metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9.Q8, C4⋊C4.5D9, C2.5(Q8×D9), (C2×C12).7D6, C6.34(S3×Q8), (C2×C4).30D18, C4⋊Dic9.6C2, C18.11(C2×Q8), C9⋊2(C42.C2), Dic9⋊C4.3C2, (C2×C36).8C22, (C4×Dic9).9C2, C3.(Dic3.Q8), C6.81(C4○D12), C18.11(C4○D4), (C2×C18).30C23, C2.11(D4⋊2D9), C6.80(D4⋊2S3), C2.13(D36⋊5C2), C22.47(C22×D9), (C2×Dic9).30C22, (C9×C4⋊C4).6C2, (C3×C4⋊C4).7S3, (C2×C6).187(C22×S3), SmallGroup(288,99)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9.Q8
G = < a,b,c,d | a18=c4=1, b2=a9, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a9b, bd=db, dcd-1=a9c-1 >
Subgroups: 292 in 84 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C9, Dic3, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C18, C2×Dic3, C2×C12, C42.C2, Dic9, Dic9, C36, C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic9, C2×C36, Dic3.Q8, C4×Dic9, Dic9⋊C4, C4⋊Dic9, C9×C4⋊C4, Dic9.Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, D9, C22×S3, C42.C2, D18, C4○D12, D4⋊2S3, S3×Q8, C22×D9, Dic3.Q8, D36⋊5C2, D4⋊2D9, Q8×D9, Dic9.Q8
►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)
(1 214 10 205)(2 213 11 204)(3 212 12 203)(4 211 13 202)(5 210 14 201)(6 209 15 200)(7 208 16 199)(8 207 17 216)(9 206 18 215)(19 145 28 154)(20 162 29 153)(21 161 30 152)(22 160 31 151)(23 159 32 150)(24 158 33 149)(25 157 34 148)(26 156 35 147)(27 155 36 146)(37 60 46 69)(38 59 47 68)(39 58 48 67)(40 57 49 66)(41 56 50 65)(42 55 51 64)(43 72 52 63)(44 71 53 62)(45 70 54 61)(73 170 82 179)(74 169 83 178)(75 168 84 177)(76 167 85 176)(77 166 86 175)(78 165 87 174)(79 164 88 173)(80 163 89 172)(81 180 90 171)(91 252 100 243)(92 251 101 242)(93 250 102 241)(94 249 103 240)(95 248 104 239)(96 247 105 238)(97 246 106 237)(98 245 107 236)(99 244 108 235)(109 277 118 286)(110 276 119 285)(111 275 120 284)(112 274 121 283)(113 273 122 282)(114 272 123 281)(115 271 124 280)(116 288 125 279)(117 287 126 278)(127 223 136 232)(128 222 137 231)(129 221 138 230)(130 220 139 229)(131 219 140 228)(132 218 141 227)(133 217 142 226)(134 234 143 225)(135 233 144 224)(181 259 190 268)(182 258 191 267)(183 257 192 266)(184 256 193 265)(185 255 194 264)(186 254 195 263)(187 253 196 262)(188 270 197 261)(189 269 198 260)
(1 114 67 29)(2 115 68 30)(3 116 69 31)(4 117 70 32)(5 118 71 33)(6 119 72 34)(7 120 55 35)(8 121 56 36)(9 122 57 19)(10 123 58 20)(11 124 59 21)(12 125 60 22)(13 126 61 23)(14 109 62 24)(15 110 63 25)(16 111 64 26)(17 112 65 27)(18 113 66 28)(37 160 212 279)(38 161 213 280)(39 162 214 281)(40 145 215 282)(41 146 216 283)(42 147 199 284)(43 148 200 285)(44 149 201 286)(45 150 202 287)(46 151 203 288)(47 152 204 271)(48 153 205 272)(49 154 206 273)(50 155 207 274)(51 156 208 275)(52 157 209 276)(53 158 210 277)(54 159 211 278)(73 182 222 107)(74 183 223 108)(75 184 224 91)(76 185 225 92)(77 186 226 93)(78 187 227 94)(79 188 228 95)(80 189 229 96)(81 190 230 97)(82 191 231 98)(83 192 232 99)(84 193 233 100)(85 194 234 101)(86 195 217 102)(87 196 218 103)(88 197 219 104)(89 198 220 105)(90 181 221 106)(127 235 178 257)(128 236 179 258)(129 237 180 259)(130 238 163 260)(131 239 164 261)(132 240 165 262)(133 241 166 263)(134 242 167 264)(135 243 168 265)(136 244 169 266)(137 245 170 267)(138 246 171 268)(139 247 172 269)(140 248 173 270)(141 249 174 253)(142 250 175 254)(143 251 176 255)(144 252 177 256)
(1 182 67 107)(2 181 68 106)(3 198 69 105)(4 197 70 104)(5 196 71 103)(6 195 72 102)(7 194 55 101)(8 193 56 100)(9 192 57 99)(10 191 58 98)(11 190 59 97)(12 189 60 96)(13 188 61 95)(14 187 62 94)(15 186 63 93)(16 185 64 92)(17 184 65 91)(18 183 66 108)(19 223 122 74)(20 222 123 73)(21 221 124 90)(22 220 125 89)(23 219 126 88)(24 218 109 87)(25 217 110 86)(26 234 111 85)(27 233 112 84)(28 232 113 83)(29 231 114 82)(30 230 115 81)(31 229 116 80)(32 228 117 79)(33 227 118 78)(34 226 119 77)(35 225 120 76)(36 224 121 75)(37 238 212 260)(38 237 213 259)(39 236 214 258)(40 235 215 257)(41 252 216 256)(42 251 199 255)(43 250 200 254)(44 249 201 253)(45 248 202 270)(46 247 203 269)(47 246 204 268)(48 245 205 267)(49 244 206 266)(50 243 207 265)(51 242 208 264)(52 241 209 263)(53 240 210 262)(54 239 211 261)(127 273 178 154)(128 272 179 153)(129 271 180 152)(130 288 163 151)(131 287 164 150)(132 286 165 149)(133 285 166 148)(134 284 167 147)(135 283 168 146)(136 282 169 145)(137 281 170 162)(138 280 171 161)(139 279 172 160)(140 278 173 159)(141 277 174 158)(142 276 175 157)(143 275 176 156)(144 274 177 155)
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), (1,214,10,205)(2,213,11,204)(3,212,12,203)(4,211,13,202)(5,210,14,201)(6,209,15,200)(7,208,16,199)(8,207,17,216)(9,206,18,215)(19,145,28,154)(20,162,29,153)(21,161,30,152)(22,160,31,151)(23,159,32,150)(24,158,33,149)(25,157,34,148)(26,156,35,147)(27,155,36,146)(37,60,46,69)(38,59,47,68)(39,58,48,67)(40,57,49,66)(41,56,50,65)(42,55,51,64)(43,72,52,63)(44,71,53,62)(45,70,54,61)(73,170,82,179)(74,169,83,178)(75,168,84,177)(76,167,85,176)(77,166,86,175)(78,165,87,174)(79,164,88,173)(80,163,89,172)(81,180,90,171)(91,252,100,243)(92,251,101,242)(93,250,102,241)(94,249,103,240)(95,248,104,239)(96,247,105,238)(97,246,106,237)(98,245,107,236)(99,244,108,235)(109,277,118,286)(110,276,119,285)(111,275,120,284)(112,274,121,283)(113,273,122,282)(114,272,123,281)(115,271,124,280)(116,288,125,279)(117,287,126,278)(127,223,136,232)(128,222,137,231)(129,221,138,230)(130,220,139,229)(131,219,140,228)(132,218,141,227)(133,217,142,226)(134,234,143,225)(135,233,144,224)(181,259,190,268)(182,258,191,267)(183,257,192,266)(184,256,193,265)(185,255,194,264)(186,254,195,263)(187,253,196,262)(188,270,197,261)(189,269,198,260), (1,114,67,29)(2,115,68,30)(3,116,69,31)(4,117,70,32)(5,118,71,33)(6,119,72,34)(7,120,55,35)(8,121,56,36)(9,122,57,19)(10,123,58,20)(11,124,59,21)(12,125,60,22)(13,126,61,23)(14,109,62,24)(15,110,63,25)(16,111,64,26)(17,112,65,27)(18,113,66,28)(37,160,212,279)(38,161,213,280)(39,162,214,281)(40,145,215,282)(41,146,216,283)(42,147,199,284)(43,148,200,285)(44,149,201,286)(45,150,202,287)(46,151,203,288)(47,152,204,271)(48,153,205,272)(49,154,206,273)(50,155,207,274)(51,156,208,275)(52,157,209,276)(53,158,210,277)(54,159,211,278)(73,182,222,107)(74,183,223,108)(75,184,224,91)(76,185,225,92)(77,186,226,93)(78,187,227,94)(79,188,228,95)(80,189,229,96)(81,190,230,97)(82,191,231,98)(83,192,232,99)(84,193,233,100)(85,194,234,101)(86,195,217,102)(87,196,218,103)(88,197,219,104)(89,198,220,105)(90,181,221,106)(127,235,178,257)(128,236,179,258)(129,237,180,259)(130,238,163,260)(131,239,164,261)(132,240,165,262)(133,241,166,263)(134,242,167,264)(135,243,168,265)(136,244,169,266)(137,245,170,267)(138,246,171,268)(139,247,172,269)(140,248,173,270)(141,249,174,253)(142,250,175,254)(143,251,176,255)(144,252,177,256), (1,182,67,107)(2,181,68,106)(3,198,69,105)(4,197,70,104)(5,196,71,103)(6,195,72,102)(7,194,55,101)(8,193,56,100)(9,192,57,99)(10,191,58,98)(11,190,59,97)(12,189,60,96)(13,188,61,95)(14,187,62,94)(15,186,63,93)(16,185,64,92)(17,184,65,91)(18,183,66,108)(19,223,122,74)(20,222,123,73)(21,221,124,90)(22,220,125,89)(23,219,126,88)(24,218,109,87)(25,217,110,86)(26,234,111,85)(27,233,112,84)(28,232,113,83)(29,231,114,82)(30,230,115,81)(31,229,116,80)(32,228,117,79)(33,227,118,78)(34,226,119,77)(35,225,120,76)(36,224,121,75)(37,238,212,260)(38,237,213,259)(39,236,214,258)(40,235,215,257)(41,252,216,256)(42,251,199,255)(43,250,200,254)(44,249,201,253)(45,248,202,270)(46,247,203,269)(47,246,204,268)(48,245,205,267)(49,244,206,266)(50,243,207,265)(51,242,208,264)(52,241,209,263)(53,240,210,262)(54,239,211,261)(127,273,178,154)(128,272,179,153)(129,271,180,152)(130,288,163,151)(131,287,164,150)(132,286,165,149)(133,285,166,148)(134,284,167,147)(135,283,168,146)(136,282,169,145)(137,281,170,162)(138,280,171,161)(139,279,172,160)(140,278,173,159)(141,277,174,158)(142,276,175,157)(143,275,176,156)(144,274,177,155)>;
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), (1,214,10,205)(2,213,11,204)(3,212,12,203)(4,211,13,202)(5,210,14,201)(6,209,15,200)(7,208,16,199)(8,207,17,216)(9,206,18,215)(19,145,28,154)(20,162,29,153)(21,161,30,152)(22,160,31,151)(23,159,32,150)(24,158,33,149)(25,157,34,148)(26,156,35,147)(27,155,36,146)(37,60,46,69)(38,59,47,68)(39,58,48,67)(40,57,49,66)(41,56,50,65)(42,55,51,64)(43,72,52,63)(44,71,53,62)(45,70,54,61)(73,170,82,179)(74,169,83,178)(75,168,84,177)(76,167,85,176)(77,166,86,175)(78,165,87,174)(79,164,88,173)(80,163,89,172)(81,180,90,171)(91,252,100,243)(92,251,101,242)(93,250,102,241)(94,249,103,240)(95,248,104,239)(96,247,105,238)(97,246,106,237)(98,245,107,236)(99,244,108,235)(109,277,118,286)(110,276,119,285)(111,275,120,284)(112,274,121,283)(113,273,122,282)(114,272,123,281)(115,271,124,280)(116,288,125,279)(117,287,126,278)(127,223,136,232)(128,222,137,231)(129,221,138,230)(130,220,139,229)(131,219,140,228)(132,218,141,227)(133,217,142,226)(134,234,143,225)(135,233,144,224)(181,259,190,268)(182,258,191,267)(183,257,192,266)(184,256,193,265)(185,255,194,264)(186,254,195,263)(187,253,196,262)(188,270,197,261)(189,269,198,260), (1,114,67,29)(2,115,68,30)(3,116,69,31)(4,117,70,32)(5,118,71,33)(6,119,72,34)(7,120,55,35)(8,121,56,36)(9,122,57,19)(10,123,58,20)(11,124,59,21)(12,125,60,22)(13,126,61,23)(14,109,62,24)(15,110,63,25)(16,111,64,26)(17,112,65,27)(18,113,66,28)(37,160,212,279)(38,161,213,280)(39,162,214,281)(40,145,215,282)(41,146,216,283)(42,147,199,284)(43,148,200,285)(44,149,201,286)(45,150,202,287)(46,151,203,288)(47,152,204,271)(48,153,205,272)(49,154,206,273)(50,155,207,274)(51,156,208,275)(52,157,209,276)(53,158,210,277)(54,159,211,278)(73,182,222,107)(74,183,223,108)(75,184,224,91)(76,185,225,92)(77,186,226,93)(78,187,227,94)(79,188,228,95)(80,189,229,96)(81,190,230,97)(82,191,231,98)(83,192,232,99)(84,193,233,100)(85,194,234,101)(86,195,217,102)(87,196,218,103)(88,197,219,104)(89,198,220,105)(90,181,221,106)(127,235,178,257)(128,236,179,258)(129,237,180,259)(130,238,163,260)(131,239,164,261)(132,240,165,262)(133,241,166,263)(134,242,167,264)(135,243,168,265)(136,244,169,266)(137,245,170,267)(138,246,171,268)(139,247,172,269)(140,248,173,270)(141,249,174,253)(142,250,175,254)(143,251,176,255)(144,252,177,256), (1,182,67,107)(2,181,68,106)(3,198,69,105)(4,197,70,104)(5,196,71,103)(6,195,72,102)(7,194,55,101)(8,193,56,100)(9,192,57,99)(10,191,58,98)(11,190,59,97)(12,189,60,96)(13,188,61,95)(14,187,62,94)(15,186,63,93)(16,185,64,92)(17,184,65,91)(18,183,66,108)(19,223,122,74)(20,222,123,73)(21,221,124,90)(22,220,125,89)(23,219,126,88)(24,218,109,87)(25,217,110,86)(26,234,111,85)(27,233,112,84)(28,232,113,83)(29,231,114,82)(30,230,115,81)(31,229,116,80)(32,228,117,79)(33,227,118,78)(34,226,119,77)(35,225,120,76)(36,224,121,75)(37,238,212,260)(38,237,213,259)(39,236,214,258)(40,235,215,257)(41,252,216,256)(42,251,199,255)(43,250,200,254)(44,249,201,253)(45,248,202,270)(46,247,203,269)(47,246,204,268)(48,245,205,267)(49,244,206,266)(50,243,207,265)(51,242,208,264)(52,241,209,263)(53,240,210,262)(54,239,211,261)(127,273,178,154)(128,272,179,153)(129,271,180,152)(130,288,163,151)(131,287,164,150)(132,286,165,149)(133,285,166,148)(134,284,167,147)(135,283,168,146)(136,282,169,145)(137,281,170,162)(138,280,171,161)(139,279,172,160)(140,278,173,159)(141,277,174,158)(142,276,175,157)(143,275,176,156)(144,274,177,155) );
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)], [(1,214,10,205),(2,213,11,204),(3,212,12,203),(4,211,13,202),(5,210,14,201),(6,209,15,200),(7,208,16,199),(8,207,17,216),(9,206,18,215),(19,145,28,154),(20,162,29,153),(21,161,30,152),(22,160,31,151),(23,159,32,150),(24,158,33,149),(25,157,34,148),(26,156,35,147),(27,155,36,146),(37,60,46,69),(38,59,47,68),(39,58,48,67),(40,57,49,66),(41,56,50,65),(42,55,51,64),(43,72,52,63),(44,71,53,62),(45,70,54,61),(73,170,82,179),(74,169,83,178),(75,168,84,177),(76,167,85,176),(77,166,86,175),(78,165,87,174),(79,164,88,173),(80,163,89,172),(81,180,90,171),(91,252,100,243),(92,251,101,242),(93,250,102,241),(94,249,103,240),(95,248,104,239),(96,247,105,238),(97,246,106,237),(98,245,107,236),(99,244,108,235),(109,277,118,286),(110,276,119,285),(111,275,120,284),(112,274,121,283),(113,273,122,282),(114,272,123,281),(115,271,124,280),(116,288,125,279),(117,287,126,278),(127,223,136,232),(128,222,137,231),(129,221,138,230),(130,220,139,229),(131,219,140,228),(132,218,141,227),(133,217,142,226),(134,234,143,225),(135,233,144,224),(181,259,190,268),(182,258,191,267),(183,257,192,266),(184,256,193,265),(185,255,194,264),(186,254,195,263),(187,253,196,262),(188,270,197,261),(189,269,198,260)], [(1,114,67,29),(2,115,68,30),(3,116,69,31),(4,117,70,32),(5,118,71,33),(6,119,72,34),(7,120,55,35),(8,121,56,36),(9,122,57,19),(10,123,58,20),(11,124,59,21),(12,125,60,22),(13,126,61,23),(14,109,62,24),(15,110,63,25),(16,111,64,26),(17,112,65,27),(18,113,66,28),(37,160,212,279),(38,161,213,280),(39,162,214,281),(40,145,215,282),(41,146,216,283),(42,147,199,284),(43,148,200,285),(44,149,201,286),(45,150,202,287),(46,151,203,288),(47,152,204,271),(48,153,205,272),(49,154,206,273),(50,155,207,274),(51,156,208,275),(52,157,209,276),(53,158,210,277),(54,159,211,278),(73,182,222,107),(74,183,223,108),(75,184,224,91),(76,185,225,92),(77,186,226,93),(78,187,227,94),(79,188,228,95),(80,189,229,96),(81,190,230,97),(82,191,231,98),(83,192,232,99),(84,193,233,100),(85,194,234,101),(86,195,217,102),(87,196,218,103),(88,197,219,104),(89,198,220,105),(90,181,221,106),(127,235,178,257),(128,236,179,258),(129,237,180,259),(130,238,163,260),(131,239,164,261),(132,240,165,262),(133,241,166,263),(134,242,167,264),(135,243,168,265),(136,244,169,266),(137,245,170,267),(138,246,171,268),(139,247,172,269),(140,248,173,270),(141,249,174,253),(142,250,175,254),(143,251,176,255),(144,252,177,256)], [(1,182,67,107),(2,181,68,106),(3,198,69,105),(4,197,70,104),(5,196,71,103),(6,195,72,102),(7,194,55,101),(8,193,56,100),(9,192,57,99),(10,191,58,98),(11,190,59,97),(12,189,60,96),(13,188,61,95),(14,187,62,94),(15,186,63,93),(16,185,64,92),(17,184,65,91),(18,183,66,108),(19,223,122,74),(20,222,123,73),(21,221,124,90),(22,220,125,89),(23,219,126,88),(24,218,109,87),(25,217,110,86),(26,234,111,85),(27,233,112,84),(28,232,113,83),(29,231,114,82),(30,230,115,81),(31,229,116,80),(32,228,117,79),(33,227,118,78),(34,226,119,77),(35,225,120,76),(36,224,121,75),(37,238,212,260),(38,237,213,259),(39,236,214,258),(40,235,215,257),(41,252,216,256),(42,251,199,255),(43,250,200,254),(44,249,201,253),(45,248,202,270),(46,247,203,269),(47,246,204,268),(48,245,205,267),(49,244,206,266),(50,243,207,265),(51,242,208,264),(52,241,209,263),(53,240,210,262),(54,239,211,261),(127,273,178,154),(128,272,179,153),(129,271,180,152),(130,288,163,151),(131,287,164,150),(132,286,165,149),(133,285,166,148),(134,284,167,147),(135,283,168,146),(136,282,169,145),(137,281,170,162),(138,280,171,161),(139,279,172,160),(140,278,173,159),(141,277,174,158),(142,276,175,157),(143,275,176,156),(144,274,177,155)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | C4○D4 | D9 | D18 | C4○D12 | D36⋊5C2 | D4⋊2S3 | S3×Q8 | D4⋊2D9 | Q8×D9 |
| kernel | Dic9.Q8 | C4×Dic9 | Dic9⋊C4 | C4⋊Dic9 | C9×C4⋊C4 | C3×C4⋊C4 | Dic9 | C2×C12 | C18 | C4⋊C4 | C2×C4 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 9 | 4 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of Dic9.Q8 ►in GL4(𝔽37) generated by
| 20 | 26 | 0 | 0 |
| 11 | 31 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 9 | 0 | 0 |
| 1 | 29 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 7 | 14 | 0 | 0 |
| 23 | 30 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 26 | 20 | 0 | 0 |
| 31 | 11 | 0 | 0 |
| 0 | 0 | 32 | 23 |
| 0 | 0 | 23 | 5 |
G:=sub<GL(4,GF(37))| [20,11,0,0,26,31,0,0,0,0,1,0,0,0,0,1],[8,1,0,0,9,29,0,0,0,0,36,0,0,0,0,36],[7,23,0,0,14,30,0,0,0,0,0,36,0,0,1,0],[26,31,0,0,20,11,0,0,0,0,32,23,0,0,23,5] >;
Dic9.Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_9.Q_8 % in TeX
G:=Group("Dic9.Q8"); // GroupNames label
G:=SmallGroup(288,99);
// by ID
G=gap.SmallGroup(288,99);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,254,219,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=c^4=1,b^2=a^9,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=a^9*c^-1>;
// generators/relations