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## G = Q8×C62order 288 = 25·32

### Direct product of C62 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C62
 Chief series C1 — C2 — C6 — C3×C6 — C3×C12 — Q8×C32 — Q8×C3×C6 — Q8×C62
 Lower central C1 — C2 — Q8×C62
 Upper central C1 — C2×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, C3, C4, C22, C6, C2×C4, Q8, C23, C32, C12, C2×C6, C22×C4, C2×Q8, C3×C6, C3×C6, C2×C12, C3×Q8, C22×C6, C22×Q8, C3×C12, C62, C22×C12, C6×Q8, C6×C12, Q8×C32, C2×C62, Q8×C2×C6, C2×C6×C12, Q8×C3×C6, Q8×C62
Quotients: C1, C2, C3, C22, C6, Q8, C23, C32, C2×C6, C2×Q8, C24, C3×C6, C3×Q8, C22×C6, C22×Q8, C62, C6×Q8, C23×C6, Q8×C32, C2×C62, Q8×C2×C6, Q8×C3×C6, 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)
(1 106 115 100 109 51)(2 107 116 101 110 52)(3 108 117 102 111 53)(4 103 118 97 112 54)(5 104 119 98 113 49)(6 105 120 99 114 50)(7 284 64 60 13 72)(8 285 65 55 14 67)(9 286 66 56 15 68)(10 287 61 57 16 69)(11 288 62 58 17 70)(12 283 63 59 18 71)(19 33 48 27 42 94)(20 34 43 28 37 95)(21 35 44 29 38 96)(22 36 45 30 39 91)(23 31 46 25 40 92)(24 32 47 26 41 93)(73 87 138 81 132 122)(74 88 133 82 127 123)(75 89 134 83 128 124)(76 90 135 84 129 125)(77 85 136 79 130 126)(78 86 137 80 131 121)(139 168 176 191 170 185)(140 163 177 192 171 186)(141 164 178 187 172 181)(142 165 179 188 173 182)(143 166 180 189 174 183)(144 167 175 190 169 184)(145 159 210 153 204 194)(146 160 205 154 199 195)(147 161 206 155 200 196)(148 162 207 156 201 197)(149 157 208 151 202 198)(150 158 209 152 203 193)(211 240 248 263 242 257)(212 235 249 264 243 258)(213 236 250 259 244 253)(214 237 251 260 245 254)(215 238 252 261 246 255)(216 239 247 262 241 256)(217 231 282 225 276 266)(218 232 277 226 271 267)(219 233 278 227 272 268)(220 234 279 228 273 269)(221 229 280 223 274 270)(222 230 281 224 275 265)
(1 169 25 74)(2 170 26 75)(3 171 27 76)(4 172 28 77)(5 173 29 78)(6 174 30 73)(7 252 234 204)(8 247 229 199)(9 248 230 200)(10 249 231 201)(11 250 232 202)(12 251 233 203)(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 84 102 163)(20 79 97 164)(21 80 98 165)(22 81 99 166)(23 82 100 167)(24 83 101 168)(31 127 109 175)(32 128 110 176)(33 129 111 177)(34 130 112 178)(35 131 113 179)(36 132 114 180)(37 85 103 181)(38 86 104 182)(39 87 105 183)(40 88 106 184)(41 89 107 185)(42 90 108 186)(43 126 54 187)(44 121 49 188)(45 122 50 189)(46 123 51 190)(47 124 52 191)(48 125 53 192)(55 256 274 160)(56 257 275 161)(57 258 276 162)(58 253 271 157)(59 254 272 158)(60 255 273 159)(61 243 225 148)(62 244 226 149)(63 245 227 150)(64 246 228 145)(65 241 223 146)(66 242 224 147)(67 239 221 154)(68 240 222 155)(69 235 217 156)(70 236 218 151)(71 237 219 152)(72 238 220 153)(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)
(1 241 25 146)(2 242 26 147)(3 243 27 148)(4 244 28 149)(5 245 29 150)(6 246 30 145)(7 132 234 180)(8 127 229 175)(9 128 230 176)(10 129 231 177)(11 130 232 178)(12 131 233 179)(13 138 269 143)(14 133 270 144)(15 134 265 139)(16 135 266 140)(17 136 267 141)(18 137 268 142)(19 156 102 235)(20 151 97 236)(21 152 98 237)(22 153 99 238)(23 154 100 239)(24 155 101 240)(31 199 109 247)(32 200 110 248)(33 201 111 249)(34 202 112 250)(35 203 113 251)(36 204 114 252)(37 157 103 253)(38 158 104 254)(39 159 105 255)(40 160 106 256)(41 161 107 257)(42 162 108 258)(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 88 274 184)(56 89 275 185)(57 90 276 186)(58 85 271 181)(59 86 272 182)(60 87 273 183)(61 76 225 171)(62 77 226 172)(63 78 227 173)(64 73 228 174)(65 74 223 169)(66 75 224 170)(67 82 221 167)(68 83 222 168)(69 84 217 163)(70 79 218 164)(71 80 219 165)(72 81 220 166)(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), (1,106,115,100,109,51)(2,107,116,101,110,52)(3,108,117,102,111,53)(4,103,118,97,112,54)(5,104,119,98,113,49)(6,105,120,99,114,50)(7,284,64,60,13,72)(8,285,65,55,14,67)(9,286,66,56,15,68)(10,287,61,57,16,69)(11,288,62,58,17,70)(12,283,63,59,18,71)(19,33,48,27,42,94)(20,34,43,28,37,95)(21,35,44,29,38,96)(22,36,45,30,39,91)(23,31,46,25,40,92)(24,32,47,26,41,93)(73,87,138,81,132,122)(74,88,133,82,127,123)(75,89,134,83,128,124)(76,90,135,84,129,125)(77,85,136,79,130,126)(78,86,137,80,131,121)(139,168,176,191,170,185)(140,163,177,192,171,186)(141,164,178,187,172,181)(142,165,179,188,173,182)(143,166,180,189,174,183)(144,167,175,190,169,184)(145,159,210,153,204,194)(146,160,205,154,199,195)(147,161,206,155,200,196)(148,162,207,156,201,197)(149,157,208,151,202,198)(150,158,209,152,203,193)(211,240,248,263,242,257)(212,235,249,264,243,258)(213,236,250,259,244,253)(214,237,251,260,245,254)(215,238,252,261,246,255)(216,239,247,262,241,256)(217,231,282,225,276,266)(218,232,277,226,271,267)(219,233,278,227,272,268)(220,234,279,228,273,269)(221,229,280,223,274,270)(222,230,281,224,275,265), (1,169,25,74)(2,170,26,75)(3,171,27,76)(4,172,28,77)(5,173,29,78)(6,174,30,73)(7,252,234,204)(8,247,229,199)(9,248,230,200)(10,249,231,201)(11,250,232,202)(12,251,233,203)(13,215,269,210)(14,216,270,205)(15,211,265,206)(16,212,266,207)(17,213,267,208)(18,214,268,209)(19,84,102,163)(20,79,97,164)(21,80,98,165)(22,81,99,166)(23,82,100,167)(24,83,101,168)(31,127,109,175)(32,128,110,176)(33,129,111,177)(34,130,112,178)(35,131,113,179)(36,132,114,180)(37,85,103,181)(38,86,104,182)(39,87,105,183)(40,88,106,184)(41,89,107,185)(42,90,108,186)(43,126,54,187)(44,121,49,188)(45,122,50,189)(46,123,51,190)(47,124,52,191)(48,125,53,192)(55,256,274,160)(56,257,275,161)(57,258,276,162)(58,253,271,157)(59,254,272,158)(60,255,273,159)(61,243,225,148)(62,244,226,149)(63,245,227,150)(64,246,228,145)(65,241,223,146)(66,242,224,147)(67,239,221,154)(68,240,222,155)(69,235,217,156)(70,236,218,151)(71,237,219,152)(72,238,220,153)(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), (1,241,25,146)(2,242,26,147)(3,243,27,148)(4,244,28,149)(5,245,29,150)(6,246,30,145)(7,132,234,180)(8,127,229,175)(9,128,230,176)(10,129,231,177)(11,130,232,178)(12,131,233,179)(13,138,269,143)(14,133,270,144)(15,134,265,139)(16,135,266,140)(17,136,267,141)(18,137,268,142)(19,156,102,235)(20,151,97,236)(21,152,98,237)(22,153,99,238)(23,154,100,239)(24,155,101,240)(31,199,109,247)(32,200,110,248)(33,201,111,249)(34,202,112,250)(35,203,113,251)(36,204,114,252)(37,157,103,253)(38,158,104,254)(39,159,105,255)(40,160,106,256)(41,161,107,257)(42,162,108,258)(43,198,54,259)(44,193,49,260)(45,194,50,261)(46,195,51,262)(47,196,52,263)(48,197,53,264)(55,88,274,184)(56,89,275,185)(57,90,276,186)(58,85,271,181)(59,86,272,182)(60,87,273,183)(61,76,225,171)(62,77,226,172)(63,78,227,173)(64,73,228,174)(65,74,223,169)(66,75,224,170)(67,82,221,167)(68,83,222,168)(69,84,217,163)(70,79,218,164)(71,80,219,165)(72,81,220,166)(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), (1,106,115,100,109,51)(2,107,116,101,110,52)(3,108,117,102,111,53)(4,103,118,97,112,54)(5,104,119,98,113,49)(6,105,120,99,114,50)(7,284,64,60,13,72)(8,285,65,55,14,67)(9,286,66,56,15,68)(10,287,61,57,16,69)(11,288,62,58,17,70)(12,283,63,59,18,71)(19,33,48,27,42,94)(20,34,43,28,37,95)(21,35,44,29,38,96)(22,36,45,30,39,91)(23,31,46,25,40,92)(24,32,47,26,41,93)(73,87,138,81,132,122)(74,88,133,82,127,123)(75,89,134,83,128,124)(76,90,135,84,129,125)(77,85,136,79,130,126)(78,86,137,80,131,121)(139,168,176,191,170,185)(140,163,177,192,171,186)(141,164,178,187,172,181)(142,165,179,188,173,182)(143,166,180,189,174,183)(144,167,175,190,169,184)(145,159,210,153,204,194)(146,160,205,154,199,195)(147,161,206,155,200,196)(148,162,207,156,201,197)(149,157,208,151,202,198)(150,158,209,152,203,193)(211,240,248,263,242,257)(212,235,249,264,243,258)(213,236,250,259,244,253)(214,237,251,260,245,254)(215,238,252,261,246,255)(216,239,247,262,241,256)(217,231,282,225,276,266)(218,232,277,226,271,267)(219,233,278,227,272,268)(220,234,279,228,273,269)(221,229,280,223,274,270)(222,230,281,224,275,265), (1,169,25,74)(2,170,26,75)(3,171,27,76)(4,172,28,77)(5,173,29,78)(6,174,30,73)(7,252,234,204)(8,247,229,199)(9,248,230,200)(10,249,231,201)(11,250,232,202)(12,251,233,203)(13,215,269,210)(14,216,270,205)(15,211,265,206)(16,212,266,207)(17,213,267,208)(18,214,268,209)(19,84,102,163)(20,79,97,164)(21,80,98,165)(22,81,99,166)(23,82,100,167)(24,83,101,168)(31,127,109,175)(32,128,110,176)(33,129,111,177)(34,130,112,178)(35,131,113,179)(36,132,114,180)(37,85,103,181)(38,86,104,182)(39,87,105,183)(40,88,106,184)(41,89,107,185)(42,90,108,186)(43,126,54,187)(44,121,49,188)(45,122,50,189)(46,123,51,190)(47,124,52,191)(48,125,53,192)(55,256,274,160)(56,257,275,161)(57,258,276,162)(58,253,271,157)(59,254,272,158)(60,255,273,159)(61,243,225,148)(62,244,226,149)(63,245,227,150)(64,246,228,145)(65,241,223,146)(66,242,224,147)(67,239,221,154)(68,240,222,155)(69,235,217,156)(70,236,218,151)(71,237,219,152)(72,238,220,153)(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), (1,241,25,146)(2,242,26,147)(3,243,27,148)(4,244,28,149)(5,245,29,150)(6,246,30,145)(7,132,234,180)(8,127,229,175)(9,128,230,176)(10,129,231,177)(11,130,232,178)(12,131,233,179)(13,138,269,143)(14,133,270,144)(15,134,265,139)(16,135,266,140)(17,136,267,141)(18,137,268,142)(19,156,102,235)(20,151,97,236)(21,152,98,237)(22,153,99,238)(23,154,100,239)(24,155,101,240)(31,199,109,247)(32,200,110,248)(33,201,111,249)(34,202,112,250)(35,203,113,251)(36,204,114,252)(37,157,103,253)(38,158,104,254)(39,159,105,255)(40,160,106,256)(41,161,107,257)(42,162,108,258)(43,198,54,259)(44,193,49,260)(45,194,50,261)(46,195,51,262)(47,196,52,263)(48,197,53,264)(55,88,274,184)(56,89,275,185)(57,90,276,186)(58,85,271,181)(59,86,272,182)(60,87,273,183)(61,76,225,171)(62,77,226,172)(63,78,227,173)(64,73,228,174)(65,74,223,169)(66,75,224,170)(67,82,221,167)(68,83,222,168)(69,84,217,163)(70,79,218,164)(71,80,219,165)(72,81,220,166)(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)], [(1,106,115,100,109,51),(2,107,116,101,110,52),(3,108,117,102,111,53),(4,103,118,97,112,54),(5,104,119,98,113,49),(6,105,120,99,114,50),(7,284,64,60,13,72),(8,285,65,55,14,67),(9,286,66,56,15,68),(10,287,61,57,16,69),(11,288,62,58,17,70),(12,283,63,59,18,71),(19,33,48,27,42,94),(20,34,43,28,37,95),(21,35,44,29,38,96),(22,36,45,30,39,91),(23,31,46,25,40,92),(24,32,47,26,41,93),(73,87,138,81,132,122),(74,88,133,82,127,123),(75,89,134,83,128,124),(76,90,135,84,129,125),(77,85,136,79,130,126),(78,86,137,80,131,121),(139,168,176,191,170,185),(140,163,177,192,171,186),(141,164,178,187,172,181),(142,165,179,188,173,182),(143,166,180,189,174,183),(144,167,175,190,169,184),(145,159,210,153,204,194),(146,160,205,154,199,195),(147,161,206,155,200,196),(148,162,207,156,201,197),(149,157,208,151,202,198),(150,158,209,152,203,193),(211,240,248,263,242,257),(212,235,249,264,243,258),(213,236,250,259,244,253),(214,237,251,260,245,254),(215,238,252,261,246,255),(216,239,247,262,241,256),(217,231,282,225,276,266),(218,232,277,226,271,267),(219,233,278,227,272,268),(220,234,279,228,273,269),(221,229,280,223,274,270),(222,230,281,224,275,265)], [(1,169,25,74),(2,170,26,75),(3,171,27,76),(4,172,28,77),(5,173,29,78),(6,174,30,73),(7,252,234,204),(8,247,229,199),(9,248,230,200),(10,249,231,201),(11,250,232,202),(12,251,233,203),(13,215,269,210),(14,216,270,205),(15,211,265,206),(16,212,266,207),(17,213,267,208),(18,214,268,209),(19,84,102,163),(20,79,97,164),(21,80,98,165),(22,81,99,166),(23,82,100,167),(24,83,101,168),(31,127,109,175),(32,128,110,176),(33,129,111,177),(34,130,112,178),(35,131,113,179),(36,132,114,180),(37,85,103,181),(38,86,104,182),(39,87,105,183),(40,88,106,184),(41,89,107,185),(42,90,108,186),(43,126,54,187),(44,121,49,188),(45,122,50,189),(46,123,51,190),(47,124,52,191),(48,125,53,192),(55,256,274,160),(56,257,275,161),(57,258,276,162),(58,253,271,157),(59,254,272,158),(60,255,273,159),(61,243,225,148),(62,244,226,149),(63,245,227,150),(64,246,228,145),(65,241,223,146),(66,242,224,147),(67,239,221,154),(68,240,222,155),(69,235,217,156),(70,236,218,151),(71,237,219,152),(72,238,220,153),(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)], [(1,241,25,146),(2,242,26,147),(3,243,27,148),(4,244,28,149),(5,245,29,150),(6,246,30,145),(7,132,234,180),(8,127,229,175),(9,128,230,176),(10,129,231,177),(11,130,232,178),(12,131,233,179),(13,138,269,143),(14,133,270,144),(15,134,265,139),(16,135,266,140),(17,136,267,141),(18,137,268,142),(19,156,102,235),(20,151,97,236),(21,152,98,237),(22,153,99,238),(23,154,100,239),(24,155,101,240),(31,199,109,247),(32,200,110,248),(33,201,111,249),(34,202,112,250),(35,203,113,251),(36,204,114,252),(37,157,103,253),(38,158,104,254),(39,159,105,255),(40,160,106,256),(41,161,107,257),(42,162,108,258),(43,198,54,259),(44,193,49,260),(45,194,50,261),(46,195,51,262),(47,196,52,263),(48,197,53,264),(55,88,274,184),(56,89,275,185),(57,90,276,186),(58,85,271,181),(59,86,272,182),(60,87,273,183),(61,76,225,171),(62,77,226,172),(63,78,227,173),(64,73,228,174),(65,74,223,169),(66,75,224,170),(67,82,221,167),(68,83,222,168),(69,84,217,163),(70,79,218,164),(71,80,219,165),(72,81,220,166),(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 ··· 2G 3A ··· 3H 4A ··· 4L 6A ··· 6BD 12A ··· 12CR order 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C3 C6 C6 Q8 C3×Q8 kernel Q8×C62 C2×C6×C12 Q8×C3×C6 Q8×C2×C6 C22×C12 C6×Q8 C62 C2×C6 # reps 1 3 12 8 24 96 4 32

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

 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 11 0 0 1 1
,
 1 0 0 0 0 12 0 0 0 0 5 10 0 0 0 8
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