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G = C4×C324Q8order 288 = 25·32

Direct product of C4 and C324Q8

direct product, metabelian, supersoluble, monomial

Aliases: C4×C324Q8, C127Dic6, C122.5C2, C62.212C23, (C3×C12)⋊11Q8, C34(C4×Dic6), C3210(C4×Q8), C12.63(C4×S3), (C4×C12).15S3, (C2×C12).354D6, C42.3(C3⋊S3), C6.38(C2×Dic6), C6.91(C4○D12), (C6×C12).353C22, C12⋊Dic3.20C2, C2.1(C12.59D6), C6.Dic6.12C2, C4.9(C4×C3⋊S3), C6.61(S3×C2×C4), (C3×C6).52(C2×Q8), (C3×C12).94(C2×C4), C2.1(C2×C324Q8), C3⋊Dic3.36(C2×C4), (C4×C3⋊Dic3).21C2, (C3×C6).92(C22×C4), C22.8(C22×C3⋊S3), (C3×C6).107(C4○D4), (C2×C6).229(C22×S3), (C2×C324Q8).17C2, (C2×C3⋊Dic3).150C22, C2.4(C2×C4×C3⋊S3), (C2×C4).95(C2×C3⋊S3), SmallGroup(288,725)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C4×C324Q8
C1C3C32C3×C6C62C2×C3⋊Dic3C2×C324Q8 — C4×C324Q8
C32C3×C6 — C4×C324Q8
C1C2×C4C42

Generators and relations for C4×C324Q8
 G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 596 in 210 conjugacy classes, 97 normal (21 characteristic)
C1, C2 [×3], C3 [×4], C4 [×4], C4 [×7], C22, C6 [×12], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C32, Dic3 [×24], C12 [×16], C12 [×4], C2×C6 [×4], C42, C42 [×2], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], Dic6 [×16], C2×Dic3 [×16], C2×C12 [×12], C4×Q8, C3⋊Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×4], C3×C12, C62, C4×Dic3 [×8], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C4×C12 [×4], C2×Dic6 [×4], C324Q8 [×4], C2×C3⋊Dic3 [×4], C6×C12 [×3], C4×Dic6 [×4], C4×C3⋊Dic3 [×2], C6.Dic6 [×2], C12⋊Dic3, C122, C2×C324Q8, C4×C324Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], Q8 [×2], C23, D6 [×12], C22×C4, C2×Q8, C4○D4, C3⋊S3, Dic6 [×8], C4×S3 [×8], C22×S3 [×4], C4×Q8, C2×C3⋊S3 [×3], C2×Dic6 [×4], S3×C2×C4 [×4], C4○D12 [×4], C324Q8 [×2], C4×C3⋊S3 [×2], C22×C3⋊S3, C4×Dic6 [×4], C2×C324Q8, C2×C4×C3⋊S3, C12.59D6, C4×C324Q8

Smallest permutation representation of C4×C324Q8
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 23 15)(2 24 16)(3 21 13)(4 22 14)(5 111 57)(6 112 58)(7 109 59)(8 110 60)(9 29 77)(10 30 78)(11 31 79)(12 32 80)(17 259 226)(18 260 227)(19 257 228)(20 258 225)(25 69 43)(26 70 44)(27 71 41)(28 72 42)(33 49 47)(34 50 48)(35 51 45)(36 52 46)(37 266 272)(38 267 269)(39 268 270)(40 265 271)(53 234 262)(54 235 263)(55 236 264)(56 233 261)(61 75 67)(62 76 68)(63 73 65)(64 74 66)(81 114 183)(82 115 184)(83 116 181)(84 113 182)(85 143 93)(86 144 94)(87 141 95)(88 142 96)(89 151 117)(90 152 118)(91 149 119)(92 150 120)(97 131 103)(98 132 104)(99 129 101)(100 130 102)(105 139 133)(106 140 134)(107 137 135)(108 138 136)(121 145 127)(122 146 128)(123 147 125)(124 148 126)(153 255 186)(154 256 187)(155 253 188)(156 254 185)(157 165 215)(158 166 216)(159 167 213)(160 168 214)(161 189 223)(162 190 224)(163 191 221)(164 192 222)(169 175 203)(170 176 204)(171 173 201)(172 174 202)(177 205 211)(178 206 212)(179 207 209)(180 208 210)(193 199 217)(194 200 218)(195 197 219)(196 198 220)(229 237 287)(230 238 288)(231 239 285)(232 240 286)(241 247 275)(242 248 276)(243 245 273)(244 246 274)(249 277 283)(250 278 284)(251 279 281)(252 280 282)
(1 41 35)(2 42 36)(3 43 33)(4 44 34)(5 67 79)(6 68 80)(7 65 77)(8 66 78)(9 109 63)(10 110 64)(11 111 61)(12 112 62)(13 69 47)(14 70 48)(15 71 45)(16 72 46)(17 262 268)(18 263 265)(19 264 266)(20 261 267)(21 25 49)(22 26 50)(23 27 51)(24 28 52)(29 59 73)(30 60 74)(31 57 75)(32 58 76)(37 228 236)(38 225 233)(39 226 234)(40 227 235)(53 270 259)(54 271 260)(55 272 257)(56 269 258)(81 147 89)(82 148 90)(83 145 91)(84 146 92)(85 137 129)(86 138 130)(87 139 131)(88 140 132)(93 107 99)(94 108 100)(95 105 97)(96 106 98)(101 143 135)(102 144 136)(103 141 133)(104 142 134)(113 128 150)(114 125 151)(115 126 152)(116 127 149)(117 183 123)(118 184 124)(119 181 121)(120 182 122)(153 161 219)(154 162 220)(155 163 217)(156 164 218)(157 201 209)(158 202 210)(159 203 211)(160 204 212)(165 171 179)(166 172 180)(167 169 177)(168 170 178)(173 207 215)(174 208 216)(175 205 213)(176 206 214)(185 222 200)(186 223 197)(187 224 198)(188 221 199)(189 195 255)(190 196 256)(191 193 253)(192 194 254)(229 273 281)(230 274 282)(231 275 283)(232 276 284)(237 243 251)(238 244 252)(239 241 249)(240 242 250)(245 279 287)(246 280 288)(247 277 285)(248 278 286)
(1 147 73 97)(2 148 74 98)(3 145 75 99)(4 146 76 100)(5 137 49 116)(6 138 50 113)(7 139 51 114)(8 140 52 115)(9 141 71 117)(10 142 72 118)(11 143 69 119)(12 144 70 120)(13 121 61 101)(14 122 62 102)(15 123 63 103)(16 124 64 104)(17 208 278 254)(18 205 279 255)(19 206 280 256)(20 207 277 253)(21 127 67 129)(22 128 68 130)(23 125 65 131)(24 126 66 132)(25 149 79 85)(26 150 80 86)(27 151 77 87)(28 152 78 88)(29 95 41 89)(30 96 42 90)(31 93 43 91)(32 94 44 92)(33 83 57 107)(34 84 58 108)(35 81 59 105)(36 82 60 106)(37 170 244 220)(38 171 241 217)(39 172 242 218)(40 169 243 219)(45 183 109 133)(46 184 110 134)(47 181 111 135)(48 182 112 136)(53 158 232 222)(54 159 229 223)(55 160 230 224)(56 157 231 221)(153 227 177 251)(154 228 178 252)(155 225 179 249)(156 226 180 250)(161 235 167 237)(162 236 168 238)(163 233 165 239)(164 234 166 240)(173 247 193 267)(174 248 194 268)(175 245 195 265)(176 246 196 266)(185 259 210 284)(186 260 211 281)(187 257 212 282)(188 258 209 283)(189 263 213 287)(190 264 214 288)(191 261 215 285)(192 262 216 286)(197 271 203 273)(198 272 204 274)(199 269 201 275)(200 270 202 276)
(1 219 73 169)(2 220 74 170)(3 217 75 171)(4 218 76 172)(5 209 49 188)(6 210 50 185)(7 211 51 186)(8 212 52 187)(9 213 71 189)(10 214 72 190)(11 215 69 191)(12 216 70 192)(13 193 61 173)(14 194 62 174)(15 195 63 175)(16 196 64 176)(17 182 278 136)(18 183 279 133)(19 184 280 134)(20 181 277 135)(21 199 67 201)(22 200 68 202)(23 197 65 203)(24 198 66 204)(25 221 79 157)(26 222 80 158)(27 223 77 159)(28 224 78 160)(29 167 41 161)(30 168 42 162)(31 165 43 163)(32 166 44 164)(33 155 57 179)(34 156 58 180)(35 153 59 177)(36 154 60 178)(37 148 244 98)(38 145 241 99)(39 146 242 100)(40 147 243 97)(45 255 109 205)(46 256 110 206)(47 253 111 207)(48 254 112 208)(53 150 232 86)(54 151 229 87)(55 152 230 88)(56 149 231 85)(81 251 105 227)(82 252 106 228)(83 249 107 225)(84 250 108 226)(89 237 95 235)(90 238 96 236)(91 239 93 233)(92 240 94 234)(101 267 121 247)(102 268 122 248)(103 265 123 245)(104 266 124 246)(113 284 138 259)(114 281 139 260)(115 282 140 257)(116 283 137 258)(117 287 141 263)(118 288 142 264)(119 285 143 261)(120 286 144 262)(125 273 131 271)(126 274 132 272)(127 275 129 269)(128 276 130 270)

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

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

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

84 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F4G4H4I···4P6A···6L12A···12AV
order12223333444444444···46···612···12
size111122221111222218···182···22···2

84 irreducible representations

dim11111112222222
type+++++++-+-
imageC1C2C2C2C2C2C4S3Q8D6C4○D4Dic6C4×S3C4○D12
kernelC4×C324Q8C4×C3⋊Dic3C6.Dic6C12⋊Dic3C122C2×C324Q8C324Q8C4×C12C3×C12C2×C12C3×C6C12C12C6
# reps122111842122161616

Matrix representation of C4×C324Q8 in GL5(𝔽13)

80000
012000
001200
00080
00008
,
10000
012100
012000
000121
000120
,
10000
01000
00100
000121
000120
,
120000
01000
00100
00037
000610
,
10000
011200
001200
000411
00029

G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,1,0],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,6,0,0,0,7,10],[1,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,4,2,0,0,0,11,9] >;

C4×C324Q8 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_4Q_8
% in TeX

G:=Group("C4xC3^2:4Q8");
// GroupNames label

G:=SmallGroup(288,725);
// by ID

G=gap.SmallGroup(288,725);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,58,2693,9414]);
// Polycyclic

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

׿
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𝔽