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## G = C4×C32⋊4Q8order 288 = 25·32

### Direct product of C4 and C32⋊4Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4×C32⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C2×C32⋊4Q8 — C4×C32⋊4Q8
 Lower central C32 — C3×C6 — C4×C32⋊4Q8
 Upper central C1 — C2×C4 — C42

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, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C4×Q8, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C324Q8, C2×C3⋊Dic3, C6×C12, C4×Dic6, C4×C3⋊Dic3, C6.Dic6, C12⋊Dic3, C122, C2×C324Q8, C4×C324Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C3⋊S3, Dic6, C4×S3, C22×S3, C4×Q8, C2×C3⋊S3, C2×Dic6, S3×C2×C4, C4○D12, C324Q8, C4×C3⋊S3, C22×C3⋊S3, C4×Dic6, 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 7 51)(2 8 52)(3 5 49)(4 6 50)(9 35 15)(10 36 16)(11 33 13)(12 34 14)(17 232 234)(18 229 235)(19 230 236)(20 231 233)(21 45 73)(22 46 74)(23 47 75)(24 48 76)(25 69 41)(26 70 42)(27 71 43)(28 72 44)(29 113 57)(30 114 58)(31 115 59)(32 116 60)(37 221 264)(38 222 261)(39 223 262)(40 224 263)(53 270 226)(54 271 227)(55 272 228)(56 269 225)(61 111 67)(62 112 68)(63 109 65)(64 110 66)(77 146 120)(78 147 117)(79 148 118)(80 145 119)(81 125 187)(82 126 188)(83 127 185)(84 128 186)(85 181 91)(86 182 92)(87 183 89)(88 184 90)(93 121 143)(94 122 144)(95 123 141)(96 124 142)(97 131 103)(98 132 104)(99 129 101)(100 130 102)(105 139 133)(106 140 134)(107 137 135)(108 138 136)(149 192 218)(150 189 219)(151 190 220)(152 191 217)(153 259 197)(154 260 198)(155 257 199)(156 258 200)(157 163 253)(158 164 254)(159 161 255)(160 162 256)(165 215 193)(166 216 194)(167 213 195)(168 214 196)(169 175 203)(170 176 204)(171 173 201)(172 174 202)(177 205 211)(178 206 212)(179 207 209)(180 208 210)(237 287 265)(238 288 266)(239 285 267)(240 286 268)(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 43 35)(2 44 36)(3 41 33)(4 42 34)(5 25 13)(6 26 14)(7 27 15)(8 28 16)(9 51 71)(10 52 72)(11 49 69)(12 50 70)(17 262 270)(18 263 271)(19 264 272)(20 261 269)(21 109 29)(22 110 30)(23 111 31)(24 112 32)(37 228 230)(38 225 231)(39 226 232)(40 227 229)(45 65 113)(46 66 114)(47 67 115)(48 68 116)(53 234 223)(54 235 224)(55 236 221)(56 233 222)(57 73 63)(58 74 64)(59 75 61)(60 76 62)(77 92 186)(78 89 187)(79 90 188)(80 91 185)(81 147 87)(82 148 88)(83 145 85)(84 146 86)(93 107 99)(94 108 100)(95 105 97)(96 106 98)(101 143 135)(102 144 136)(103 141 133)(104 142 134)(117 183 125)(118 184 126)(119 181 127)(120 182 128)(121 137 129)(122 138 130)(123 139 131)(124 140 132)(149 258 164)(150 259 161)(151 260 162)(152 257 163)(153 159 219)(154 160 220)(155 157 217)(156 158 218)(165 171 179)(166 172 180)(167 169 177)(168 170 178)(173 207 215)(174 208 216)(175 205 213)(176 206 214)(189 197 255)(190 198 256)(191 199 253)(192 200 254)(193 201 209)(194 202 210)(195 203 211)(196 204 212)(237 243 251)(238 244 252)(239 241 249)(240 242 250)(245 279 287)(246 280 288)(247 277 285)(248 278 286)(265 273 281)(266 274 282)(267 275 283)(268 276 284)
(1 183 109 97)(2 184 110 98)(3 181 111 99)(4 182 112 100)(5 91 67 129)(6 92 68 130)(7 89 65 131)(8 90 66 132)(9 147 73 133)(10 148 74 134)(11 145 75 135)(12 146 76 136)(13 80 47 137)(14 77 48 138)(15 78 45 139)(16 79 46 140)(17 172 242 254)(18 169 243 255)(19 170 244 256)(20 171 241 253)(21 105 35 117)(22 106 36 118)(23 107 33 119)(24 108 34 120)(25 185 115 121)(26 186 116 122)(27 187 113 123)(28 188 114 124)(29 95 43 125)(30 96 44 126)(31 93 41 127)(32 94 42 128)(37 206 280 220)(38 207 277 217)(39 208 278 218)(40 205 279 219)(49 85 61 101)(50 86 62 102)(51 87 63 103)(52 88 64 104)(53 194 268 258)(54 195 265 259)(55 196 266 260)(56 193 267 257)(57 141 71 81)(58 142 72 82)(59 143 69 83)(60 144 70 84)(149 223 210 284)(150 224 211 281)(151 221 212 282)(152 222 209 283)(153 227 213 287)(154 228 214 288)(155 225 215 285)(156 226 216 286)(157 231 173 247)(158 232 174 248)(159 229 175 245)(160 230 176 246)(161 235 203 273)(162 236 204 274)(163 233 201 275)(164 234 202 276)(165 239 199 269)(166 240 200 270)(167 237 197 271)(168 238 198 272)(177 251 189 263)(178 252 190 264)(179 249 191 261)(180 250 192 262)
(1 255 109 169)(2 256 110 170)(3 253 111 171)(4 254 112 172)(5 163 67 201)(6 164 68 202)(7 161 65 203)(8 162 66 204)(9 219 73 205)(10 220 74 206)(11 217 75 207)(12 218 76 208)(13 152 47 209)(14 149 48 210)(15 150 45 211)(16 151 46 212)(17 182 242 100)(18 183 243 97)(19 184 244 98)(20 181 241 99)(21 177 35 189)(22 178 36 190)(23 179 33 191)(24 180 34 192)(25 257 115 193)(26 258 116 194)(27 259 113 195)(28 260 114 196)(29 167 43 197)(30 168 44 198)(31 165 41 199)(32 166 42 200)(37 148 280 134)(38 145 277 135)(39 146 278 136)(40 147 279 133)(49 157 61 173)(50 158 62 174)(51 159 63 175)(52 160 64 176)(53 186 268 122)(54 187 265 123)(55 188 266 124)(56 185 267 121)(57 213 71 153)(58 214 72 154)(59 215 69 155)(60 216 70 156)(77 284 138 223)(78 281 139 224)(79 282 140 221)(80 283 137 222)(81 287 141 227)(82 288 142 228)(83 285 143 225)(84 286 144 226)(85 247 101 231)(86 248 102 232)(87 245 103 229)(88 246 104 230)(89 273 131 235)(90 274 132 236)(91 275 129 233)(92 276 130 234)(93 269 127 239)(94 270 128 240)(95 271 125 237)(96 272 126 238)(105 263 117 251)(106 264 118 252)(107 261 119 249)(108 262 120 250)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A ··· 6L 12A ··· 12AV order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 18 ··· 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D6 C4○D4 Dic6 C4×S3 C4○D12 kernel C4×C32⋊4Q8 C4×C3⋊Dic3 C6.Dic6 C12⋊Dic3 C122 C2×C32⋊4Q8 C32⋊4Q8 C4×C12 C3×C12 C2×C12 C3×C6 C12 C12 C6 # reps 1 2 2 1 1 1 8 4 2 12 2 16 16 16

Matrix representation of C4×C324Q8 in GL5(𝔽13)

 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 1 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 1 0 0 0 12 0
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 7 0 0 0 6 10
,
 1 0 0 0 0 0 1 12 0 0 0 0 12 0 0 0 0 0 4 11 0 0 0 2 9

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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