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## G = C32×C8⋊C4order 288 = 25·32

### Direct product of C32 and C8⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×C8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×C12 — C6×C24 — C32×C8⋊C4
 Lower central C1 — C2 — C32×C8⋊C4
 Upper central C1 — C6×C12 — C32×C8⋊C4

Generators and relations for C32×C8⋊C4
G = < a,b,c,d | a3=b3=c8=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 132 in 120 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×4], C2×C4, C2×C4 [×2], C32, C12 [×8], C12 [×8], C2×C6 [×4], C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C24 [×16], C2×C12 [×12], C8⋊C4, C3×C12 [×2], C3×C12 [×2], C62, C4×C12 [×4], C2×C24 [×8], C3×C24 [×4], C6×C12, C6×C12 [×2], C3×C8⋊C4 [×4], C122, C6×C24 [×2], C32×C8⋊C4
Quotients: C1, C2 [×3], C3 [×4], C4 [×6], C22, C6 [×12], C2×C4 [×3], C32, C12 [×24], C2×C6 [×4], C42, M4(2) [×2], C3×C6 [×3], C2×C12 [×12], C8⋊C4, C3×C12 [×6], C62, C4×C12 [×4], C3×M4(2) [×8], C6×C12 [×3], C3×C8⋊C4 [×4], C122, C32×M4(2) [×2], C32×C8⋊C4

Smallest permutation representation of C32×C8⋊C4
Regular action on 288 points
Generators in S288
(1 170 154)(2 171 155)(3 172 156)(4 173 157)(5 174 158)(6 175 159)(7 176 160)(8 169 153)(9 243 25)(10 244 26)(11 245 27)(12 246 28)(13 247 29)(14 248 30)(15 241 31)(16 242 32)(17 227 33)(18 228 34)(19 229 35)(20 230 36)(21 231 37)(22 232 38)(23 225 39)(24 226 40)(41 224 57)(42 217 58)(43 218 59)(44 219 60)(45 220 61)(46 221 62)(47 222 63)(48 223 64)(49 283 267)(50 284 268)(51 285 269)(52 286 270)(53 287 271)(54 288 272)(55 281 265)(56 282 266)(65 275 259)(66 276 260)(67 277 261)(68 278 262)(69 279 263)(70 280 264)(71 273 257)(72 274 258)(73 161 89)(74 162 90)(75 163 91)(76 164 92)(77 165 93)(78 166 94)(79 167 95)(80 168 96)(81 145 97)(82 146 98)(83 147 99)(84 148 100)(85 149 101)(86 150 102)(87 151 103)(88 152 104)(105 139 123)(106 140 124)(107 141 125)(108 142 126)(109 143 127)(110 144 128)(111 137 121)(112 138 122)(113 202 186)(114 203 187)(115 204 188)(116 205 189)(117 206 190)(118 207 191)(119 208 192)(120 201 185)(129 193 179)(130 194 180)(131 195 181)(132 196 182)(133 197 183)(134 198 184)(135 199 177)(136 200 178)(209 249 233)(210 250 234)(211 251 235)(212 252 236)(213 253 237)(214 254 238)(215 255 239)(216 256 240)
(1 146 74)(2 147 75)(3 148 76)(4 149 77)(5 150 78)(6 151 79)(7 152 80)(8 145 73)(9 211 227)(10 212 228)(11 213 229)(12 214 230)(13 215 231)(14 216 232)(15 209 225)(16 210 226)(17 25 235)(18 26 236)(19 27 237)(20 28 238)(21 29 239)(22 30 240)(23 31 233)(24 32 234)(33 243 251)(34 244 252)(35 245 253)(36 246 254)(37 247 255)(38 248 256)(39 241 249)(40 242 250)(41 49 259)(42 50 260)(43 51 261)(44 52 262)(45 53 263)(46 54 264)(47 55 257)(48 56 258)(57 267 275)(58 268 276)(59 269 277)(60 270 278)(61 271 279)(62 272 280)(63 265 273)(64 266 274)(65 224 283)(66 217 284)(67 218 285)(68 219 286)(69 220 287)(70 221 288)(71 222 281)(72 223 282)(81 89 153)(82 90 154)(83 91 155)(84 92 156)(85 93 157)(86 94 158)(87 95 159)(88 96 160)(97 161 169)(98 162 170)(99 163 171)(100 164 172)(101 165 173)(102 166 174)(103 167 175)(104 168 176)(105 114 181)(106 115 182)(107 116 183)(108 117 184)(109 118 177)(110 119 178)(111 120 179)(112 113 180)(121 185 193)(122 186 194)(123 187 195)(124 188 196)(125 189 197)(126 190 198)(127 191 199)(128 192 200)(129 137 201)(130 138 202)(131 139 203)(132 140 204)(133 141 205)(134 142 206)(135 143 207)(136 144 208)
(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)
(1 191 41 211)(2 188 42 216)(3 185 43 213)(4 190 44 210)(5 187 45 215)(6 192 46 212)(7 189 47 209)(8 186 48 214)(9 74 127 259)(10 79 128 264)(11 76 121 261)(12 73 122 258)(13 78 123 263)(14 75 124 260)(15 80 125 257)(16 77 126 262)(17 82 135 267)(18 87 136 272)(19 84 129 269)(20 81 130 266)(21 86 131 271)(22 83 132 268)(23 88 133 265)(24 85 134 270)(25 90 143 275)(26 95 144 280)(27 92 137 277)(28 89 138 274)(29 94 139 279)(30 91 140 276)(31 96 141 273)(32 93 142 278)(33 98 177 283)(34 103 178 288)(35 100 179 285)(36 97 180 282)(37 102 181 287)(38 99 182 284)(39 104 183 281)(40 101 184 286)(49 227 146 199)(50 232 147 196)(51 229 148 193)(52 226 149 198)(53 231 150 195)(54 228 151 200)(55 225 152 197)(56 230 145 194)(57 235 154 207)(58 240 155 204)(59 237 156 201)(60 234 157 206)(61 239 158 203)(62 236 159 208)(63 233 160 205)(64 238 153 202)(65 243 162 109)(66 248 163 106)(67 245 164 111)(68 242 165 108)(69 247 166 105)(70 244 167 110)(71 241 168 107)(72 246 161 112)(113 223 254 169)(114 220 255 174)(115 217 256 171)(116 222 249 176)(117 219 250 173)(118 224 251 170)(119 221 252 175)(120 218 253 172)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6X 8A ··· 8H 12A ··· 12AF 12AG ··· 12BL 24A ··· 24BL order 1 2 2 2 3 ··· 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 ··· 1 1 1 1 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) kernel C32×C8⋊C4 C122 C6×C24 C3×C8⋊C4 C3×C24 C6×C12 C4×C12 C2×C24 C24 C2×C12 C3×C6 C6 # reps 1 1 2 8 8 4 8 16 64 32 4 32

Matrix representation of C32×C8⋊C4 in GL4(𝔽73) generated by

 64 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 72 0 0 0 0 61 62 0 0 62 12
,
 72 0 0 0 0 27 0 0 0 0 0 1 0 0 72 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,72,0,0,0,0,61,62,0,0,62,12],[72,0,0,0,0,27,0,0,0,0,0,72,0,0,1,0] >;

C32×C8⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times C_8\rtimes C_4
% in TeX

G:=Group("C3^2xC8:C4");
// GroupNames label

G:=SmallGroup(288,315);
// by ID

G=gap.SmallGroup(288,315);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,252,2045,512,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^4=1,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^5>;
// generators/relations

׿
×
𝔽