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## G = Q16×C3×C6order 288 = 25·32

### Direct product of C3×C6 and Q16

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — Q16×C3×C6
 Chief series C1 — C2 — C4 — C12 — C3×C12 — Q8×C32 — C32×Q16 — Q16×C3×C6
 Lower central C1 — C2 — C4 — Q16×C3×C6
 Upper central C1 — C62 — C6×C12 — Q16×C3×C6

Generators and relations for Q16×C3×C6
G = < a,b,c,d | a3=b6=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 180 conjugacy classes, 132 normal (16 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×4], Q8 [×2], C32, C12 [×8], C12 [×16], C2×C6 [×4], C2×C8, Q16 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], C24 [×8], C2×C12 [×4], C2×C12 [×8], C3×Q8 [×16], C3×Q8 [×8], C2×Q16, C3×C12 [×2], C3×C12 [×4], C62, C2×C24 [×4], C3×Q16 [×16], C6×Q8 [×8], C3×C24 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×4], Q8×C32 [×2], C6×Q16 [×4], C6×C24, C32×Q16 [×4], Q8×C3×C6 [×2], Q16×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], Q16 [×2], C2×D4, C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C2×Q16, C62 [×7], C3×Q16 [×8], C6×D4 [×4], D4×C32 [×2], C2×C62, C6×Q16 [×4], C32×Q16 [×2], D4×C3×C6, Q16×C3×C6

Smallest permutation representation of Q16×C3×C6
Regular action on 288 points
Generators in S288
(1 163 155)(2 164 156)(3 165 157)(4 166 158)(5 167 159)(6 168 160)(7 161 153)(8 162 154)(9 116 17)(10 117 18)(11 118 19)(12 119 20)(13 120 21)(14 113 22)(15 114 23)(16 115 24)(25 111 205)(26 112 206)(27 105 207)(28 106 208)(29 107 201)(30 108 202)(31 109 203)(32 110 204)(33 140 132)(34 141 133)(35 142 134)(36 143 135)(37 144 136)(38 137 129)(39 138 130)(40 139 131)(41 96 49)(42 89 50)(43 90 51)(44 91 52)(45 92 53)(46 93 54)(47 94 55)(48 95 56)(57 220 65)(58 221 66)(59 222 67)(60 223 68)(61 224 69)(62 217 70)(63 218 71)(64 219 72)(73 101 286)(74 102 287)(75 103 288)(76 104 281)(77 97 282)(78 98 283)(79 99 284)(80 100 285)(81 266 258)(82 267 259)(83 268 260)(84 269 261)(85 270 262)(86 271 263)(87 272 264)(88 265 257)(121 230 214)(122 231 215)(123 232 216)(124 225 209)(125 226 210)(126 227 211)(127 228 212)(128 229 213)(145 173 277)(146 174 278)(147 175 279)(148 176 280)(149 169 273)(150 170 274)(151 171 275)(152 172 276)(177 193 185)(178 194 186)(179 195 187)(180 196 188)(181 197 189)(182 198 190)(183 199 191)(184 200 192)(233 249 241)(234 250 242)(235 251 243)(236 252 244)(237 253 245)(238 254 246)(239 255 247)(240 256 248)
(1 102 82 63 43 279)(2 103 83 64 44 280)(3 104 84 57 45 273)(4 97 85 58 46 274)(5 98 86 59 47 275)(6 99 87 60 48 276)(7 100 88 61 41 277)(8 101 81 62 42 278)(9 245 225 201 189 144)(10 246 226 202 190 137)(11 247 227 203 191 138)(12 248 228 204 192 139)(13 241 229 205 185 140)(14 242 230 206 186 141)(15 243 231 207 187 142)(16 244 232 208 188 143)(17 253 124 107 197 37)(18 254 125 108 198 38)(19 255 126 109 199 39)(20 256 127 110 200 40)(21 249 128 111 193 33)(22 250 121 112 194 34)(23 251 122 105 195 35)(24 252 123 106 196 36)(25 177 132 120 233 213)(26 178 133 113 234 214)(27 179 134 114 235 215)(28 180 135 115 236 216)(29 181 136 116 237 209)(30 182 129 117 238 210)(31 183 130 118 239 211)(32 184 131 119 240 212)(49 173 153 80 257 69)(50 174 154 73 258 70)(51 175 155 74 259 71)(52 176 156 75 260 72)(53 169 157 76 261 65)(54 170 158 77 262 66)(55 171 159 78 263 67)(56 172 160 79 264 68)(89 146 162 286 266 217)(90 147 163 287 267 218)(91 148 164 288 268 219)(92 149 165 281 269 220)(93 150 166 282 270 221)(94 151 167 283 271 222)(95 152 168 284 272 223)(96 145 161 285 265 224)
(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)
(1 239 5 235)(2 238 6 234)(3 237 7 233)(4 236 8 240)(9 173 13 169)(10 172 14 176)(11 171 15 175)(12 170 16 174)(17 145 21 149)(18 152 22 148)(19 151 23 147)(20 150 24 146)(25 84 29 88)(26 83 30 87)(27 82 31 86)(28 81 32 85)(33 92 37 96)(34 91 38 95)(35 90 39 94)(36 89 40 93)(41 132 45 136)(42 131 46 135)(43 130 47 134)(44 129 48 133)(49 140 53 144)(50 139 54 143)(51 138 55 142)(52 137 56 141)(57 181 61 177)(58 180 62 184)(59 179 63 183)(60 178 64 182)(65 189 69 185)(66 188 70 192)(67 187 71 191)(68 186 72 190)(73 228 77 232)(74 227 78 231)(75 226 79 230)(76 225 80 229)(97 216 101 212)(98 215 102 211)(99 214 103 210)(100 213 104 209)(105 267 109 271)(106 266 110 270)(107 265 111 269)(108 272 112 268)(113 280 117 276)(114 279 118 275)(115 278 119 274)(116 277 120 273)(121 288 125 284)(122 287 126 283)(123 286 127 282)(124 285 128 281)(153 241 157 245)(154 248 158 244)(155 247 159 243)(156 246 160 242)(161 249 165 253)(162 256 166 252)(163 255 167 251)(164 254 168 250)(193 220 197 224)(194 219 198 223)(195 218 199 222)(196 217 200 221)(201 257 205 261)(202 264 206 260)(203 263 207 259)(204 262 208 258)

G:=sub<Sym(288)| (1,163,155)(2,164,156)(3,165,157)(4,166,158)(5,167,159)(6,168,160)(7,161,153)(8,162,154)(9,116,17)(10,117,18)(11,118,19)(12,119,20)(13,120,21)(14,113,22)(15,114,23)(16,115,24)(25,111,205)(26,112,206)(27,105,207)(28,106,208)(29,107,201)(30,108,202)(31,109,203)(32,110,204)(33,140,132)(34,141,133)(35,142,134)(36,143,135)(37,144,136)(38,137,129)(39,138,130)(40,139,131)(41,96,49)(42,89,50)(43,90,51)(44,91,52)(45,92,53)(46,93,54)(47,94,55)(48,95,56)(57,220,65)(58,221,66)(59,222,67)(60,223,68)(61,224,69)(62,217,70)(63,218,71)(64,219,72)(73,101,286)(74,102,287)(75,103,288)(76,104,281)(77,97,282)(78,98,283)(79,99,284)(80,100,285)(81,266,258)(82,267,259)(83,268,260)(84,269,261)(85,270,262)(86,271,263)(87,272,264)(88,265,257)(121,230,214)(122,231,215)(123,232,216)(124,225,209)(125,226,210)(126,227,211)(127,228,212)(128,229,213)(145,173,277)(146,174,278)(147,175,279)(148,176,280)(149,169,273)(150,170,274)(151,171,275)(152,172,276)(177,193,185)(178,194,186)(179,195,187)(180,196,188)(181,197,189)(182,198,190)(183,199,191)(184,200,192)(233,249,241)(234,250,242)(235,251,243)(236,252,244)(237,253,245)(238,254,246)(239,255,247)(240,256,248), (1,102,82,63,43,279)(2,103,83,64,44,280)(3,104,84,57,45,273)(4,97,85,58,46,274)(5,98,86,59,47,275)(6,99,87,60,48,276)(7,100,88,61,41,277)(8,101,81,62,42,278)(9,245,225,201,189,144)(10,246,226,202,190,137)(11,247,227,203,191,138)(12,248,228,204,192,139)(13,241,229,205,185,140)(14,242,230,206,186,141)(15,243,231,207,187,142)(16,244,232,208,188,143)(17,253,124,107,197,37)(18,254,125,108,198,38)(19,255,126,109,199,39)(20,256,127,110,200,40)(21,249,128,111,193,33)(22,250,121,112,194,34)(23,251,122,105,195,35)(24,252,123,106,196,36)(25,177,132,120,233,213)(26,178,133,113,234,214)(27,179,134,114,235,215)(28,180,135,115,236,216)(29,181,136,116,237,209)(30,182,129,117,238,210)(31,183,130,118,239,211)(32,184,131,119,240,212)(49,173,153,80,257,69)(50,174,154,73,258,70)(51,175,155,74,259,71)(52,176,156,75,260,72)(53,169,157,76,261,65)(54,170,158,77,262,66)(55,171,159,78,263,67)(56,172,160,79,264,68)(89,146,162,286,266,217)(90,147,163,287,267,218)(91,148,164,288,268,219)(92,149,165,281,269,220)(93,150,166,282,270,221)(94,151,167,283,271,222)(95,152,168,284,272,223)(96,145,161,285,265,224), (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), (1,239,5,235)(2,238,6,234)(3,237,7,233)(4,236,8,240)(9,173,13,169)(10,172,14,176)(11,171,15,175)(12,170,16,174)(17,145,21,149)(18,152,22,148)(19,151,23,147)(20,150,24,146)(25,84,29,88)(26,83,30,87)(27,82,31,86)(28,81,32,85)(33,92,37,96)(34,91,38,95)(35,90,39,94)(36,89,40,93)(41,132,45,136)(42,131,46,135)(43,130,47,134)(44,129,48,133)(49,140,53,144)(50,139,54,143)(51,138,55,142)(52,137,56,141)(57,181,61,177)(58,180,62,184)(59,179,63,183)(60,178,64,182)(65,189,69,185)(66,188,70,192)(67,187,71,191)(68,186,72,190)(73,228,77,232)(74,227,78,231)(75,226,79,230)(76,225,80,229)(97,216,101,212)(98,215,102,211)(99,214,103,210)(100,213,104,209)(105,267,109,271)(106,266,110,270)(107,265,111,269)(108,272,112,268)(113,280,117,276)(114,279,118,275)(115,278,119,274)(116,277,120,273)(121,288,125,284)(122,287,126,283)(123,286,127,282)(124,285,128,281)(153,241,157,245)(154,248,158,244)(155,247,159,243)(156,246,160,242)(161,249,165,253)(162,256,166,252)(163,255,167,251)(164,254,168,250)(193,220,197,224)(194,219,198,223)(195,218,199,222)(196,217,200,221)(201,257,205,261)(202,264,206,260)(203,263,207,259)(204,262,208,258)>;

G:=Group( (1,163,155)(2,164,156)(3,165,157)(4,166,158)(5,167,159)(6,168,160)(7,161,153)(8,162,154)(9,116,17)(10,117,18)(11,118,19)(12,119,20)(13,120,21)(14,113,22)(15,114,23)(16,115,24)(25,111,205)(26,112,206)(27,105,207)(28,106,208)(29,107,201)(30,108,202)(31,109,203)(32,110,204)(33,140,132)(34,141,133)(35,142,134)(36,143,135)(37,144,136)(38,137,129)(39,138,130)(40,139,131)(41,96,49)(42,89,50)(43,90,51)(44,91,52)(45,92,53)(46,93,54)(47,94,55)(48,95,56)(57,220,65)(58,221,66)(59,222,67)(60,223,68)(61,224,69)(62,217,70)(63,218,71)(64,219,72)(73,101,286)(74,102,287)(75,103,288)(76,104,281)(77,97,282)(78,98,283)(79,99,284)(80,100,285)(81,266,258)(82,267,259)(83,268,260)(84,269,261)(85,270,262)(86,271,263)(87,272,264)(88,265,257)(121,230,214)(122,231,215)(123,232,216)(124,225,209)(125,226,210)(126,227,211)(127,228,212)(128,229,213)(145,173,277)(146,174,278)(147,175,279)(148,176,280)(149,169,273)(150,170,274)(151,171,275)(152,172,276)(177,193,185)(178,194,186)(179,195,187)(180,196,188)(181,197,189)(182,198,190)(183,199,191)(184,200,192)(233,249,241)(234,250,242)(235,251,243)(236,252,244)(237,253,245)(238,254,246)(239,255,247)(240,256,248), (1,102,82,63,43,279)(2,103,83,64,44,280)(3,104,84,57,45,273)(4,97,85,58,46,274)(5,98,86,59,47,275)(6,99,87,60,48,276)(7,100,88,61,41,277)(8,101,81,62,42,278)(9,245,225,201,189,144)(10,246,226,202,190,137)(11,247,227,203,191,138)(12,248,228,204,192,139)(13,241,229,205,185,140)(14,242,230,206,186,141)(15,243,231,207,187,142)(16,244,232,208,188,143)(17,253,124,107,197,37)(18,254,125,108,198,38)(19,255,126,109,199,39)(20,256,127,110,200,40)(21,249,128,111,193,33)(22,250,121,112,194,34)(23,251,122,105,195,35)(24,252,123,106,196,36)(25,177,132,120,233,213)(26,178,133,113,234,214)(27,179,134,114,235,215)(28,180,135,115,236,216)(29,181,136,116,237,209)(30,182,129,117,238,210)(31,183,130,118,239,211)(32,184,131,119,240,212)(49,173,153,80,257,69)(50,174,154,73,258,70)(51,175,155,74,259,71)(52,176,156,75,260,72)(53,169,157,76,261,65)(54,170,158,77,262,66)(55,171,159,78,263,67)(56,172,160,79,264,68)(89,146,162,286,266,217)(90,147,163,287,267,218)(91,148,164,288,268,219)(92,149,165,281,269,220)(93,150,166,282,270,221)(94,151,167,283,271,222)(95,152,168,284,272,223)(96,145,161,285,265,224), (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), (1,239,5,235)(2,238,6,234)(3,237,7,233)(4,236,8,240)(9,173,13,169)(10,172,14,176)(11,171,15,175)(12,170,16,174)(17,145,21,149)(18,152,22,148)(19,151,23,147)(20,150,24,146)(25,84,29,88)(26,83,30,87)(27,82,31,86)(28,81,32,85)(33,92,37,96)(34,91,38,95)(35,90,39,94)(36,89,40,93)(41,132,45,136)(42,131,46,135)(43,130,47,134)(44,129,48,133)(49,140,53,144)(50,139,54,143)(51,138,55,142)(52,137,56,141)(57,181,61,177)(58,180,62,184)(59,179,63,183)(60,178,64,182)(65,189,69,185)(66,188,70,192)(67,187,71,191)(68,186,72,190)(73,228,77,232)(74,227,78,231)(75,226,79,230)(76,225,80,229)(97,216,101,212)(98,215,102,211)(99,214,103,210)(100,213,104,209)(105,267,109,271)(106,266,110,270)(107,265,111,269)(108,272,112,268)(113,280,117,276)(114,279,118,275)(115,278,119,274)(116,277,120,273)(121,288,125,284)(122,287,126,283)(123,286,127,282)(124,285,128,281)(153,241,157,245)(154,248,158,244)(155,247,159,243)(156,246,160,242)(161,249,165,253)(162,256,166,252)(163,255,167,251)(164,254,168,250)(193,220,197,224)(194,219,198,223)(195,218,199,222)(196,217,200,221)(201,257,205,261)(202,264,206,260)(203,263,207,259)(204,262,208,258) );

G=PermutationGroup([(1,163,155),(2,164,156),(3,165,157),(4,166,158),(5,167,159),(6,168,160),(7,161,153),(8,162,154),(9,116,17),(10,117,18),(11,118,19),(12,119,20),(13,120,21),(14,113,22),(15,114,23),(16,115,24),(25,111,205),(26,112,206),(27,105,207),(28,106,208),(29,107,201),(30,108,202),(31,109,203),(32,110,204),(33,140,132),(34,141,133),(35,142,134),(36,143,135),(37,144,136),(38,137,129),(39,138,130),(40,139,131),(41,96,49),(42,89,50),(43,90,51),(44,91,52),(45,92,53),(46,93,54),(47,94,55),(48,95,56),(57,220,65),(58,221,66),(59,222,67),(60,223,68),(61,224,69),(62,217,70),(63,218,71),(64,219,72),(73,101,286),(74,102,287),(75,103,288),(76,104,281),(77,97,282),(78,98,283),(79,99,284),(80,100,285),(81,266,258),(82,267,259),(83,268,260),(84,269,261),(85,270,262),(86,271,263),(87,272,264),(88,265,257),(121,230,214),(122,231,215),(123,232,216),(124,225,209),(125,226,210),(126,227,211),(127,228,212),(128,229,213),(145,173,277),(146,174,278),(147,175,279),(148,176,280),(149,169,273),(150,170,274),(151,171,275),(152,172,276),(177,193,185),(178,194,186),(179,195,187),(180,196,188),(181,197,189),(182,198,190),(183,199,191),(184,200,192),(233,249,241),(234,250,242),(235,251,243),(236,252,244),(237,253,245),(238,254,246),(239,255,247),(240,256,248)], [(1,102,82,63,43,279),(2,103,83,64,44,280),(3,104,84,57,45,273),(4,97,85,58,46,274),(5,98,86,59,47,275),(6,99,87,60,48,276),(7,100,88,61,41,277),(8,101,81,62,42,278),(9,245,225,201,189,144),(10,246,226,202,190,137),(11,247,227,203,191,138),(12,248,228,204,192,139),(13,241,229,205,185,140),(14,242,230,206,186,141),(15,243,231,207,187,142),(16,244,232,208,188,143),(17,253,124,107,197,37),(18,254,125,108,198,38),(19,255,126,109,199,39),(20,256,127,110,200,40),(21,249,128,111,193,33),(22,250,121,112,194,34),(23,251,122,105,195,35),(24,252,123,106,196,36),(25,177,132,120,233,213),(26,178,133,113,234,214),(27,179,134,114,235,215),(28,180,135,115,236,216),(29,181,136,116,237,209),(30,182,129,117,238,210),(31,183,130,118,239,211),(32,184,131,119,240,212),(49,173,153,80,257,69),(50,174,154,73,258,70),(51,175,155,74,259,71),(52,176,156,75,260,72),(53,169,157,76,261,65),(54,170,158,77,262,66),(55,171,159,78,263,67),(56,172,160,79,264,68),(89,146,162,286,266,217),(90,147,163,287,267,218),(91,148,164,288,268,219),(92,149,165,281,269,220),(93,150,166,282,270,221),(94,151,167,283,271,222),(95,152,168,284,272,223),(96,145,161,285,265,224)], [(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)], [(1,239,5,235),(2,238,6,234),(3,237,7,233),(4,236,8,240),(9,173,13,169),(10,172,14,176),(11,171,15,175),(12,170,16,174),(17,145,21,149),(18,152,22,148),(19,151,23,147),(20,150,24,146),(25,84,29,88),(26,83,30,87),(27,82,31,86),(28,81,32,85),(33,92,37,96),(34,91,38,95),(35,90,39,94),(36,89,40,93),(41,132,45,136),(42,131,46,135),(43,130,47,134),(44,129,48,133),(49,140,53,144),(50,139,54,143),(51,138,55,142),(52,137,56,141),(57,181,61,177),(58,180,62,184),(59,179,63,183),(60,178,64,182),(65,189,69,185),(66,188,70,192),(67,187,71,191),(68,186,72,190),(73,228,77,232),(74,227,78,231),(75,226,79,230),(76,225,80,229),(97,216,101,212),(98,215,102,211),(99,214,103,210),(100,213,104,209),(105,267,109,271),(106,266,110,270),(107,265,111,269),(108,272,112,268),(113,280,117,276),(114,279,118,275),(115,278,119,274),(116,277,120,273),(121,288,125,284),(122,287,126,283),(123,286,127,282),(124,285,128,281),(153,241,157,245),(154,248,158,244),(155,247,159,243),(156,246,160,242),(161,249,165,253),(162,256,166,252),(163,255,167,251),(164,254,168,250),(193,220,197,224),(194,219,198,223),(195,218,199,222),(196,217,200,221),(201,257,205,261),(202,264,206,260),(203,263,207,259),(204,262,208,258)])

126 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 4E 4F 6A ··· 6X 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AV 24A ··· 24AF order 1 2 2 2 3 ··· 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 ··· 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 Q16 C3×D4 C3×D4 C3×Q16 kernel Q16×C3×C6 C6×C24 C32×Q16 Q8×C3×C6 C6×Q16 C2×C24 C3×Q16 C6×Q8 C3×C12 C62 C3×C6 C12 C2×C6 C6 # reps 1 1 4 2 8 8 32 16 1 1 4 8 8 32

Matrix representation of Q16×C3×C6 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 17 12 0 0 61 56 0 0 0 0 32 32 0 0 57 0
,
 0 72 0 0 72 0 0 0 0 0 56 57 0 0 9 17
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[17,61,0,0,12,56,0,0,0,0,32,57,0,0,32,0],[0,72,0,0,72,0,0,0,0,0,56,9,0,0,57,17] >;

Q16×C3×C6 in GAP, Magma, Sage, TeX

Q_{16}\times C_3\times C_6
% in TeX

G:=Group("Q16xC3xC6");
// GroupNames label

G:=SmallGroup(288,831);
// by ID

G=gap.SmallGroup(288,831);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,1016,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽