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G = Q8×C62order 288 = 25·32

Direct product of C62 and Q8

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

Aliases: Q8×C62, C23.17C62, C62.298C23, C4.7(C2×C62), (C3×C6).69C24, (C2×C4).29C62, C6.22(C23×C6), (C22×C12).38C6, C12.61(C22×C6), C22.9(C2×C62), C2.2(C22×C62), (C3×C12).191C23, (C6×C12).378C22, (C2×C62).127C22, (C2×C6×C12).27C2, (C22×C4).9(C3×C6), (C2×C12).165(C2×C6), (C22×C6).79(C2×C6), (C2×C6).104(C22×C6), SmallGroup(288,1020)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C62
C1C2C6C3×C6C3×C12Q8×C32Q8×C3×C6 — Q8×C62
C1C2 — Q8×C62
C1C2×C62 — Q8×C62

Generators and relations for Q8×C62
 G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 468, all normal (8 characteristic)
C1, C2, C2 [×6], C3 [×4], C4 [×12], C22 [×7], C6 [×28], C2×C4 [×18], Q8 [×16], C23, C32, C12 [×48], C2×C6 [×28], C22×C4 [×3], C2×Q8 [×12], C3×C6, C3×C6 [×6], C2×C12 [×72], C3×Q8 [×64], C22×C6 [×4], C22×Q8, C3×C12 [×12], C62 [×7], C22×C12 [×12], C6×Q8 [×48], C6×C12 [×18], Q8×C32 [×16], C2×C62, Q8×C2×C6 [×4], C2×C6×C12 [×3], Q8×C3×C6 [×12], Q8×C62
Quotients: C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], Q8 [×4], C23 [×15], C32, C2×C6 [×140], C2×Q8 [×6], C24, C3×C6 [×15], C3×Q8 [×16], C22×C6 [×60], C22×Q8, C62 [×35], C6×Q8 [×24], C23×C6 [×4], Q8×C32 [×4], C2×C62 [×15], Q8×C2×C6 [×4], Q8×C3×C6 [×6], C22×C62, Q8×C62

Smallest permutation representation of Q8×C62
Regular action on 288 points
Generators in S288
(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 98 115 111 106 51)(2 99 116 112 107 52)(3 100 117 113 108 53)(4 101 118 114 103 54)(5 102 119 109 104 49)(6 97 120 110 105 50)(7 58 288 62 68 17)(8 59 283 63 69 18)(9 60 284 64 70 13)(10 55 285 65 71 14)(11 56 286 66 72 15)(12 57 287 61 67 16)(19 25 48 38 33 94)(20 26 43 39 34 95)(21 27 44 40 35 96)(22 28 45 41 36 91)(23 29 46 42 31 92)(24 30 47 37 32 93)(73 79 138 128 87 122)(74 80 133 129 88 123)(75 81 134 130 89 124)(76 82 135 131 90 125)(77 83 136 132 85 126)(78 84 137 127 86 121)(139 168 174 191 181 176)(140 163 169 192 182 177)(141 164 170 187 183 178)(142 165 171 188 184 179)(143 166 172 189 185 180)(144 167 173 190 186 175)(145 151 210 200 159 194)(146 152 205 201 160 195)(147 153 206 202 161 196)(148 154 207 203 162 197)(149 155 208 204 157 198)(150 156 209 199 158 193)(211 240 246 263 253 248)(212 235 241 264 254 249)(213 236 242 259 255 250)(214 237 243 260 256 251)(215 238 244 261 257 252)(216 239 245 262 258 247)(217 223 282 272 231 266)(218 224 277 273 232 267)(219 225 278 274 233 268)(220 226 279 275 234 269)(221 227 280 276 229 270)(222 228 281 271 230 265)
(1 186 42 74)(2 181 37 75)(3 182 38 76)(4 183 39 77)(5 184 40 78)(6 185 41 73)(7 236 218 204)(8 237 219 199)(9 238 220 200)(10 239 221 201)(11 240 222 202)(12 235 217 203)(13 215 269 210)(14 216 270 205)(15 211 265 206)(16 212 266 207)(17 213 267 208)(18 214 268 209)(19 131 113 163)(20 132 114 164)(21 127 109 165)(22 128 110 166)(23 129 111 167)(24 130 112 168)(25 90 108 169)(26 85 103 170)(27 86 104 171)(28 87 105 172)(29 88 106 173)(30 89 107 174)(31 80 98 175)(32 81 99 176)(33 82 100 177)(34 83 101 178)(35 84 102 179)(36 79 97 180)(43 126 54 187)(44 121 49 188)(45 122 50 189)(46 123 51 190)(47 124 52 191)(48 125 53 192)(55 245 227 160)(56 246 228 161)(57 241 223 162)(58 242 224 157)(59 243 225 158)(60 244 226 159)(61 254 272 148)(62 255 273 149)(63 256 274 150)(64 257 275 145)(65 258 276 146)(66 253 271 147)(67 249 231 154)(68 250 232 155)(69 251 233 156)(70 252 234 151)(71 247 229 152)(72 248 230 153)(91 138 120 143)(92 133 115 144)(93 134 116 139)(94 135 117 140)(95 136 118 141)(96 137 119 142)(193 283 260 278)(194 284 261 279)(195 285 262 280)(196 286 263 281)(197 287 264 282)(198 288 259 277)
(1 258 42 146)(2 253 37 147)(3 254 38 148)(4 255 39 149)(5 256 40 150)(6 257 41 145)(7 132 218 164)(8 127 219 165)(9 128 220 166)(10 129 221 167)(11 130 222 168)(12 131 217 163)(13 138 269 143)(14 133 270 144)(15 134 265 139)(16 135 266 140)(17 136 267 141)(18 137 268 142)(19 203 113 235)(20 204 114 236)(21 199 109 237)(22 200 110 238)(23 201 111 239)(24 202 112 240)(25 162 108 241)(26 157 103 242)(27 158 104 243)(28 159 105 244)(29 160 106 245)(30 161 107 246)(31 152 98 247)(32 153 99 248)(33 154 100 249)(34 155 101 250)(35 156 102 251)(36 151 97 252)(43 198 54 259)(44 193 49 260)(45 194 50 261)(46 195 51 262)(47 196 52 263)(48 197 53 264)(55 88 227 173)(56 89 228 174)(57 90 223 169)(58 85 224 170)(59 86 225 171)(60 87 226 172)(61 76 272 182)(62 77 273 183)(63 78 274 184)(64 73 275 185)(65 74 276 186)(66 75 271 181)(67 82 231 177)(68 83 232 178)(69 84 233 179)(70 79 234 180)(71 80 229 175)(72 81 230 176)(91 210 120 215)(92 205 115 216)(93 206 116 211)(94 207 117 212)(95 208 118 213)(96 209 119 214)(121 278 188 283)(122 279 189 284)(123 280 190 285)(124 281 191 286)(125 282 192 287)(126 277 187 288)

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

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

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

180 conjugacy classes

class 1 2A···2G3A···3H4A···4L6A···6BD12A···12CR
order12···23···34···46···612···12
size11···11···12···21···12···2

180 irreducible representations

dim11111122
type+++-
imageC1C2C2C3C6C6Q8C3×Q8
kernelQ8×C62C2×C6×C12Q8×C3×C6Q8×C2×C6C22×C12C6×Q8C62C2×C6
# reps131282496432

Matrix representation of Q8×C62 in GL4(𝔽13) generated by

4000
01200
0010
0001
,
1000
0400
0040
0004
,
1000
0100
001211
0011
,
1000
01200
00510
0008
G:=sub<GL(4,GF(13))| [4,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,11,1],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,10,8] >;

Q8×C62 in GAP, Magma, Sage, TeX

Q_8\times C_6^2
% in TeX

G:=Group("Q8xC6^2");
// GroupNames label

G:=SmallGroup(288,1020);
// by ID

G=gap.SmallGroup(288,1020);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,1008,2045,1016]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=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