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## G = Q8×C3×C12order 288 = 25·32

### Direct product of C3×C12 and Q8

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C3×C12
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C32×C4⋊C4 — Q8×C3×C12
 Lower central C1 — C2 — Q8×C3×C12
 Upper central C1 — C6×C12 — Q8×C3×C12

Generators and relations for Q8×C3×C12
G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 210 conjugacy classes, 192 normal (16 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C32, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C2×C12, C3×Q8, C4×Q8, C3×C12, C3×C12, C62, C4×C12, C3×C4⋊C4, C6×Q8, C6×C12, C6×C12, Q8×C32, Q8×C12, C122, C32×C4⋊C4, Q8×C3×C6, Q8×C3×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, Q8, C23, C32, C12, C2×C6, C22×C4, C2×Q8, C4○D4, C3×C6, C2×C12, C3×Q8, C22×C6, C4×Q8, C3×C12, C62, C22×C12, C6×Q8, C3×C4○D4, C6×C12, Q8×C32, C2×C62, Q8×C12, C2×C6×C12, Q8×C3×C6, C32×C4○D4, Q8×C3×C12

Smallest permutation representation of Q8×C3×C12
Regular action on 288 points
Generators in S288
(1 61 30)(2 62 31)(3 63 32)(4 64 33)(5 65 34)(6 66 35)(7 67 36)(8 68 25)(9 69 26)(10 70 27)(11 71 28)(12 72 29)(13 47 269)(14 48 270)(15 37 271)(16 38 272)(17 39 273)(18 40 274)(19 41 275)(20 42 276)(21 43 265)(22 44 266)(23 45 267)(24 46 268)(49 263 175)(50 264 176)(51 253 177)(52 254 178)(53 255 179)(54 256 180)(55 257 169)(56 258 170)(57 259 171)(58 260 172)(59 261 173)(60 262 174)(73 101 165)(74 102 166)(75 103 167)(76 104 168)(77 105 157)(78 106 158)(79 107 159)(80 108 160)(81 97 161)(82 98 162)(83 99 163)(84 100 164)(85 183 205)(86 184 206)(87 185 207)(88 186 208)(89 187 209)(90 188 210)(91 189 211)(92 190 212)(93 191 213)(94 192 214)(95 181 215)(96 182 216)(109 237 202)(110 238 203)(111 239 204)(112 240 193)(113 229 194)(114 230 195)(115 231 196)(116 232 197)(117 233 198)(118 234 199)(119 235 200)(120 236 201)(121 226 146)(122 227 147)(123 228 148)(124 217 149)(125 218 150)(126 219 151)(127 220 152)(128 221 153)(129 222 154)(130 223 155)(131 224 156)(132 225 145)(133 241 288)(134 242 277)(135 243 278)(136 244 279)(137 245 280)(138 246 281)(139 247 282)(140 248 283)(141 249 284)(142 250 285)(143 251 286)(144 252 287)
(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)
(1 282 203 176)(2 283 204 177)(3 284 193 178)(4 285 194 179)(5 286 195 180)(6 287 196 169)(7 288 197 170)(8 277 198 171)(9 278 199 172)(10 279 200 173)(11 280 201 174)(12 281 202 175)(13 161 182 154)(14 162 183 155)(15 163 184 156)(16 164 185 145)(17 165 186 146)(18 166 187 147)(19 167 188 148)(20 168 189 149)(21 157 190 150)(22 158 191 151)(23 159 192 152)(24 160 181 153)(25 242 233 259)(26 243 234 260)(27 244 235 261)(28 245 236 262)(29 246 237 263)(30 247 238 264)(31 248 239 253)(32 249 240 254)(33 250 229 255)(34 251 230 256)(35 252 231 257)(36 241 232 258)(37 83 206 131)(38 84 207 132)(39 73 208 121)(40 74 209 122)(41 75 210 123)(42 76 211 124)(43 77 212 125)(44 78 213 126)(45 79 214 127)(46 80 215 128)(47 81 216 129)(48 82 205 130)(49 72 138 109)(50 61 139 110)(51 62 140 111)(52 63 141 112)(53 64 142 113)(54 65 143 114)(55 66 144 115)(56 67 133 116)(57 68 134 117)(58 69 135 118)(59 70 136 119)(60 71 137 120)(85 223 270 98)(86 224 271 99)(87 225 272 100)(88 226 273 101)(89 227 274 102)(90 228 275 103)(91 217 276 104)(92 218 265 105)(93 219 266 106)(94 220 267 107)(95 221 268 108)(96 222 269 97)
(1 152 203 159)(2 153 204 160)(3 154 193 161)(4 155 194 162)(5 156 195 163)(6 145 196 164)(7 146 197 165)(8 147 198 166)(9 148 199 167)(10 149 200 168)(11 150 201 157)(12 151 202 158)(13 284 182 178)(14 285 183 179)(15 286 184 180)(16 287 185 169)(17 288 186 170)(18 277 187 171)(19 278 188 172)(20 279 189 173)(21 280 190 174)(22 281 191 175)(23 282 192 176)(24 283 181 177)(25 227 233 102)(26 228 234 103)(27 217 235 104)(28 218 236 105)(29 219 237 106)(30 220 238 107)(31 221 239 108)(32 222 240 97)(33 223 229 98)(34 224 230 99)(35 225 231 100)(36 226 232 101)(37 143 206 54)(38 144 207 55)(39 133 208 56)(40 134 209 57)(41 135 210 58)(42 136 211 59)(43 137 212 60)(44 138 213 49)(45 139 214 50)(46 140 215 51)(47 141 216 52)(48 142 205 53)(61 127 110 79)(62 128 111 80)(63 129 112 81)(64 130 113 82)(65 131 114 83)(66 132 115 84)(67 121 116 73)(68 122 117 74)(69 123 118 75)(70 124 119 76)(71 125 120 77)(72 126 109 78)(85 255 270 250)(86 256 271 251)(87 257 272 252)(88 258 273 241)(89 259 274 242)(90 260 275 243)(91 261 276 244)(92 262 265 245)(93 263 266 246)(94 264 267 247)(95 253 268 248)(96 254 269 249)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 4E ··· 4P 6A ··· 6X 12A ··· 12AF 12AG ··· 12DX order 1 2 2 2 3 ··· 3 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 Q8 C4○D4 C3×Q8 C3×C4○D4 kernel Q8×C3×C12 C122 C32×C4⋊C4 Q8×C3×C6 Q8×C12 Q8×C32 C4×C12 C3×C4⋊C4 C6×Q8 C3×Q8 C3×C12 C3×C6 C12 C6 # reps 1 3 3 1 8 8 24 24 8 64 2 2 16 16

Matrix representation of Q8×C3×C12 in GL4(𝔽13) generated by

 3 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 6 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 12 0 0 0 0 1 0 0 0 0 1 2 0 0 12 12
,
 12 0 0 0 0 1 0 0 0 0 6 7 0 0 4 7
G:=sub<GL(4,GF(13))| [3,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[6,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,1,0,0,0,0,1,12,0,0,2,12],[12,0,0,0,0,1,0,0,0,0,6,4,0,0,7,7] >;

Q8×C3×C12 in GAP, Magma, Sage, TeX

Q_8\times C_3\times C_{12}
% in TeX

G:=Group("Q8xC3xC12");
// GroupNames label

G:=SmallGroup(288,816);
// by ID

G=gap.SmallGroup(288,816);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,512,1150]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,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