metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9⋊3Q8, Dic18⋊5C4, C9⋊2(C4×Q8), C4⋊C4.7D9, C4.4(C4×D9), C2.1(Q8×D9), C12.7(C4×S3), C6.32(S3×Q8), C36.10(C2×C4), (C2×C4).29D18, C18.10(C2×Q8), (C2×C12).179D6, Dic9⋊C4.5C2, C18.8(C22×C4), (C4×Dic9).8C2, Dic9.2(C2×C4), C18.24(C4○D4), C2.3(D4⋊2D9), C3.(Dic6⋊C4), (C2×C18).28C23, (C2×C36).55C22, (C2×Dic18).7C2, C6.79(D4⋊2S3), C22.15(C22×D9), (C2×Dic9).28C22, C6.47(S3×C2×C4), C2.10(C2×C4×D9), (C9×C4⋊C4).4C2, (C3×C4⋊C4).5S3, (C2×C6).185(C22×S3), SmallGroup(288,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9⋊3Q8
G = < a,b,c,d | a18=c4=1, b2=a9, d2=c2, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 344 in 105 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, C9, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, C18, Dic6, C2×Dic3, C2×C12, C2×C12, C4×Q8, Dic9, Dic9, C36, C36, C2×C18, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, Dic18, C2×Dic9, C2×Dic9, C2×C36, C2×C36, Dic6⋊C4, C4×Dic9, C4×Dic9, Dic9⋊C4, C9×C4⋊C4, C2×Dic18, Dic9⋊3Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, D9, C4×S3, C22×S3, C4×Q8, D18, S3×C2×C4, D4⋊2S3, S3×Q8, C4×D9, C22×D9, Dic6⋊C4, C2×C4×D9, D4⋊2D9, Q8×D9, Dic9⋊3Q8
►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)
(1 159 10 150)(2 158 11 149)(3 157 12 148)(4 156 13 147)(5 155 14 146)(6 154 15 145)(7 153 16 162)(8 152 17 161)(9 151 18 160)(19 174 28 165)(20 173 29 164)(21 172 30 163)(22 171 31 180)(23 170 32 179)(24 169 33 178)(25 168 34 177)(26 167 35 176)(27 166 36 175)(37 195 46 186)(38 194 47 185)(39 193 48 184)(40 192 49 183)(41 191 50 182)(42 190 51 181)(43 189 52 198)(44 188 53 197)(45 187 54 196)(55 204 64 213)(56 203 65 212)(57 202 66 211)(58 201 67 210)(59 200 68 209)(60 199 69 208)(61 216 70 207)(62 215 71 206)(63 214 72 205)(73 230 82 221)(74 229 83 220)(75 228 84 219)(76 227 85 218)(77 226 86 217)(78 225 87 234)(79 224 88 233)(80 223 89 232)(81 222 90 231)(91 251 100 242)(92 250 101 241)(93 249 102 240)(94 248 103 239)(95 247 104 238)(96 246 105 237)(97 245 106 236)(98 244 107 235)(99 243 108 252)(109 256 118 265)(110 255 119 264)(111 254 120 263)(112 253 121 262)(113 270 122 261)(114 269 123 260)(115 268 124 259)(116 267 125 258)(117 266 126 257)(127 279 136 288)(128 278 137 287)(129 277 138 286)(130 276 139 285)(131 275 140 284)(132 274 141 283)(133 273 142 282)(134 272 143 281)(135 271 144 280)
(1 199 34 186)(2 216 35 185)(3 215 36 184)(4 214 19 183)(5 213 20 182)(6 212 21 181)(7 211 22 198)(8 210 23 197)(9 209 24 196)(10 208 25 195)(11 207 26 194)(12 206 27 193)(13 205 28 192)(14 204 29 191)(15 203 30 190)(16 202 31 189)(17 201 32 188)(18 200 33 187)(37 159 69 177)(38 158 70 176)(39 157 71 175)(40 156 72 174)(41 155 55 173)(42 154 56 172)(43 153 57 171)(44 152 58 170)(45 151 59 169)(46 150 60 168)(47 149 61 167)(48 148 62 166)(49 147 63 165)(50 146 64 164)(51 145 65 163)(52 162 66 180)(53 161 67 179)(54 160 68 178)(73 254 102 272)(74 253 103 271)(75 270 104 288)(76 269 105 287)(77 268 106 286)(78 267 107 285)(79 266 108 284)(80 265 91 283)(81 264 92 282)(82 263 93 281)(83 262 94 280)(84 261 95 279)(85 260 96 278)(86 259 97 277)(87 258 98 276)(88 257 99 275)(89 256 100 274)(90 255 101 273)(109 251 132 223)(110 250 133 222)(111 249 134 221)(112 248 135 220)(113 247 136 219)(114 246 137 218)(115 245 138 217)(116 244 139 234)(117 243 140 233)(118 242 141 232)(119 241 142 231)(120 240 143 230)(121 239 144 229)(122 238 127 228)(123 237 128 227)(124 236 129 226)(125 235 130 225)(126 252 131 224)
(1 93 34 82)(2 94 35 83)(3 95 36 84)(4 96 19 85)(5 97 20 86)(6 98 21 87)(7 99 22 88)(8 100 23 89)(9 101 24 90)(10 102 25 73)(11 103 26 74)(12 104 27 75)(13 105 28 76)(14 106 29 77)(15 107 30 78)(16 108 31 79)(17 91 32 80)(18 92 33 81)(37 134 69 111)(38 135 70 112)(39 136 71 113)(40 137 72 114)(41 138 55 115)(42 139 56 116)(43 140 57 117)(44 141 58 118)(45 142 59 119)(46 143 60 120)(47 144 61 121)(48 127 62 122)(49 128 63 123)(50 129 64 124)(51 130 65 125)(52 131 66 126)(53 132 67 109)(54 133 68 110)(145 235 163 225)(146 236 164 226)(147 237 165 227)(148 238 166 228)(149 239 167 229)(150 240 168 230)(151 241 169 231)(152 242 170 232)(153 243 171 233)(154 244 172 234)(155 245 173 217)(156 246 174 218)(157 247 175 219)(158 248 176 220)(159 249 177 221)(160 250 178 222)(161 251 179 223)(162 252 180 224)(181 276 212 258)(182 277 213 259)(183 278 214 260)(184 279 215 261)(185 280 216 262)(186 281 199 263)(187 282 200 264)(188 283 201 265)(189 284 202 266)(190 285 203 267)(191 286 204 268)(192 287 205 269)(193 288 206 270)(194 271 207 253)(195 272 208 254)(196 273 209 255)(197 274 210 256)(198 275 211 257)
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), (1,159,10,150)(2,158,11,149)(3,157,12,148)(4,156,13,147)(5,155,14,146)(6,154,15,145)(7,153,16,162)(8,152,17,161)(9,151,18,160)(19,174,28,165)(20,173,29,164)(21,172,30,163)(22,171,31,180)(23,170,32,179)(24,169,33,178)(25,168,34,177)(26,167,35,176)(27,166,36,175)(37,195,46,186)(38,194,47,185)(39,193,48,184)(40,192,49,183)(41,191,50,182)(42,190,51,181)(43,189,52,198)(44,188,53,197)(45,187,54,196)(55,204,64,213)(56,203,65,212)(57,202,66,211)(58,201,67,210)(59,200,68,209)(60,199,69,208)(61,216,70,207)(62,215,71,206)(63,214,72,205)(73,230,82,221)(74,229,83,220)(75,228,84,219)(76,227,85,218)(77,226,86,217)(78,225,87,234)(79,224,88,233)(80,223,89,232)(81,222,90,231)(91,251,100,242)(92,250,101,241)(93,249,102,240)(94,248,103,239)(95,247,104,238)(96,246,105,237)(97,245,106,236)(98,244,107,235)(99,243,108,252)(109,256,118,265)(110,255,119,264)(111,254,120,263)(112,253,121,262)(113,270,122,261)(114,269,123,260)(115,268,124,259)(116,267,125,258)(117,266,126,257)(127,279,136,288)(128,278,137,287)(129,277,138,286)(130,276,139,285)(131,275,140,284)(132,274,141,283)(133,273,142,282)(134,272,143,281)(135,271,144,280), (1,199,34,186)(2,216,35,185)(3,215,36,184)(4,214,19,183)(5,213,20,182)(6,212,21,181)(7,211,22,198)(8,210,23,197)(9,209,24,196)(10,208,25,195)(11,207,26,194)(12,206,27,193)(13,205,28,192)(14,204,29,191)(15,203,30,190)(16,202,31,189)(17,201,32,188)(18,200,33,187)(37,159,69,177)(38,158,70,176)(39,157,71,175)(40,156,72,174)(41,155,55,173)(42,154,56,172)(43,153,57,171)(44,152,58,170)(45,151,59,169)(46,150,60,168)(47,149,61,167)(48,148,62,166)(49,147,63,165)(50,146,64,164)(51,145,65,163)(52,162,66,180)(53,161,67,179)(54,160,68,178)(73,254,102,272)(74,253,103,271)(75,270,104,288)(76,269,105,287)(77,268,106,286)(78,267,107,285)(79,266,108,284)(80,265,91,283)(81,264,92,282)(82,263,93,281)(83,262,94,280)(84,261,95,279)(85,260,96,278)(86,259,97,277)(87,258,98,276)(88,257,99,275)(89,256,100,274)(90,255,101,273)(109,251,132,223)(110,250,133,222)(111,249,134,221)(112,248,135,220)(113,247,136,219)(114,246,137,218)(115,245,138,217)(116,244,139,234)(117,243,140,233)(118,242,141,232)(119,241,142,231)(120,240,143,230)(121,239,144,229)(122,238,127,228)(123,237,128,227)(124,236,129,226)(125,235,130,225)(126,252,131,224), (1,93,34,82)(2,94,35,83)(3,95,36,84)(4,96,19,85)(5,97,20,86)(6,98,21,87)(7,99,22,88)(8,100,23,89)(9,101,24,90)(10,102,25,73)(11,103,26,74)(12,104,27,75)(13,105,28,76)(14,106,29,77)(15,107,30,78)(16,108,31,79)(17,91,32,80)(18,92,33,81)(37,134,69,111)(38,135,70,112)(39,136,71,113)(40,137,72,114)(41,138,55,115)(42,139,56,116)(43,140,57,117)(44,141,58,118)(45,142,59,119)(46,143,60,120)(47,144,61,121)(48,127,62,122)(49,128,63,123)(50,129,64,124)(51,130,65,125)(52,131,66,126)(53,132,67,109)(54,133,68,110)(145,235,163,225)(146,236,164,226)(147,237,165,227)(148,238,166,228)(149,239,167,229)(150,240,168,230)(151,241,169,231)(152,242,170,232)(153,243,171,233)(154,244,172,234)(155,245,173,217)(156,246,174,218)(157,247,175,219)(158,248,176,220)(159,249,177,221)(160,250,178,222)(161,251,179,223)(162,252,180,224)(181,276,212,258)(182,277,213,259)(183,278,214,260)(184,279,215,261)(185,280,216,262)(186,281,199,263)(187,282,200,264)(188,283,201,265)(189,284,202,266)(190,285,203,267)(191,286,204,268)(192,287,205,269)(193,288,206,270)(194,271,207,253)(195,272,208,254)(196,273,209,255)(197,274,210,256)(198,275,211,257)>;
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), (1,159,10,150)(2,158,11,149)(3,157,12,148)(4,156,13,147)(5,155,14,146)(6,154,15,145)(7,153,16,162)(8,152,17,161)(9,151,18,160)(19,174,28,165)(20,173,29,164)(21,172,30,163)(22,171,31,180)(23,170,32,179)(24,169,33,178)(25,168,34,177)(26,167,35,176)(27,166,36,175)(37,195,46,186)(38,194,47,185)(39,193,48,184)(40,192,49,183)(41,191,50,182)(42,190,51,181)(43,189,52,198)(44,188,53,197)(45,187,54,196)(55,204,64,213)(56,203,65,212)(57,202,66,211)(58,201,67,210)(59,200,68,209)(60,199,69,208)(61,216,70,207)(62,215,71,206)(63,214,72,205)(73,230,82,221)(74,229,83,220)(75,228,84,219)(76,227,85,218)(77,226,86,217)(78,225,87,234)(79,224,88,233)(80,223,89,232)(81,222,90,231)(91,251,100,242)(92,250,101,241)(93,249,102,240)(94,248,103,239)(95,247,104,238)(96,246,105,237)(97,245,106,236)(98,244,107,235)(99,243,108,252)(109,256,118,265)(110,255,119,264)(111,254,120,263)(112,253,121,262)(113,270,122,261)(114,269,123,260)(115,268,124,259)(116,267,125,258)(117,266,126,257)(127,279,136,288)(128,278,137,287)(129,277,138,286)(130,276,139,285)(131,275,140,284)(132,274,141,283)(133,273,142,282)(134,272,143,281)(135,271,144,280), (1,199,34,186)(2,216,35,185)(3,215,36,184)(4,214,19,183)(5,213,20,182)(6,212,21,181)(7,211,22,198)(8,210,23,197)(9,209,24,196)(10,208,25,195)(11,207,26,194)(12,206,27,193)(13,205,28,192)(14,204,29,191)(15,203,30,190)(16,202,31,189)(17,201,32,188)(18,200,33,187)(37,159,69,177)(38,158,70,176)(39,157,71,175)(40,156,72,174)(41,155,55,173)(42,154,56,172)(43,153,57,171)(44,152,58,170)(45,151,59,169)(46,150,60,168)(47,149,61,167)(48,148,62,166)(49,147,63,165)(50,146,64,164)(51,145,65,163)(52,162,66,180)(53,161,67,179)(54,160,68,178)(73,254,102,272)(74,253,103,271)(75,270,104,288)(76,269,105,287)(77,268,106,286)(78,267,107,285)(79,266,108,284)(80,265,91,283)(81,264,92,282)(82,263,93,281)(83,262,94,280)(84,261,95,279)(85,260,96,278)(86,259,97,277)(87,258,98,276)(88,257,99,275)(89,256,100,274)(90,255,101,273)(109,251,132,223)(110,250,133,222)(111,249,134,221)(112,248,135,220)(113,247,136,219)(114,246,137,218)(115,245,138,217)(116,244,139,234)(117,243,140,233)(118,242,141,232)(119,241,142,231)(120,240,143,230)(121,239,144,229)(122,238,127,228)(123,237,128,227)(124,236,129,226)(125,235,130,225)(126,252,131,224), (1,93,34,82)(2,94,35,83)(3,95,36,84)(4,96,19,85)(5,97,20,86)(6,98,21,87)(7,99,22,88)(8,100,23,89)(9,101,24,90)(10,102,25,73)(11,103,26,74)(12,104,27,75)(13,105,28,76)(14,106,29,77)(15,107,30,78)(16,108,31,79)(17,91,32,80)(18,92,33,81)(37,134,69,111)(38,135,70,112)(39,136,71,113)(40,137,72,114)(41,138,55,115)(42,139,56,116)(43,140,57,117)(44,141,58,118)(45,142,59,119)(46,143,60,120)(47,144,61,121)(48,127,62,122)(49,128,63,123)(50,129,64,124)(51,130,65,125)(52,131,66,126)(53,132,67,109)(54,133,68,110)(145,235,163,225)(146,236,164,226)(147,237,165,227)(148,238,166,228)(149,239,167,229)(150,240,168,230)(151,241,169,231)(152,242,170,232)(153,243,171,233)(154,244,172,234)(155,245,173,217)(156,246,174,218)(157,247,175,219)(158,248,176,220)(159,249,177,221)(160,250,178,222)(161,251,179,223)(162,252,180,224)(181,276,212,258)(182,277,213,259)(183,278,214,260)(184,279,215,261)(185,280,216,262)(186,281,199,263)(187,282,200,264)(188,283,201,265)(189,284,202,266)(190,285,203,267)(191,286,204,268)(192,287,205,269)(193,288,206,270)(194,271,207,253)(195,272,208,254)(196,273,209,255)(197,274,210,256)(198,275,211,257) );
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)], [(1,159,10,150),(2,158,11,149),(3,157,12,148),(4,156,13,147),(5,155,14,146),(6,154,15,145),(7,153,16,162),(8,152,17,161),(9,151,18,160),(19,174,28,165),(20,173,29,164),(21,172,30,163),(22,171,31,180),(23,170,32,179),(24,169,33,178),(25,168,34,177),(26,167,35,176),(27,166,36,175),(37,195,46,186),(38,194,47,185),(39,193,48,184),(40,192,49,183),(41,191,50,182),(42,190,51,181),(43,189,52,198),(44,188,53,197),(45,187,54,196),(55,204,64,213),(56,203,65,212),(57,202,66,211),(58,201,67,210),(59,200,68,209),(60,199,69,208),(61,216,70,207),(62,215,71,206),(63,214,72,205),(73,230,82,221),(74,229,83,220),(75,228,84,219),(76,227,85,218),(77,226,86,217),(78,225,87,234),(79,224,88,233),(80,223,89,232),(81,222,90,231),(91,251,100,242),(92,250,101,241),(93,249,102,240),(94,248,103,239),(95,247,104,238),(96,246,105,237),(97,245,106,236),(98,244,107,235),(99,243,108,252),(109,256,118,265),(110,255,119,264),(111,254,120,263),(112,253,121,262),(113,270,122,261),(114,269,123,260),(115,268,124,259),(116,267,125,258),(117,266,126,257),(127,279,136,288),(128,278,137,287),(129,277,138,286),(130,276,139,285),(131,275,140,284),(132,274,141,283),(133,273,142,282),(134,272,143,281),(135,271,144,280)], [(1,199,34,186),(2,216,35,185),(3,215,36,184),(4,214,19,183),(5,213,20,182),(6,212,21,181),(7,211,22,198),(8,210,23,197),(9,209,24,196),(10,208,25,195),(11,207,26,194),(12,206,27,193),(13,205,28,192),(14,204,29,191),(15,203,30,190),(16,202,31,189),(17,201,32,188),(18,200,33,187),(37,159,69,177),(38,158,70,176),(39,157,71,175),(40,156,72,174),(41,155,55,173),(42,154,56,172),(43,153,57,171),(44,152,58,170),(45,151,59,169),(46,150,60,168),(47,149,61,167),(48,148,62,166),(49,147,63,165),(50,146,64,164),(51,145,65,163),(52,162,66,180),(53,161,67,179),(54,160,68,178),(73,254,102,272),(74,253,103,271),(75,270,104,288),(76,269,105,287),(77,268,106,286),(78,267,107,285),(79,266,108,284),(80,265,91,283),(81,264,92,282),(82,263,93,281),(83,262,94,280),(84,261,95,279),(85,260,96,278),(86,259,97,277),(87,258,98,276),(88,257,99,275),(89,256,100,274),(90,255,101,273),(109,251,132,223),(110,250,133,222),(111,249,134,221),(112,248,135,220),(113,247,136,219),(114,246,137,218),(115,245,138,217),(116,244,139,234),(117,243,140,233),(118,242,141,232),(119,241,142,231),(120,240,143,230),(121,239,144,229),(122,238,127,228),(123,237,128,227),(124,236,129,226),(125,235,130,225),(126,252,131,224)], [(1,93,34,82),(2,94,35,83),(3,95,36,84),(4,96,19,85),(5,97,20,86),(6,98,21,87),(7,99,22,88),(8,100,23,89),(9,101,24,90),(10,102,25,73),(11,103,26,74),(12,104,27,75),(13,105,28,76),(14,106,29,77),(15,107,30,78),(16,108,31,79),(17,91,32,80),(18,92,33,81),(37,134,69,111),(38,135,70,112),(39,136,71,113),(40,137,72,114),(41,138,55,115),(42,139,56,116),(43,140,57,117),(44,141,58,118),(45,142,59,119),(46,143,60,120),(47,144,61,121),(48,127,62,122),(49,128,63,123),(50,129,64,124),(51,130,65,125),(52,131,66,126),(53,132,67,109),(54,133,68,110),(145,235,163,225),(146,236,164,226),(147,237,165,227),(148,238,166,228),(149,239,167,229),(150,240,168,230),(151,241,169,231),(152,242,170,232),(153,243,171,233),(154,244,172,234),(155,245,173,217),(156,246,174,218),(157,247,175,219),(158,248,176,220),(159,249,177,221),(160,250,178,222),(161,251,179,223),(162,252,180,224),(181,276,212,258),(182,277,213,259),(183,278,214,260),(184,279,215,261),(185,280,216,262),(186,281,199,263),(187,282,200,264),(188,283,201,265),(189,284,202,266),(190,285,203,267),(191,286,204,268),(192,287,205,269),(193,288,206,270),(194,271,207,253),(195,272,208,254),(196,273,209,255),(197,274,210,256),(198,275,211,257)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D6 | C4○D4 | D9 | C4×S3 | D18 | C4×D9 | D4⋊2S3 | S3×Q8 | D4⋊2D9 | Q8×D9 |
| kernel | Dic9⋊3Q8 | C4×Dic9 | Dic9⋊C4 | C9×C4⋊C4 | C2×Dic18 | Dic18 | C3×C4⋊C4 | Dic9 | C2×C12 | C18 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 8 | 1 | 2 | 3 | 2 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of Dic9⋊3Q8 ►in GL4(𝔽37) generated by
| 6 | 17 | 0 | 0 |
| 20 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 8 | 0 | 0 |
| 9 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 11 | 0 | 0 |
| 17 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 8 | 34 |
| 0 | 0 | 34 | 29 |
G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,1,0,0,0,0,1],[36,9,0,0,8,1,0,0,0,0,1,0,0,0,0,1],[31,17,0,0,11,6,0,0,0,0,0,36,0,0,1,0],[36,0,0,0,0,36,0,0,0,0,8,34,0,0,34,29] >;
Dic9⋊3Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_9\rtimes_3Q_8 % in TeX
G:=Group("Dic9:3Q8"); // GroupNames label
G:=SmallGroup(288,97);
// by ID
G=gap.SmallGroup(288,97);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,135,142,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=c^4=1,b^2=a^9,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations