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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 [×3], C3 [×4], C4 [×8], C4 [×3], C22, C6 [×12], C2×C4, C2×C4 [×6], Q8 [×4], C32, C12 [×32], C12 [×12], C2×C6 [×4], C42 [×3], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], C2×C12 [×28], C3×Q8 [×16], C4×Q8, C3×C12 [×8], C3×C12 [×3], C62, C4×C12 [×12], C3×C4⋊C4 [×12], C6×Q8 [×4], C6×C12, C6×C12 [×6], Q8×C32 [×4], Q8×C12 [×4], C122 [×3], C32×C4⋊C4 [×3], Q8×C3×C6, Q8×C3×C12
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], Q8 [×2], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C2×Q8, C4○D4, C3×C6 [×7], C2×C12 [×24], C3×Q8 [×8], C22×C6 [×4], C4×Q8, C3×C12 [×4], C62 [×7], C22×C12 [×4], C6×Q8 [×4], C3×C4○D4 [×4], C6×C12 [×6], Q8×C32 [×2], C2×C62, Q8×C12 [×4], 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
(1 192 272)(2 181 273)(3 182 274)(4 183 275)(5 184 276)(6 185 265)(7 186 266)(8 187 267)(9 188 268)(10 189 269)(11 190 270)(12 191 271)(13 225 235)(14 226 236)(15 227 237)(16 228 238)(17 217 239)(18 218 240)(19 219 229)(20 220 230)(21 221 231)(22 222 232)(23 223 233)(24 224 234)(25 216 90)(26 205 91)(27 206 92)(28 207 93)(29 208 94)(30 209 95)(31 210 96)(32 211 85)(33 212 86)(34 213 87)(35 214 88)(36 215 89)(37 103 244)(38 104 245)(39 105 246)(40 106 247)(41 107 248)(42 108 249)(43 97 250)(44 98 251)(45 99 252)(46 100 241)(47 101 242)(48 102 243)(49 180 62)(50 169 63)(51 170 64)(52 171 65)(53 172 66)(54 173 67)(55 174 68)(56 175 69)(57 176 70)(58 177 71)(59 178 72)(60 179 61)(73 280 111)(74 281 112)(75 282 113)(76 283 114)(77 284 115)(78 285 116)(79 286 117)(80 287 118)(81 288 119)(82 277 120)(83 278 109)(84 279 110)(121 195 262)(122 196 263)(123 197 264)(124 198 253)(125 199 254)(126 200 255)(127 201 256)(128 202 257)(129 203 258)(130 204 259)(131 193 260)(132 194 261)(133 150 163)(134 151 164)(135 152 165)(136 153 166)(137 154 167)(138 155 168)(139 156 157)(140 145 158)(141 146 159)(142 147 160)(143 148 161)(144 149 162)
(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 16 35 84)(2 17 36 73)(3 18 25 74)(4 19 26 75)(5 20 27 76)(6 21 28 77)(7 22 29 78)(8 23 30 79)(9 24 31 80)(10 13 32 81)(11 14 33 82)(12 15 34 83)(37 52 162 261)(38 53 163 262)(39 54 164 263)(40 55 165 264)(41 56 166 253)(42 57 167 254)(43 58 168 255)(44 59 157 256)(45 60 158 257)(46 49 159 258)(47 50 160 259)(48 51 161 260)(61 145 202 252)(62 146 203 241)(63 147 204 242)(64 148 193 243)(65 149 194 244)(66 150 195 245)(67 151 196 246)(68 152 197 247)(69 153 198 248)(70 154 199 249)(71 155 200 250)(72 156 201 251)(85 119 269 235)(86 120 270 236)(87 109 271 237)(88 110 272 238)(89 111 273 239)(90 112 274 240)(91 113 275 229)(92 114 276 230)(93 115 265 231)(94 116 266 232)(95 117 267 233)(96 118 268 234)(97 177 138 126)(98 178 139 127)(99 179 140 128)(100 180 141 129)(101 169 142 130)(102 170 143 131)(103 171 144 132)(104 172 133 121)(105 173 134 122)(106 174 135 123)(107 175 136 124)(108 176 137 125)(181 217 215 280)(182 218 216 281)(183 219 205 282)(184 220 206 283)(185 221 207 284)(186 222 208 285)(187 223 209 286)(188 224 210 287)(189 225 211 288)(190 226 212 277)(191 227 213 278)(192 228 214 279)
(1 105 35 134)(2 106 36 135)(3 107 25 136)(4 108 26 137)(5 97 27 138)(6 98 28 139)(7 99 29 140)(8 100 30 141)(9 101 31 142)(10 102 32 143)(11 103 33 144)(12 104 34 133)(13 131 81 170)(14 132 82 171)(15 121 83 172)(16 122 84 173)(17 123 73 174)(18 124 74 175)(19 125 75 176)(20 126 76 177)(21 127 77 178)(22 128 78 179)(23 129 79 180)(24 130 80 169)(37 86 162 270)(38 87 163 271)(39 88 164 272)(40 89 165 273)(41 90 166 274)(42 91 167 275)(43 92 168 276)(44 93 157 265)(45 94 158 266)(46 95 159 267)(47 96 160 268)(48 85 161 269)(49 233 258 117)(50 234 259 118)(51 235 260 119)(52 236 261 120)(53 237 262 109)(54 238 263 110)(55 239 264 111)(56 240 253 112)(57 229 254 113)(58 230 255 114)(59 231 256 115)(60 232 257 116)(61 222 202 285)(62 223 203 286)(63 224 204 287)(64 225 193 288)(65 226 194 277)(66 227 195 278)(67 228 196 279)(68 217 197 280)(69 218 198 281)(70 219 199 282)(71 220 200 283)(72 221 201 284)(145 186 252 208)(146 187 241 209)(147 188 242 210)(148 189 243 211)(149 190 244 212)(150 191 245 213)(151 192 246 214)(152 181 247 215)(153 182 248 216)(154 183 249 205)(155 184 250 206)(156 185 251 207)

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

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

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

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