(45,195,246,164)(46,196,247,165)(47,197,248,166)(48,198,249,167)(49,199,250,168)(50,200,251,169)(51,201,252,170)(52,202,253,171)(53,203,254,172)(54,204,255,173)(55,205,256,174)(56,206,257,175)(57,207,258,176)(58,208,259,177)(59,209,260,178)(60,210,261,179)(61,211,262,180)(62,212,263,181)(63,213,264,182)(64,214,265,183)(65,215,266,184)(66,216,229,185)(67,217,230,186)(68,218,231,187)(69,219,232,188)(70,220,233,189)(71,221,234,190)(72,222,235,153)(73,223,236,154)(74,224,237,155)(75,225,238,156)(76,226,239,157), (1,59,273,260)(2,60,274,261)(3,61,275,262)(4,62,276,263)(5,63,277,264)(6,64,278,265)(7,65,279,266)(8,66,280,229)(9,67,281,230)(10,68,282,231)(11,69,283,232)(12,70,284,233)(13,71,285,234)(14,72,286,235)(15,73,287,236)(16,74,288,237)(17,75,289,238)(18,76,290,239)(19,39,291,240)(20,40,292,241)(21,41,293,242)(22,42,294,243)(23,43,295,244)(24,44,296,245)(25,45,297,246)(26,46,298,247)(27,47,299,248)(28,48,300,249)(29,49,301,250)(30,50,302,251)(31,51,303,252)(32,52,304,253)(33,53,267,254)(34,54,268,255)(35,55,269,256)(36,56,270,257)(37,57,271,258)(38,58,272,259)(77,183,137,214)(78,184,138,215)(79,185,139,216)(80,186,140,217)(81,187,141,218)(82,188,142,219)(83,189,143,220)(84,190,144,221)(85,153,145,222)(86,154,146,223)(87,155,147,224)(88,156,148,225)(89,157,149,226)(90,158,150,227)(91,159,151,228)(92,160,152,191)(93,161,115,192)(94,162,116,193)(95,163,117,194)(96,164,118,195)(97,165,119,196)(98,166,120,197)(99,167,121,198)(100,168,122,199)(101,169,123,200)(102,170,124,201)(103,171,125,202)(104,172,126,203)(105,173,127,204)(106,174,128,205)(107,175,129,206)(108,176,130,207)(109,177,131,208)(110,178,132,209)(111,179,133,210)(112,180,134,211)(113,181,135,212)(114,182,136,213) );
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),(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)], [(1,110,273,132),(2,111,274,133),(3,112,275,134),(4,113,276,135),(5,114,277,136),(6,77,278,137),(7,78,279,138),(8,79,280,139),(9,80,281,140),(10,81,282,141),(11,82,283,142),(12,83,284,143),(13,84,285,144),(14,85,286,145),(15,86,287,146),(16,87,288,147),(17,88,289,148),(18,89,290,149),(19,90,291,150),(20,91,292,151),(21,92,293,152),(22,93,294,115),(23,94,295,116),(24,95,296,117),(25,96,297,118),(26,97,298,119),(27,98,299,120),(28,99,300,121),(29,100,301,122),(30,101,302,123),(31,102,303,124),(32,103,304,125),(33,104,267,126),(34,105,268,127),(35,106,269,128),(36,107,270,129),(37,108,271,130),(38,109,272,131),(39,227,240,158),(40,228,241,159),(41,191,242,160),(42,192,243,161),(43,193,244,162),(44,194,245,163),(45,195,246,164),(46,196,247,165),(47,197,248,166),(48,198,249,167),(49,199,250,168),(50,200,251,169),(51,201,252,170),(52,202,253,171),(53,203,254,172),(54,204,255,173),(55,205,256,174),(56,206,257,175),(57,207,258,176),(58,208,259,177),(59,209,260,178),(60,210,261,179),(61,211,262,180),(62,212,263,181),(63,213,264,182),(64,214,265,183),(65,215,266,184),(66,216,229,185),(67,217,230,186),(68,218,231,187),(69,219,232,188),(70,220,233,189),(71,221,234,190),(72,222,235,153),(73,223,236,154),(74,224,237,155),(75,225,238,156),(76,226,239,157)], [(1,59,273,260),(2,60,274,261),(3,61,275,262),(4,62,276,263),(5,63,277,264),(6,64,278,265),(7,65,279,266),(8,66,280,229),(9,67,281,230),(10,68,282,231),(11,69,283,232),(12,70,284,233),(13,71,285,234),(14,72,286,235),(15,73,287,236),(16,74,288,237),(17,75,289,238),(18,76,290,239),(19,39,291,240),(20,40,292,241),(21,41,293,242),(22,42,294,243),(23,43,295,244),(24,44,296,245),(25,45,297,246),(26,46,298,247),(27,47,299,248),(28,48,300,249),(29,49,301,250),(30,50,302,251),(31,51,303,252),(32,52,304,253),(33,53,267,254),(34,54,268,255),(35,55,269,256),(36,56,270,257),(37,57,271,258),(38,58,272,259),(77,183,137,214),(78,184,138,215),(79,185,139,216),(80,186,140,217),(81,187,141,218),(82,188,142,219),(83,189,143,220),(84,190,144,221),(85,153,145,222),(86,154,146,223),(87,155,147,224),(88,156,148,225),(89,157,149,226),(90,158,150,227),(91,159,151,228),(92,160,152,191),(93,161,115,192),(94,162,116,193),(95,163,117,194),(96,164,118,195),(97,165,119,196),(98,166,120,197),(99,167,121,198),(100,168,122,199),(101,169,123,200),(102,170,124,201),(103,171,125,202),(104,172,126,203),(105,173,127,204),(106,174,128,205),(107,175,129,206),(108,176,130,207),(109,177,131,208),(110,178,132,209),(111,179,133,210),(112,180,134,211),(113,181,135,212),(114,182,136,213)]])
190 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 19A | ··· | 19R | 38A | ··· | 38BB | 76A | ··· | 76DD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
190 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | Q8 | Q8×C19 |
| kernel | Q8×C38 | C2×C76 | Q8×C19 | C2×Q8 | C2×C4 | Q8 | C38 | C2 |
| # reps | 1 | 3 | 4 | 18 | 54 | 72 | 2 | 36 |
Matrix representation of Q8×C38 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 172 | 0 |
| 0 | 0 | 172 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 228 | 0 |
| 228 | 0 | 0 |
| 0 | 208 | 4 |
| 0 | 4 | 21 |
G:=sub<GL(3,GF(229))| [228,0,0,0,172,0,0,0,172],[1,0,0,0,0,228,0,1,0],[228,0,0,0,208,4,0,4,21] >;
Q8×C38 in GAP, Magma, Sage, TeX
Q_8\times C_{38} % in TeX
G:=Group("Q8xC38"); // GroupNames label
G:=SmallGroup(304,39);
// by ID
G=gap.SmallGroup(304,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-19,-2,760,1541,766]);
// Polycyclic
G:=Group<a,b,c|a^38=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export