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## G = C32×C4⋊Q8order 288 = 25·32

### Direct product of C32 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C4⋊Q8
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — Q8×C3×C6 — C32×C4⋊Q8
 Lower central C1 — C22 — C32×C4⋊Q8
 Upper central C1 — C62 — C32×C4⋊Q8

Generators and relations for C32×C4⋊Q8
G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=d2, 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: 252 in 204 conjugacy classes, 156 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C32, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C3×C6, C2×C12, C3×Q8, C4⋊Q8, C3×C12, C3×C12, C62, C4×C12, C3×C4⋊C4, C6×Q8, C6×C12, C6×C12, Q8×C32, C3×C4⋊Q8, C122, C32×C4⋊C4, Q8×C3×C6, C32×C4⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C32, C2×C6, C2×D4, C2×Q8, C3×C6, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C62, C6×D4, C6×Q8, D4×C32, Q8×C32, C2×C62, C3×C4⋊Q8, D4×C3×C6, Q8×C3×C6, C32×C4⋊Q8

Smallest permutation representation of C32×C4⋊Q8
Regular action on 288 points
Generators in S288
(1 86 78)(2 87 79)(3 88 80)(4 85 77)(5 122 13)(6 123 14)(7 124 15)(8 121 16)(9 114 17)(10 115 18)(11 116 19)(12 113 20)(21 183 29)(22 184 30)(23 181 31)(24 182 32)(25 142 134)(26 143 135)(27 144 136)(28 141 133)(33 138 130)(34 139 131)(35 140 132)(36 137 129)(37 81 45)(38 82 46)(39 83 47)(40 84 48)(41 73 49)(42 74 50)(43 75 51)(44 76 52)(53 98 90)(54 99 91)(55 100 92)(56 97 89)(57 101 65)(58 102 66)(59 103 67)(60 104 68)(61 93 69)(62 94 70)(63 95 71)(64 96 72)(105 125 117)(106 126 118)(107 127 119)(108 128 120)(109 176 168)(110 173 165)(111 174 166)(112 175 167)(145 251 243)(146 252 244)(147 249 241)(148 250 242)(149 267 157)(150 268 158)(151 265 159)(152 266 160)(153 259 161)(154 260 162)(155 257 163)(156 258 164)(169 287 279)(170 288 280)(171 285 277)(172 286 278)(177 283 275)(178 284 276)(179 281 273)(180 282 274)(185 232 193)(186 229 194)(187 230 195)(188 231 196)(189 224 197)(190 221 198)(191 222 199)(192 223 200)(201 246 211)(202 247 212)(203 248 209)(204 245 210)(205 237 213)(206 238 214)(207 239 215)(208 240 216)(217 233 225)(218 234 226)(219 235 227)(220 236 228)(253 269 261)(254 270 262)(255 271 263)(256 272 264)
(1 74 38)(2 75 39)(3 76 40)(4 73 37)(5 106 114)(6 107 115)(7 108 116)(8 105 113)(9 13 118)(10 14 119)(11 15 120)(12 16 117)(17 122 126)(18 123 127)(19 124 128)(20 121 125)(21 25 130)(22 26 131)(23 27 132)(24 28 129)(29 134 138)(30 135 139)(31 136 140)(32 133 137)(33 183 142)(34 184 143)(35 181 144)(36 182 141)(41 45 77)(42 46 78)(43 47 79)(44 48 80)(49 81 85)(50 82 86)(51 83 87)(52 84 88)(53 70 102)(54 71 103)(55 72 104)(56 69 101)(57 89 93)(58 90 94)(59 91 95)(60 92 96)(61 65 97)(62 66 98)(63 67 99)(64 68 100)(109 286 180)(110 287 177)(111 288 178)(112 285 179)(145 240 202)(146 237 203)(147 238 204)(148 239 201)(149 255 259)(150 256 260)(151 253 257)(152 254 258)(153 157 263)(154 158 264)(155 159 261)(156 160 262)(161 267 271)(162 268 272)(163 265 269)(164 266 270)(165 169 275)(166 170 276)(167 171 273)(168 172 274)(173 279 283)(174 280 284)(175 277 281)(176 278 282)(185 220 224)(186 217 221)(187 218 222)(188 219 223)(189 193 228)(190 194 225)(191 195 226)(192 196 227)(197 232 236)(198 229 233)(199 230 234)(200 231 235)(205 209 244)(206 210 241)(207 211 242)(208 212 243)(213 248 252)(214 245 249)(215 246 250)(216 247 251)
(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 92 24 106)(2 89 21 107)(3 90 22 108)(4 91 23 105)(5 38 60 129)(6 39 57 130)(7 40 58 131)(8 37 59 132)(9 42 64 133)(10 43 61 134)(11 44 62 135)(12 41 63 136)(13 46 68 137)(14 47 65 138)(15 48 66 139)(16 45 67 140)(17 50 72 141)(18 51 69 142)(19 52 70 143)(20 49 71 144)(25 115 75 93)(26 116 76 94)(27 113 73 95)(28 114 74 96)(29 119 79 97)(30 120 80 98)(31 117 77 99)(32 118 78 100)(33 123 83 101)(34 124 84 102)(35 121 81 103)(36 122 82 104)(53 184 128 88)(54 181 125 85)(55 182 126 86)(56 183 127 87)(109 238 233 269)(110 239 234 270)(111 240 235 271)(112 237 236 272)(145 231 267 178)(146 232 268 179)(147 229 265 180)(148 230 266 177)(149 276 243 188)(150 273 244 185)(151 274 241 186)(152 275 242 187)(153 280 247 192)(154 277 248 189)(155 278 245 190)(156 279 246 191)(157 284 251 196)(158 281 252 193)(159 282 249 194)(160 283 250 195)(161 288 202 200)(162 285 203 197)(163 286 204 198)(164 287 201 199)(165 207 218 254)(166 208 219 255)(167 205 220 256)(168 206 217 253)(169 211 222 258)(170 212 223 259)(171 209 224 260)(172 210 221 257)(173 215 226 262)(174 216 227 263)(175 213 228 264)(176 214 225 261)
(1 165 24 218)(2 168 21 217)(3 167 22 220)(4 166 23 219)(5 242 60 152)(6 241 57 151)(7 244 58 150)(8 243 59 149)(9 246 64 156)(10 245 61 155)(11 248 62 154)(12 247 63 153)(13 250 68 160)(14 249 65 159)(15 252 66 158)(16 251 67 157)(17 201 72 164)(18 204 69 163)(19 203 70 162)(20 202 71 161)(25 221 75 172)(26 224 76 171)(27 223 73 170)(28 222 74 169)(29 225 79 176)(30 228 80 175)(31 227 77 174)(32 226 78 173)(33 229 83 180)(34 232 84 179)(35 231 81 178)(36 230 82 177)(37 276 132 188)(38 275 129 187)(39 274 130 186)(40 273 131 185)(41 280 136 192)(42 279 133 191)(43 278 134 190)(44 277 135 189)(45 284 140 196)(46 283 137 195)(47 282 138 194)(48 281 139 193)(49 288 144 200)(50 287 141 199)(51 286 142 198)(52 285 143 197)(53 272 128 237)(54 271 125 240)(55 270 126 239)(56 269 127 238)(85 111 181 235)(86 110 182 234)(87 109 183 233)(88 112 184 236)(89 253 107 206)(90 256 108 205)(91 255 105 208)(92 254 106 207)(93 257 115 210)(94 260 116 209)(95 259 113 212)(96 258 114 211)(97 261 119 214)(98 264 120 213)(99 263 117 216)(100 262 118 215)(101 265 123 147)(102 268 124 146)(103 267 121 145)(104 266 122 148)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A ··· 4F 4G 4H 4I 4J 6A ··· 6X 12A ··· 12AV 12AW ··· 12CB order 1 2 2 2 3 ··· 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 ··· 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 Q8 C3×D4 C3×Q8 kernel C32×C4⋊Q8 C122 C32×C4⋊C4 Q8×C3×C6 C3×C4⋊Q8 C4×C12 C3×C4⋊C4 C6×Q8 C3×C12 C3×C12 C12 C12 # reps 1 1 4 2 8 8 32 16 2 4 16 32

Matrix representation of C32×C4⋊Q8 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 1 11 0 0 1 12
,
 0 1 0 0 12 0 0 0 0 0 12 2 0 0 12 1
,
 3 4 0 0 4 10 0 0 0 0 5 0 0 0 5 8
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,11,12],[0,12,0,0,1,0,0,0,0,0,12,12,0,0,2,1],[3,4,0,0,4,10,0,0,0,0,5,5,0,0,0,8] >;

C32×C4⋊Q8 in GAP, Magma, Sage, TeX

C_3^2\times C_4\rtimes Q_8
% in TeX

G:=Group("C3^2xC4:Q8");
// GroupNames label

G:=SmallGroup(288,825);
// by ID

G=gap.SmallGroup(288,825);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,1037,512,3110,772]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=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,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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