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## G = Q8×C3⋊Dic3order 288 = 25·32

### Direct product of Q8 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Q8×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — Q8×C3⋊Dic3
 Lower central C32 — C3×C6 — Q8×C3⋊Dic3
 Upper central C1 — C22 — C2×Q8

Generators and relations for Q8×C3⋊Dic3
G = < a,b,c,d,e | a4=c3=d6=1, b2=a2, e2=d3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 532 in 210 conjugacy classes, 127 normal (14 characteristic)
C1, C2 [×3], C3 [×4], C4 [×6], C4 [×5], C22, C6 [×12], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C32, Dic3 [×20], C12 [×24], C2×C6 [×4], C42 [×3], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], C2×Dic3 [×16], C2×C12 [×12], C3×Q8 [×16], C4×Q8, C3⋊Dic3 [×2], C3⋊Dic3 [×3], C3×C12 [×6], C62, C4×Dic3 [×12], C4⋊Dic3 [×12], C6×Q8 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×3], C6×C12 [×3], Q8×C32 [×4], Q8×Dic3 [×4], C4×C3⋊Dic3 [×3], C12⋊Dic3 [×3], Q8×C3×C6, Q8×C3⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], Q8 [×2], C23, Dic3 [×16], D6 [×12], C22×C4, C2×Q8, C4○D4, C3⋊S3, C2×Dic3 [×24], C22×S3 [×4], C4×Q8, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], S3×Q8 [×4], Q83S3 [×4], C22×Dic3 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, Q8×Dic3 [×4], Q8×C3⋊S3, C12.26D6, C22×C3⋊Dic3, Q8×C3⋊Dic3

Smallest permutation representation of Q8×C3⋊Dic3
Regular action on 288 points
Generators in S288
(1 69 33 50)(2 70 34 51)(3 71 35 52)(4 72 36 53)(5 67 31 54)(6 68 32 49)(7 278 27 275)(8 279 28 276)(9 280 29 271)(10 281 30 272)(11 282 25 273)(12 277 26 274)(13 93 57 66)(14 94 58 61)(15 95 59 62)(16 96 60 63)(17 91 55 64)(18 92 56 65)(19 224 260 254)(20 225 261 255)(21 226 262 256)(22 227 263 257)(23 228 264 258)(24 223 259 253)(37 79 43 73)(38 80 44 74)(39 81 45 75)(40 82 46 76)(41 83 47 77)(42 84 48 78)(85 121 104 140)(86 122 105 141)(87 123 106 142)(88 124 107 143)(89 125 108 144)(90 126 103 139)(97 133 130 168)(98 134 131 163)(99 135 132 164)(100 136 127 165)(101 137 128 166)(102 138 129 167)(109 147 115 169)(110 148 116 170)(111 149 117 171)(112 150 118 172)(113 145 119 173)(114 146 120 174)(151 217 181 187)(152 218 182 188)(153 219 183 189)(154 220 184 190)(155 221 185 191)(156 222 186 192)(157 216 180 193)(158 211 175 194)(159 212 176 195)(160 213 177 196)(161 214 178 197)(162 215 179 198)(199 244 208 235)(200 245 209 236)(201 246 210 237)(202 241 205 238)(203 242 206 239)(204 243 207 240)(229 265 252 288)(230 266 247 283)(231 267 248 284)(232 268 249 285)(233 269 250 286)(234 270 251 287)
(1 105 33 86)(2 106 34 87)(3 107 35 88)(4 108 36 89)(5 103 31 90)(6 104 32 85)(7 239 27 242)(8 240 28 243)(9 235 29 244)(10 236 30 245)(11 237 25 246)(12 238 26 241)(13 129 57 102)(14 130 58 97)(15 131 59 98)(16 132 60 99)(17 127 55 100)(18 128 56 101)(19 218 260 188)(20 219 261 189)(21 220 262 190)(22 221 263 191)(23 222 264 192)(24 217 259 187)(37 115 43 109)(38 116 44 110)(39 117 45 111)(40 118 46 112)(41 119 47 113)(42 120 48 114)(49 140 68 121)(50 141 69 122)(51 142 70 123)(52 143 71 124)(53 144 72 125)(54 139 67 126)(61 168 94 133)(62 163 95 134)(63 164 96 135)(64 165 91 136)(65 166 92 137)(66 167 93 138)(73 169 79 147)(74 170 80 148)(75 171 81 149)(76 172 82 150)(77 173 83 145)(78 174 84 146)(151 253 181 223)(152 254 182 224)(153 255 183 225)(154 256 184 226)(155 257 185 227)(156 258 186 228)(157 252 180 229)(158 247 175 230)(159 248 176 231)(160 249 177 232)(161 250 178 233)(162 251 179 234)(193 288 216 265)(194 283 211 266)(195 284 212 267)(196 285 213 268)(197 286 214 269)(198 287 215 270)(199 280 208 271)(200 281 209 272)(201 282 210 273)(202 277 205 274)(203 278 206 275)(204 279 207 276)
(1 41 60)(2 42 55)(3 37 56)(4 38 57)(5 39 58)(6 40 59)(7 269 21)(8 270 22)(9 265 23)(10 266 24)(11 267 19)(12 268 20)(13 36 44)(14 31 45)(15 32 46)(16 33 47)(17 34 48)(18 35 43)(25 284 260)(26 285 261)(27 286 262)(28 287 263)(29 288 264)(30 283 259)(49 76 95)(50 77 96)(51 78 91)(52 73 92)(53 74 93)(54 75 94)(61 67 81)(62 68 82)(63 69 83)(64 70 84)(65 71 79)(66 72 80)(85 112 131)(86 113 132)(87 114 127)(88 109 128)(89 110 129)(90 111 130)(97 103 117)(98 104 118)(99 105 119)(100 106 120)(101 107 115)(102 108 116)(121 150 163)(122 145 164)(123 146 165)(124 147 166)(125 148 167)(126 149 168)(133 139 171)(134 140 172)(135 141 173)(136 142 174)(137 143 169)(138 144 170)(151 209 175)(152 210 176)(153 205 177)(154 206 178)(155 207 179)(156 208 180)(157 186 199)(158 181 200)(159 182 201)(160 183 202)(161 184 203)(162 185 204)(187 245 211)(188 246 212)(189 241 213)(190 242 214)(191 243 215)(192 244 216)(193 222 235)(194 217 236)(195 218 237)(196 219 238)(197 220 239)(198 221 240)(223 281 247)(224 282 248)(225 277 249)(226 278 250)(227 279 251)(228 280 252)(229 258 271)(230 253 272)(231 254 273)(232 255 274)(233 256 275)(234 257 276)
(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)
(1 157 4 160)(2 162 5 159)(3 161 6 158)(7 150 10 147)(8 149 11 146)(9 148 12 145)(13 153 16 156)(14 152 17 155)(15 151 18 154)(19 165 22 168)(20 164 23 167)(21 163 24 166)(25 174 28 171)(26 173 29 170)(27 172 30 169)(31 176 34 179)(32 175 35 178)(33 180 36 177)(37 203 40 200)(38 202 41 199)(39 201 42 204)(43 206 46 209)(44 205 47 208)(45 210 48 207)(49 194 52 197)(50 193 53 196)(51 198 54 195)(55 185 58 182)(56 184 59 181)(57 183 60 186)(61 188 64 191)(62 187 65 190)(63 192 66 189)(67 212 70 215)(68 211 71 214)(69 216 72 213)(73 239 76 236)(74 238 77 235)(75 237 78 240)(79 242 82 245)(80 241 83 244)(81 246 84 243)(85 230 88 233)(86 229 89 232)(87 234 90 231)(91 221 94 218)(92 220 95 217)(93 219 96 222)(97 224 100 227)(98 223 101 226)(99 228 102 225)(103 248 106 251)(104 247 107 250)(105 252 108 249)(109 275 112 272)(110 274 113 271)(111 273 114 276)(115 278 118 281)(116 277 119 280)(117 282 120 279)(121 266 124 269)(122 265 125 268)(123 270 126 267)(127 257 130 254)(128 256 131 253)(129 255 132 258)(133 260 136 263)(134 259 137 262)(135 264 138 261)(139 284 142 287)(140 283 143 286)(141 288 144 285)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 6A ··· 6L 12A ··· 12X order 1 2 2 2 3 3 3 3 4 ··· 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 ··· 2 9 9 9 9 18 ··· 18 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + - - + image C1 C2 C2 C2 C4 S3 Q8 D6 Dic3 C4○D4 S3×Q8 Q8⋊3S3 kernel Q8×C3⋊Dic3 C4×C3⋊Dic3 C12⋊Dic3 Q8×C3×C6 Q8×C32 C6×Q8 C3⋊Dic3 C2×C12 C3×Q8 C3×C6 C6 C6 # reps 1 3 3 1 8 4 2 12 16 2 4 4

Matrix representation of Q8×C3⋊Dic3 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 10 3

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,7,3] >;

Q8×C3⋊Dic3 in GAP, Magma, Sage, TeX

Q_8\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("Q8xC3:Dic3");
// GroupNames label

G:=SmallGroup(288,802);
// by ID

G=gap.SmallGroup(288,802);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,219,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^6=1,b^2=a^2,e^2=d^3,b*a*b^-1=a^-1,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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