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## G = C22×C32⋊4C8order 288 = 25·32

### Direct product of C22 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C32⋊4C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C2×C32⋊4C8 — C22×C32⋊4C8
 Lower central C32 — C22×C32⋊4C8
 Upper central C1 — C22×C4

Generators and relations for C22×C324C8
G = < a,b,c,d,e | a2=b2=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 404 in 228 conjugacy classes, 173 normal (11 characteristic)
C1, C2, C2 [×6], C3 [×4], C4, C4 [×3], C22 [×7], C6 [×28], C8 [×4], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C2×C8 [×6], C22×C4, C3×C6, C3×C6 [×6], C3⋊C8 [×16], C2×C12 [×24], C22×C6 [×4], C22×C8, C3×C12, C3×C12 [×3], C62 [×7], C2×C3⋊C8 [×24], C22×C12 [×4], C324C8 [×4], C6×C12 [×6], C2×C62, C22×C3⋊C8 [×4], C2×C324C8 [×6], C2×C6×C12, C22×C324C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C8 [×4], C2×C4 [×6], C23, Dic3 [×16], D6 [×12], C2×C8 [×6], C22×C4, C3⋊S3, C3⋊C8 [×16], C2×Dic3 [×24], C22×S3 [×4], C22×C8, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C2×C3⋊C8 [×24], C22×Dic3 [×4], C324C8 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, C22×C3⋊C8 [×4], C2×C324C8 [×6], C22×C3⋊Dic3, C22×C324C8

Smallest permutation representation of C22×C324C8
Regular action on 288 points
Generators in S288
(1 200)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 201)(10 202)(11 203)(12 204)(13 205)(14 206)(15 207)(16 208)(17 190)(18 191)(19 192)(20 185)(21 186)(22 187)(23 188)(24 189)(25 211)(26 212)(27 213)(28 214)(29 215)(30 216)(31 209)(32 210)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 116)(42 117)(43 118)(44 119)(45 120)(46 113)(47 114)(48 115)(49 124)(50 125)(51 126)(52 127)(53 128)(54 121)(55 122)(56 123)(57 132)(58 133)(59 134)(60 135)(61 136)(62 129)(63 130)(64 131)(65 140)(66 141)(67 142)(68 143)(69 144)(70 137)(71 138)(72 139)(81 250)(82 251)(83 252)(84 253)(85 254)(86 255)(87 256)(88 249)(89 258)(90 259)(91 260)(92 261)(93 262)(94 263)(95 264)(96 257)(97 266)(98 267)(99 268)(100 269)(101 270)(102 271)(103 272)(104 265)(105 274)(106 275)(107 276)(108 277)(109 278)(110 279)(111 280)(112 273)(145 222)(146 223)(147 224)(148 217)(149 218)(150 219)(151 220)(152 221)(153 230)(154 231)(155 232)(156 225)(157 226)(158 227)(159 228)(160 229)(161 238)(162 239)(163 240)(164 233)(165 234)(166 235)(167 236)(168 237)(169 246)(170 247)(171 248)(172 241)(173 242)(174 243)(175 244)(176 245)(177 281)(178 282)(179 283)(180 284)(181 285)(182 286)(183 287)(184 288)
(1 142)(2 143)(3 144)(4 137)(5 138)(6 139)(7 140)(8 141)(9 124)(10 125)(11 126)(12 127)(13 128)(14 121)(15 122)(16 123)(17 132)(18 133)(19 134)(20 135)(21 136)(22 129)(23 130)(24 131)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(41 184)(42 177)(43 178)(44 179)(45 180)(46 181)(47 182)(48 183)(49 201)(50 202)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(57 190)(58 191)(59 192)(60 185)(61 186)(62 187)(63 188)(64 189)(65 198)(66 199)(67 200)(68 193)(69 194)(70 195)(71 196)(72 197)(73 213)(74 214)(75 215)(76 216)(77 209)(78 210)(79 211)(80 212)(81 218)(82 219)(83 220)(84 221)(85 222)(86 223)(87 224)(88 217)(89 226)(90 227)(91 228)(92 229)(93 230)(94 231)(95 232)(96 225)(97 234)(98 235)(99 236)(100 237)(101 238)(102 239)(103 240)(104 233)(105 242)(106 243)(107 244)(108 245)(109 246)(110 247)(111 248)(112 241)(113 285)(114 286)(115 287)(116 288)(117 281)(118 282)(119 283)(120 284)(145 254)(146 255)(147 256)(148 249)(149 250)(150 251)(151 252)(152 253)(153 262)(154 263)(155 264)(156 257)(157 258)(158 259)(159 260)(160 261)(161 270)(162 271)(163 272)(164 265)(165 266)(166 267)(167 268)(168 269)(169 278)(170 279)(171 280)(172 273)(173 274)(174 275)(175 276)(176 277)
(1 117 272)(2 265 118)(3 119 266)(4 267 120)(5 113 268)(6 269 114)(7 115 270)(8 271 116)(9 273 145)(10 146 274)(11 275 147)(12 148 276)(13 277 149)(14 150 278)(15 279 151)(16 152 280)(17 36 159)(18 160 37)(19 38 153)(20 154 39)(21 40 155)(22 156 33)(23 34 157)(24 158 35)(25 135 263)(26 264 136)(27 129 257)(28 258 130)(29 131 259)(30 260 132)(31 133 261)(32 262 134)(41 199 102)(42 103 200)(43 193 104)(44 97 194)(45 195 98)(46 99 196)(47 197 100)(48 101 198)(49 241 85)(50 86 242)(51 243 87)(52 88 244)(53 245 81)(54 82 246)(55 247 83)(56 84 248)(57 216 91)(58 92 209)(59 210 93)(60 94 211)(61 212 95)(62 96 213)(63 214 89)(64 90 215)(65 183 238)(66 239 184)(67 177 240)(68 233 178)(69 179 234)(70 235 180)(71 181 236)(72 237 182)(73 187 225)(74 226 188)(75 189 227)(76 228 190)(77 191 229)(78 230 192)(79 185 231)(80 232 186)(105 202 223)(106 224 203)(107 204 217)(108 218 205)(109 206 219)(110 220 207)(111 208 221)(112 222 201)(121 251 169)(122 170 252)(123 253 171)(124 172 254)(125 255 173)(126 174 256)(127 249 175)(128 176 250)(137 166 284)(138 285 167)(139 168 286)(140 287 161)(141 162 288)(142 281 163)(143 164 282)(144 283 165)
(1 15 257)(2 258 16)(3 9 259)(4 260 10)(5 11 261)(6 262 12)(7 13 263)(8 264 14)(17 255 166)(18 167 256)(19 249 168)(20 161 250)(21 251 162)(22 163 252)(23 253 164)(24 165 254)(25 115 277)(26 278 116)(27 117 279)(28 280 118)(29 119 273)(30 274 120)(31 113 275)(32 276 114)(33 281 170)(34 171 282)(35 283 172)(36 173 284)(37 285 174)(38 175 286)(39 287 176)(40 169 288)(41 212 109)(42 110 213)(43 214 111)(44 112 215)(45 216 105)(46 106 209)(47 210 107)(48 108 211)(49 227 69)(50 70 228)(51 229 71)(52 72 230)(53 231 65)(54 66 232)(55 225 67)(56 68 226)(57 223 98)(58 99 224)(59 217 100)(60 101 218)(61 219 102)(62 103 220)(63 221 104)(64 97 222)(73 177 247)(74 248 178)(75 179 241)(76 242 180)(77 181 243)(78 244 182)(79 183 245)(80 246 184)(81 185 238)(82 239 186)(83 187 240)(84 233 188)(85 189 234)(86 235 190)(87 191 236)(88 237 192)(89 208 193)(90 194 201)(91 202 195)(92 196 203)(93 204 197)(94 198 205)(95 206 199)(96 200 207)(121 141 155)(122 156 142)(123 143 157)(124 158 144)(125 137 159)(126 160 138)(127 139 153)(128 154 140)(129 272 151)(130 152 265)(131 266 145)(132 146 267)(133 268 147)(134 148 269)(135 270 149)(136 150 271)
(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)| (1,200)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,201)(10,202)(11,203)(12,204)(13,205)(14,206)(15,207)(16,208)(17,190)(18,191)(19,192)(20,185)(21,186)(22,187)(23,188)(24,189)(25,211)(26,212)(27,213)(28,214)(29,215)(30,216)(31,209)(32,210)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,116)(42,117)(43,118)(44,119)(45,120)(46,113)(47,114)(48,115)(49,124)(50,125)(51,126)(52,127)(53,128)(54,121)(55,122)(56,123)(57,132)(58,133)(59,134)(60,135)(61,136)(62,129)(63,130)(64,131)(65,140)(66,141)(67,142)(68,143)(69,144)(70,137)(71,138)(72,139)(81,250)(82,251)(83,252)(84,253)(85,254)(86,255)(87,256)(88,249)(89,258)(90,259)(91,260)(92,261)(93,262)(94,263)(95,264)(96,257)(97,266)(98,267)(99,268)(100,269)(101,270)(102,271)(103,272)(104,265)(105,274)(106,275)(107,276)(108,277)(109,278)(110,279)(111,280)(112,273)(145,222)(146,223)(147,224)(148,217)(149,218)(150,219)(151,220)(152,221)(153,230)(154,231)(155,232)(156,225)(157,226)(158,227)(159,228)(160,229)(161,238)(162,239)(163,240)(164,233)(165,234)(166,235)(167,236)(168,237)(169,246)(170,247)(171,248)(172,241)(173,242)(174,243)(175,244)(176,245)(177,281)(178,282)(179,283)(180,284)(181,285)(182,286)(183,287)(184,288), (1,142)(2,143)(3,144)(4,137)(5,138)(6,139)(7,140)(8,141)(9,124)(10,125)(11,126)(12,127)(13,128)(14,121)(15,122)(16,123)(17,132)(18,133)(19,134)(20,135)(21,136)(22,129)(23,130)(24,131)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(41,184)(42,177)(43,178)(44,179)(45,180)(46,181)(47,182)(48,183)(49,201)(50,202)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,190)(58,191)(59,192)(60,185)(61,186)(62,187)(63,188)(64,189)(65,198)(66,199)(67,200)(68,193)(69,194)(70,195)(71,196)(72,197)(73,213)(74,214)(75,215)(76,216)(77,209)(78,210)(79,211)(80,212)(81,218)(82,219)(83,220)(84,221)(85,222)(86,223)(87,224)(88,217)(89,226)(90,227)(91,228)(92,229)(93,230)(94,231)(95,232)(96,225)(97,234)(98,235)(99,236)(100,237)(101,238)(102,239)(103,240)(104,233)(105,242)(106,243)(107,244)(108,245)(109,246)(110,247)(111,248)(112,241)(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)(120,284)(145,254)(146,255)(147,256)(148,249)(149,250)(150,251)(151,252)(152,253)(153,262)(154,263)(155,264)(156,257)(157,258)(158,259)(159,260)(160,261)(161,270)(162,271)(163,272)(164,265)(165,266)(166,267)(167,268)(168,269)(169,278)(170,279)(171,280)(172,273)(173,274)(174,275)(175,276)(176,277), (1,117,272)(2,265,118)(3,119,266)(4,267,120)(5,113,268)(6,269,114)(7,115,270)(8,271,116)(9,273,145)(10,146,274)(11,275,147)(12,148,276)(13,277,149)(14,150,278)(15,279,151)(16,152,280)(17,36,159)(18,160,37)(19,38,153)(20,154,39)(21,40,155)(22,156,33)(23,34,157)(24,158,35)(25,135,263)(26,264,136)(27,129,257)(28,258,130)(29,131,259)(30,260,132)(31,133,261)(32,262,134)(41,199,102)(42,103,200)(43,193,104)(44,97,194)(45,195,98)(46,99,196)(47,197,100)(48,101,198)(49,241,85)(50,86,242)(51,243,87)(52,88,244)(53,245,81)(54,82,246)(55,247,83)(56,84,248)(57,216,91)(58,92,209)(59,210,93)(60,94,211)(61,212,95)(62,96,213)(63,214,89)(64,90,215)(65,183,238)(66,239,184)(67,177,240)(68,233,178)(69,179,234)(70,235,180)(71,181,236)(72,237,182)(73,187,225)(74,226,188)(75,189,227)(76,228,190)(77,191,229)(78,230,192)(79,185,231)(80,232,186)(105,202,223)(106,224,203)(107,204,217)(108,218,205)(109,206,219)(110,220,207)(111,208,221)(112,222,201)(121,251,169)(122,170,252)(123,253,171)(124,172,254)(125,255,173)(126,174,256)(127,249,175)(128,176,250)(137,166,284)(138,285,167)(139,168,286)(140,287,161)(141,162,288)(142,281,163)(143,164,282)(144,283,165), (1,15,257)(2,258,16)(3,9,259)(4,260,10)(5,11,261)(6,262,12)(7,13,263)(8,264,14)(17,255,166)(18,167,256)(19,249,168)(20,161,250)(21,251,162)(22,163,252)(23,253,164)(24,165,254)(25,115,277)(26,278,116)(27,117,279)(28,280,118)(29,119,273)(30,274,120)(31,113,275)(32,276,114)(33,281,170)(34,171,282)(35,283,172)(36,173,284)(37,285,174)(38,175,286)(39,287,176)(40,169,288)(41,212,109)(42,110,213)(43,214,111)(44,112,215)(45,216,105)(46,106,209)(47,210,107)(48,108,211)(49,227,69)(50,70,228)(51,229,71)(52,72,230)(53,231,65)(54,66,232)(55,225,67)(56,68,226)(57,223,98)(58,99,224)(59,217,100)(60,101,218)(61,219,102)(62,103,220)(63,221,104)(64,97,222)(73,177,247)(74,248,178)(75,179,241)(76,242,180)(77,181,243)(78,244,182)(79,183,245)(80,246,184)(81,185,238)(82,239,186)(83,187,240)(84,233,188)(85,189,234)(86,235,190)(87,191,236)(88,237,192)(89,208,193)(90,194,201)(91,202,195)(92,196,203)(93,204,197)(94,198,205)(95,206,199)(96,200,207)(121,141,155)(122,156,142)(123,143,157)(124,158,144)(125,137,159)(126,160,138)(127,139,153)(128,154,140)(129,272,151)(130,152,265)(131,266,145)(132,146,267)(133,268,147)(134,148,269)(135,270,149)(136,150,271), (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( (1,200)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,201)(10,202)(11,203)(12,204)(13,205)(14,206)(15,207)(16,208)(17,190)(18,191)(19,192)(20,185)(21,186)(22,187)(23,188)(24,189)(25,211)(26,212)(27,213)(28,214)(29,215)(30,216)(31,209)(32,210)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,116)(42,117)(43,118)(44,119)(45,120)(46,113)(47,114)(48,115)(49,124)(50,125)(51,126)(52,127)(53,128)(54,121)(55,122)(56,123)(57,132)(58,133)(59,134)(60,135)(61,136)(62,129)(63,130)(64,131)(65,140)(66,141)(67,142)(68,143)(69,144)(70,137)(71,138)(72,139)(81,250)(82,251)(83,252)(84,253)(85,254)(86,255)(87,256)(88,249)(89,258)(90,259)(91,260)(92,261)(93,262)(94,263)(95,264)(96,257)(97,266)(98,267)(99,268)(100,269)(101,270)(102,271)(103,272)(104,265)(105,274)(106,275)(107,276)(108,277)(109,278)(110,279)(111,280)(112,273)(145,222)(146,223)(147,224)(148,217)(149,218)(150,219)(151,220)(152,221)(153,230)(154,231)(155,232)(156,225)(157,226)(158,227)(159,228)(160,229)(161,238)(162,239)(163,240)(164,233)(165,234)(166,235)(167,236)(168,237)(169,246)(170,247)(171,248)(172,241)(173,242)(174,243)(175,244)(176,245)(177,281)(178,282)(179,283)(180,284)(181,285)(182,286)(183,287)(184,288), (1,142)(2,143)(3,144)(4,137)(5,138)(6,139)(7,140)(8,141)(9,124)(10,125)(11,126)(12,127)(13,128)(14,121)(15,122)(16,123)(17,132)(18,133)(19,134)(20,135)(21,136)(22,129)(23,130)(24,131)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(41,184)(42,177)(43,178)(44,179)(45,180)(46,181)(47,182)(48,183)(49,201)(50,202)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,190)(58,191)(59,192)(60,185)(61,186)(62,187)(63,188)(64,189)(65,198)(66,199)(67,200)(68,193)(69,194)(70,195)(71,196)(72,197)(73,213)(74,214)(75,215)(76,216)(77,209)(78,210)(79,211)(80,212)(81,218)(82,219)(83,220)(84,221)(85,222)(86,223)(87,224)(88,217)(89,226)(90,227)(91,228)(92,229)(93,230)(94,231)(95,232)(96,225)(97,234)(98,235)(99,236)(100,237)(101,238)(102,239)(103,240)(104,233)(105,242)(106,243)(107,244)(108,245)(109,246)(110,247)(111,248)(112,241)(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)(120,284)(145,254)(146,255)(147,256)(148,249)(149,250)(150,251)(151,252)(152,253)(153,262)(154,263)(155,264)(156,257)(157,258)(158,259)(159,260)(160,261)(161,270)(162,271)(163,272)(164,265)(165,266)(166,267)(167,268)(168,269)(169,278)(170,279)(171,280)(172,273)(173,274)(174,275)(175,276)(176,277), (1,117,272)(2,265,118)(3,119,266)(4,267,120)(5,113,268)(6,269,114)(7,115,270)(8,271,116)(9,273,145)(10,146,274)(11,275,147)(12,148,276)(13,277,149)(14,150,278)(15,279,151)(16,152,280)(17,36,159)(18,160,37)(19,38,153)(20,154,39)(21,40,155)(22,156,33)(23,34,157)(24,158,35)(25,135,263)(26,264,136)(27,129,257)(28,258,130)(29,131,259)(30,260,132)(31,133,261)(32,262,134)(41,199,102)(42,103,200)(43,193,104)(44,97,194)(45,195,98)(46,99,196)(47,197,100)(48,101,198)(49,241,85)(50,86,242)(51,243,87)(52,88,244)(53,245,81)(54,82,246)(55,247,83)(56,84,248)(57,216,91)(58,92,209)(59,210,93)(60,94,211)(61,212,95)(62,96,213)(63,214,89)(64,90,215)(65,183,238)(66,239,184)(67,177,240)(68,233,178)(69,179,234)(70,235,180)(71,181,236)(72,237,182)(73,187,225)(74,226,188)(75,189,227)(76,228,190)(77,191,229)(78,230,192)(79,185,231)(80,232,186)(105,202,223)(106,224,203)(107,204,217)(108,218,205)(109,206,219)(110,220,207)(111,208,221)(112,222,201)(121,251,169)(122,170,252)(123,253,171)(124,172,254)(125,255,173)(126,174,256)(127,249,175)(128,176,250)(137,166,284)(138,285,167)(139,168,286)(140,287,161)(141,162,288)(142,281,163)(143,164,282)(144,283,165), (1,15,257)(2,258,16)(3,9,259)(4,260,10)(5,11,261)(6,262,12)(7,13,263)(8,264,14)(17,255,166)(18,167,256)(19,249,168)(20,161,250)(21,251,162)(22,163,252)(23,253,164)(24,165,254)(25,115,277)(26,278,116)(27,117,279)(28,280,118)(29,119,273)(30,274,120)(31,113,275)(32,276,114)(33,281,170)(34,171,282)(35,283,172)(36,173,284)(37,285,174)(38,175,286)(39,287,176)(40,169,288)(41,212,109)(42,110,213)(43,214,111)(44,112,215)(45,216,105)(46,106,209)(47,210,107)(48,108,211)(49,227,69)(50,70,228)(51,229,71)(52,72,230)(53,231,65)(54,66,232)(55,225,67)(56,68,226)(57,223,98)(58,99,224)(59,217,100)(60,101,218)(61,219,102)(62,103,220)(63,221,104)(64,97,222)(73,177,247)(74,248,178)(75,179,241)(76,242,180)(77,181,243)(78,244,182)(79,183,245)(80,246,184)(81,185,238)(82,239,186)(83,187,240)(84,233,188)(85,189,234)(86,235,190)(87,191,236)(88,237,192)(89,208,193)(90,194,201)(91,202,195)(92,196,203)(93,204,197)(94,198,205)(95,206,199)(96,200,207)(121,141,155)(122,156,142)(123,143,157)(124,158,144)(125,137,159)(126,160,138)(127,139,153)(128,154,140)(129,272,151)(130,152,265)(131,266,145)(132,146,267)(133,268,147)(134,148,269)(135,270,149)(136,150,271), (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([(1,200),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,199),(9,201),(10,202),(11,203),(12,204),(13,205),(14,206),(15,207),(16,208),(17,190),(18,191),(19,192),(20,185),(21,186),(22,187),(23,188),(24,189),(25,211),(26,212),(27,213),(28,214),(29,215),(30,216),(31,209),(32,210),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,116),(42,117),(43,118),(44,119),(45,120),(46,113),(47,114),(48,115),(49,124),(50,125),(51,126),(52,127),(53,128),(54,121),(55,122),(56,123),(57,132),(58,133),(59,134),(60,135),(61,136),(62,129),(63,130),(64,131),(65,140),(66,141),(67,142),(68,143),(69,144),(70,137),(71,138),(72,139),(81,250),(82,251),(83,252),(84,253),(85,254),(86,255),(87,256),(88,249),(89,258),(90,259),(91,260),(92,261),(93,262),(94,263),(95,264),(96,257),(97,266),(98,267),(99,268),(100,269),(101,270),(102,271),(103,272),(104,265),(105,274),(106,275),(107,276),(108,277),(109,278),(110,279),(111,280),(112,273),(145,222),(146,223),(147,224),(148,217),(149,218),(150,219),(151,220),(152,221),(153,230),(154,231),(155,232),(156,225),(157,226),(158,227),(159,228),(160,229),(161,238),(162,239),(163,240),(164,233),(165,234),(166,235),(167,236),(168,237),(169,246),(170,247),(171,248),(172,241),(173,242),(174,243),(175,244),(176,245),(177,281),(178,282),(179,283),(180,284),(181,285),(182,286),(183,287),(184,288)], [(1,142),(2,143),(3,144),(4,137),(5,138),(6,139),(7,140),(8,141),(9,124),(10,125),(11,126),(12,127),(13,128),(14,121),(15,122),(16,123),(17,132),(18,133),(19,134),(20,135),(21,136),(22,129),(23,130),(24,131),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(41,184),(42,177),(43,178),(44,179),(45,180),(46,181),(47,182),(48,183),(49,201),(50,202),(51,203),(52,204),(53,205),(54,206),(55,207),(56,208),(57,190),(58,191),(59,192),(60,185),(61,186),(62,187),(63,188),(64,189),(65,198),(66,199),(67,200),(68,193),(69,194),(70,195),(71,196),(72,197),(73,213),(74,214),(75,215),(76,216),(77,209),(78,210),(79,211),(80,212),(81,218),(82,219),(83,220),(84,221),(85,222),(86,223),(87,224),(88,217),(89,226),(90,227),(91,228),(92,229),(93,230),(94,231),(95,232),(96,225),(97,234),(98,235),(99,236),(100,237),(101,238),(102,239),(103,240),(104,233),(105,242),(106,243),(107,244),(108,245),(109,246),(110,247),(111,248),(112,241),(113,285),(114,286),(115,287),(116,288),(117,281),(118,282),(119,283),(120,284),(145,254),(146,255),(147,256),(148,249),(149,250),(150,251),(151,252),(152,253),(153,262),(154,263),(155,264),(156,257),(157,258),(158,259),(159,260),(160,261),(161,270),(162,271),(163,272),(164,265),(165,266),(166,267),(167,268),(168,269),(169,278),(170,279),(171,280),(172,273),(173,274),(174,275),(175,276),(176,277)], [(1,117,272),(2,265,118),(3,119,266),(4,267,120),(5,113,268),(6,269,114),(7,115,270),(8,271,116),(9,273,145),(10,146,274),(11,275,147),(12,148,276),(13,277,149),(14,150,278),(15,279,151),(16,152,280),(17,36,159),(18,160,37),(19,38,153),(20,154,39),(21,40,155),(22,156,33),(23,34,157),(24,158,35),(25,135,263),(26,264,136),(27,129,257),(28,258,130),(29,131,259),(30,260,132),(31,133,261),(32,262,134),(41,199,102),(42,103,200),(43,193,104),(44,97,194),(45,195,98),(46,99,196),(47,197,100),(48,101,198),(49,241,85),(50,86,242),(51,243,87),(52,88,244),(53,245,81),(54,82,246),(55,247,83),(56,84,248),(57,216,91),(58,92,209),(59,210,93),(60,94,211),(61,212,95),(62,96,213),(63,214,89),(64,90,215),(65,183,238),(66,239,184),(67,177,240),(68,233,178),(69,179,234),(70,235,180),(71,181,236),(72,237,182),(73,187,225),(74,226,188),(75,189,227),(76,228,190),(77,191,229),(78,230,192),(79,185,231),(80,232,186),(105,202,223),(106,224,203),(107,204,217),(108,218,205),(109,206,219),(110,220,207),(111,208,221),(112,222,201),(121,251,169),(122,170,252),(123,253,171),(124,172,254),(125,255,173),(126,174,256),(127,249,175),(128,176,250),(137,166,284),(138,285,167),(139,168,286),(140,287,161),(141,162,288),(142,281,163),(143,164,282),(144,283,165)], [(1,15,257),(2,258,16),(3,9,259),(4,260,10),(5,11,261),(6,262,12),(7,13,263),(8,264,14),(17,255,166),(18,167,256),(19,249,168),(20,161,250),(21,251,162),(22,163,252),(23,253,164),(24,165,254),(25,115,277),(26,278,116),(27,117,279),(28,280,118),(29,119,273),(30,274,120),(31,113,275),(32,276,114),(33,281,170),(34,171,282),(35,283,172),(36,173,284),(37,285,174),(38,175,286),(39,287,176),(40,169,288),(41,212,109),(42,110,213),(43,214,111),(44,112,215),(45,216,105),(46,106,209),(47,210,107),(48,108,211),(49,227,69),(50,70,228),(51,229,71),(52,72,230),(53,231,65),(54,66,232),(55,225,67),(56,68,226),(57,223,98),(58,99,224),(59,217,100),(60,101,218),(61,219,102),(62,103,220),(63,221,104),(64,97,222),(73,177,247),(74,248,178),(75,179,241),(76,242,180),(77,181,243),(78,244,182),(79,183,245),(80,246,184),(81,185,238),(82,239,186),(83,187,240),(84,233,188),(85,189,234),(86,235,190),(87,191,236),(88,237,192),(89,208,193),(90,194,201),(91,202,195),(92,196,203),(93,204,197),(94,198,205),(95,206,199),(96,200,207),(121,141,155),(122,156,142),(123,143,157),(124,158,144),(125,137,159),(126,160,138),(127,139,153),(128,154,140),(129,272,151),(130,152,265),(131,266,145),(132,146,267),(133,268,147),(134,148,269),(135,270,149),(136,150,271)], [(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 ··· 2G 3A 3B 3C 3D 4A ··· 4H 6A ··· 6AB 8A ··· 8P 12A ··· 12AF order 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 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 Dic3 C3⋊C8 kernel C22×C32⋊4C8 C2×C32⋊4C8 C2×C6×C12 C6×C12 C2×C62 C62 C22×C12 C2×C12 C2×C12 C22×C6 C2×C6 # reps 1 6 1 6 2 16 4 12 12 4 32

Matrix representation of C22×C324C8 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 72 1 0 0 0 72 0 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 1 0 0 0 72 72
,
 1 0 0 0 0 0 0 51 0 0 0 51 0 0 0 0 0 0 72 45 0 0 0 46 1

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,72,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,72,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,1,72],[1,0,0,0,0,0,0,51,0,0,0,51,0,0,0,0,0,0,72,46,0,0,0,45,1] >;

C22×C324C8 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("C2^2xC3^2:4C8");
// GroupNames label

G:=SmallGroup(288,777);
// by ID

G=gap.SmallGroup(288,777);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^8=1,a*b=b*a,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

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