Copied to
clipboard

## G = C32×C4⋊C8order 288 = 25·32

### Direct product of C32 and C4⋊C8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32×C4⋊C8
G = < a,b,c,d | a3=b3=c4=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 132 in 114 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C4, C22, C6 [×12], C8 [×2], C2×C4 [×3], C32, C12 [×16], C12 [×4], C2×C6 [×4], C42, C2×C8 [×2], C3×C6 [×3], C24 [×8], C2×C12 [×12], C4⋊C8, C3×C12 [×2], C3×C12 [×2], C3×C12, C62, C4×C12 [×4], C2×C24 [×8], C3×C24 [×2], C6×C12 [×3], C3×C4⋊C8 [×4], C122, C6×C24 [×2], C32×C4⋊C8
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, D4, Q8, C32, C12 [×8], C2×C6 [×4], C4⋊C4, C2×C8, M4(2), C3×C6 [×3], C24 [×8], C2×C12 [×4], C3×D4 [×4], C3×Q8 [×4], C4⋊C8, C3×C12 [×2], C62, C3×C4⋊C4 [×4], C2×C24 [×4], C3×M4(2) [×4], C3×C24 [×2], C6×C12, D4×C32, Q8×C32, C3×C4⋊C8 [×4], C32×C4⋊C4, C6×C24, C32×M4(2), C32×C4⋊C8

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

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 2 2 2 2 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C8 C12 C24 D4 Q8 M4(2) C3×D4 C3×Q8 C3×M4(2) kernel C32×C4⋊C8 C122 C6×C24 C3×C4⋊C8 C6×C12 C4×C12 C2×C24 C3×C12 C2×C12 C12 C3×C12 C3×C12 C3×C6 C12 C12 C6 # reps 1 1 2 8 4 8 16 8 32 64 1 1 2 8 8 16

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

 8 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 46 0 0 0 0 27
,
 22 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,0,27],[22,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C32×C4⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3^2xC4:C8");
// GroupNames label

G:=SmallGroup(288,323);
// by ID

G=gap.SmallGroup(288,323);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,260,124]);
// Polycyclic

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

׿
×
𝔽