Q8xC3:Dic3 - GroupNames

G = Q₈×C₃⋊Dic₃ order 288 = 2⁵·3²

Direct product of Q₈ and C₃⋊Dic₃

direct product, metabelian, supersoluble, monomial

Aliases: Q₈×C₃⋊Dic₃, C₆².₂₆₀C₂³, C₃⋊₃(Q₈×Dic₃), C₆.₄₉(S₃×Q₈), C₃²⋊₁₄(C₄×Q₈), (C₃×Q₈)⋊₅Dic₃, (C₆×Q₈).₂₅S₃, (Q₈×C₃²)⋊₉C₄, (C₂×C₁₂).₁₅₉D₆, C₁₂.₂₀(C₂×Dic₃), (C₆×C₁₂).₁₅₀C₂², C₆.₅₃(Q₈⋊₃S₃), C₆.₃₇(C₂²×Dic₃), C₁₂⋊Dic₃.₁₉C₂, C₂.₃(C₁₂.₂₆D₆), C₂.₃(Q₈×C₃⋊S₃), (Q₈×C₃×C₆).₁₀C₂, C₄.₄(C₂×C₃⋊Dic₃), (C₃×C₆).₇₆(C₂×Q₈), (C₃×C₁₂).₇₅(C₂×C₄), (C₂×Q₈).₇(C₃⋊S₃), (C₄×C₃⋊Dic₃).₈C₂, C₂.₇(C₂²×C₃⋊Dic₃), (C₃×C₆).₁₆₂(C₄○D₄), (C₃×C₆).₁₂₅(C₂²×C₄), (C₂×C₆).₂₇₇(C₂²×S₃), C₂².₂₆(C₂²×C₃⋊S₃), (C₂×C₃⋊Dic₃).₁₆₉C₂², (C₂×C₄).₅₅(C₂×C₃⋊S₃), SmallGroup(288,802)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — Q₈×C₃⋊Dic₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₂×C₃⋊Dic₃ — C₄×C₃⋊Dic₃ — Q₈×C₃⋊Dic₃

Lower central

C₃² — C₃×C₆ — Q₈×C₃⋊Dic₃

Upper central

C₁ — C₂² — C₂×Q₈

Generators and relations for Q₈×C₃⋊Dic₃
G = < a,b,c,d,e | a⁴=c³=d⁶=1, b²=a², e²=d³, bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d^-1 >

Subgroups: 532 in 210 conjugacy classes, 127 normal (14 characteristic)
C₁, C₂, C₃, C₄, C₄, C₂², C₆, C₂×C₄, C₂×C₄, Q₈, C₃², Dic₃, C₁₂, C₂×C₆, C₄², C₄⋊C₄, C₂×Q₈, C₃×C₆, C₂×Dic₃, C₂×C₁₂, C₃×Q₈, C₄×Q₈, C₃⋊Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₆², C₄×Dic₃, C₄⋊Dic₃, C₆×Q₈, C₂×C₃⋊Dic₃, C₂×C₃⋊Dic₃, C₆×C₁₂, Q₈×C₃², Q₈×Dic₃, C₄×C₃⋊Dic₃, C₁₂⋊Dic₃, Q₈×C₃×C₆, Q₈×C₃⋊Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Q₈, C₂³, Dic₃, D₆, C₂²×C₄, C₂×Q₈, C₄○D₄, C₃⋊S₃, C₂×Dic₃, C₂²×S₃, C₄×Q₈, C₃⋊Dic₃, C₂×C₃⋊S₃, S₃×Q₈, Q₈⋊₃S₃, C₂²×Dic₃, C₂×C₃⋊Dic₃, C₂²×C₃⋊S₃, Q₈×Dic₃, Q₈×C₃⋊S₃, C₁₂.₂₆D₆, C₂²×C₃⋊Dic₃, Q₈×C₃⋊Dic₃

Smallest permutation representation of Q₈×C₃⋊Dic₃
►Regular action on 288 points

Generators in S₂₈₈

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

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

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

(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)

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	4A	···	4F	4G	4H	4I	4J	4K	···	4P	6A	···	6L	12A	···	12X
order	1	2	2	2	3	3	3	3	4	···	4	4	4	4	4	4	···	4	6	···	6	12	···	12
size	1	1	1	1	2	2	2	2	2	···	2	9	9	9	9	18	···	18	2	···	2	4	···	4

60 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+		+	-	+	-		-	+
image	C₁	C₂	C₂	C₂	C₄	S₃	Q₈	D₆	Dic₃	C₄○D₄	S₃×Q₈	Q₈⋊₃S₃
kernel	Q₈×C₃⋊Dic₃	C₄×C₃⋊Dic₃	C₁₂⋊Dic₃	Q₈×C₃×C₆	Q₈×C₃²	C₆×Q₈	C₃⋊Dic₃	C₂×C₁₂	C₃×Q₈	C₃×C₆	C₆	C₆
# reps	1	3	3	1	8	4	2	12	16	2	4	4

Matrix representation of Q₈×C₃⋊Dic₃ ►in GL₇(𝔽₁₃)

12	0	0	0	0	0	0
0	0	1	0	0	0	0
0	12	0	0	0	0	0
0	0	0	12	0	0	0
0	0	0	0	12	0	0
0	0	0	0	0	12	0
0	0	0	0	0	0	12

1	0	0	0	0	0	0
0	0	5	0	0	0	0
0	5	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	1	0	0
0	0	0	12	12	0	0
0	0	0	0	0	0	1
0	0	0	0	0	12	12

12	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	12	12
0	0	0	0	0	1	0

5	0	0	0	0	0	0
0	12	0	0	0	0	0
0	0	12	0	0	0	0
0	0	0	0	1	0	0
0	0	0	1	0	0	0
0	0	0	0	0	10	7
0	0	0	0	0	10	3

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,7,3] >;

Q₈×C₃⋊Dic₃ in GAP, Magma, Sage, TeX

Q_8\times C_3\rtimes {\rm Dic}_3

% in TeX

G:=Group("Q8xC3:Dic3");

// GroupNames label

G:=SmallGroup(288,802);

// by ID

G=gap.SmallGroup(288,802);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,219,100,2693,9414]);

// Polycyclic

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

// generators/relations

G = Q8×C3⋊Dic3 order 288 = 25·32

Direct product of Q8 and C3⋊Dic3

G = Q₈×C₃⋊Dic₃ order 288 = 2⁵·3²

Direct product of Q₈ and C₃⋊Dic₃