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## G = C4×C32⋊4C8order 288 = 25·32

### Direct product of C4 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C32⋊4C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C32⋊4C8 — C4×C32⋊4C8
 Lower central C32 — C4×C32⋊4C8
 Upper central C1 — C42

Generators and relations for C4×C324C8
G = < a,b,c,d | a4=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 244 in 132 conjugacy classes, 97 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×6], C22, C6 [×12], C8 [×4], C2×C4, C2×C4 [×2], C32, C12 [×24], C2×C6 [×4], C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×16], C2×C12 [×12], C4×C8, C3×C12 [×6], C62, C2×C3⋊C8 [×8], C4×C12 [×4], C324C8 [×4], C6×C12, C6×C12 [×2], C4×C3⋊C8 [×4], C2×C324C8 [×2], C122, C4×C324C8
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×4], C8 [×4], C2×C4 [×3], Dic3 [×8], D6 [×4], C42, C2×C8 [×2], C3⋊S3, C3⋊C8 [×16], C4×S3 [×8], C2×Dic3 [×4], C4×C8, C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×8], C4×Dic3 [×4], C324C8 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C4×C3⋊C8 [×4], C2×C324C8 [×2], C4×C3⋊Dic3, C4×C324C8

Smallest permutation representation of C4×C324C8
Regular action on 288 points
Generators in S288
(1 260 20 47)(2 261 21 48)(3 262 22 41)(4 263 23 42)(5 264 24 43)(6 257 17 44)(7 258 18 45)(8 259 19 46)(9 114 230 66)(10 115 231 67)(11 116 232 68)(12 117 225 69)(13 118 226 70)(14 119 227 71)(15 120 228 72)(16 113 229 65)(25 167 265 218)(26 168 266 219)(27 161 267 220)(28 162 268 221)(29 163 269 222)(30 164 270 223)(31 165 271 224)(32 166 272 217)(33 83 136 54)(34 84 129 55)(35 85 130 56)(36 86 131 49)(37 87 132 50)(38 88 133 51)(39 81 134 52)(40 82 135 53)(57 288 107 211)(58 281 108 212)(59 282 109 213)(60 283 110 214)(61 284 111 215)(62 285 112 216)(63 286 105 209)(64 287 106 210)(73 178 205 255)(74 179 206 256)(75 180 207 249)(76 181 208 250)(77 182 201 251)(78 183 202 252)(79 184 203 253)(80 177 204 254)(89 152 123 238)(90 145 124 239)(91 146 125 240)(92 147 126 233)(93 148 127 234)(94 149 128 235)(95 150 121 236)(96 151 122 237)(97 157 199 241)(98 158 200 242)(99 159 193 243)(100 160 194 244)(101 153 195 245)(102 154 196 246)(103 155 197 247)(104 156 198 248)(137 189 176 273)(138 190 169 274)(139 191 170 275)(140 192 171 276)(141 185 172 277)(142 186 173 278)(143 187 174 279)(144 188 175 280)
(1 60 158)(2 159 61)(3 62 160)(4 153 63)(5 64 154)(6 155 57)(7 58 156)(8 157 59)(9 79 219)(10 220 80)(11 73 221)(12 222 74)(13 75 223)(14 224 76)(15 77 217)(16 218 78)(17 247 107)(18 108 248)(19 241 109)(20 110 242)(21 243 111)(22 112 244)(23 245 105)(24 106 246)(25 183 113)(26 114 184)(27 177 115)(28 116 178)(29 179 117)(30 118 180)(31 181 119)(32 120 182)(33 96 174)(34 175 89)(35 90 176)(36 169 91)(37 92 170)(38 171 93)(39 94 172)(40 173 95)(41 216 100)(42 101 209)(43 210 102)(44 103 211)(45 212 104)(46 97 213)(47 214 98)(48 99 215)(49 190 240)(50 233 191)(51 192 234)(52 235 185)(53 186 236)(54 237 187)(55 188 238)(56 239 189)(65 265 252)(66 253 266)(67 267 254)(68 255 268)(69 269 256)(70 249 270)(71 271 250)(72 251 272)(81 149 277)(82 278 150)(83 151 279)(84 280 152)(85 145 273)(86 274 146)(87 147 275)(88 276 148)(121 135 142)(122 143 136)(123 129 144)(124 137 130)(125 131 138)(126 139 132)(127 133 140)(128 141 134)(161 204 231)(162 232 205)(163 206 225)(164 226 207)(165 208 227)(166 228 201)(167 202 229)(168 230 203)(193 284 261)(194 262 285)(195 286 263)(196 264 287)(197 288 257)(198 258 281)(199 282 259)(200 260 283)
(1 72 274)(2 275 65)(3 66 276)(4 277 67)(5 68 278)(6 279 69)(7 70 280)(8 273 71)(9 140 262)(10 263 141)(11 142 264)(12 257 143)(13 144 258)(14 259 137)(15 138 260)(16 261 139)(17 187 117)(18 118 188)(19 189 119)(20 120 190)(21 191 113)(22 114 192)(23 185 115)(24 116 186)(25 243 50)(26 51 244)(27 245 52)(28 53 246)(29 247 54)(30 55 248)(31 241 56)(32 49 242)(33 163 103)(34 104 164)(35 165 97)(36 98 166)(37 167 99)(38 100 168)(39 161 101)(40 102 162)(41 230 171)(42 172 231)(43 232 173)(44 174 225)(45 226 175)(46 176 227)(47 228 169)(48 170 229)(57 151 256)(58 249 152)(59 145 250)(60 251 146)(61 147 252)(62 253 148)(63 149 254)(64 255 150)(73 121 287)(74 288 122)(75 123 281)(76 282 124)(77 125 283)(78 284 126)(79 127 285)(80 286 128)(81 267 153)(82 154 268)(83 269 155)(84 156 270)(85 271 157)(86 158 272)(87 265 159)(88 160 266)(89 212 207)(90 208 213)(91 214 201)(92 202 215)(93 216 203)(94 204 209)(95 210 205)(96 206 211)(105 235 177)(106 178 236)(107 237 179)(108 180 238)(109 239 181)(110 182 240)(111 233 183)(112 184 234)(129 198 223)(130 224 199)(131 200 217)(132 218 193)(133 194 219)(134 220 195)(135 196 221)(136 222 197)
(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)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A ··· 4L 6A ··· 6L 8A ··· 8P 12A ··· 12AV order 1 2 2 2 3 3 3 3 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 C3⋊C8 C4×S3 kernel C4×C32⋊4C8 C2×C32⋊4C8 C122 C32⋊4C8 C6×C12 C3×C12 C4×C12 C2×C12 C2×C12 C12 C12 # reps 1 2 1 8 4 16 4 8 4 32 16

Matrix representation of C4×C324C8 in GL5(𝔽73)

 46 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 72 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 72 72
,
 1 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 72 72
,
 27 0 0 0 0 0 41 48 0 0 0 7 32 0 0 0 0 0 46 45 0 0 0 72 27

G:=sub<GL(5,GF(73))| [46,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,72],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,1,72],[27,0,0,0,0,0,41,7,0,0,0,48,32,0,0,0,0,0,46,72,0,0,0,45,27] >;

C4×C324C8 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("C4xC3^2:4C8");
// GroupNames label

G:=SmallGroup(288,277);
// by ID

G=gap.SmallGroup(288,277);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,64,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽