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G = Dic9.Q8order 288 = 25·32

The non-split extension by Dic9 of Q8 acting via Q8/C4=C2

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), C92(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(D42D9), C6.80(D42S3), C2.13(D365C2), 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

C1C2×C18 — Dic9.Q8
C1C3C9C18C2×C18C2×Dic9C4×Dic9 — Dic9.Q8
C9C2×C18 — Dic9.Q8
C1C22C4⋊C4

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 [×3], C3, C4 [×8], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], C9, Dic3 [×5], C12 [×3], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C18 [×3], C2×Dic3 [×4], C2×C12 [×3], C42.C2, Dic9 [×2], Dic9 [×3], C36 [×3], C2×C18, C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3, C3×C4⋊C4, C2×Dic9 [×4], C2×C36 [×3], Dic3.Q8, C4×Dic9, Dic9⋊C4 [×4], C4⋊Dic9, C9×C4⋊C4, Dic9.Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, C4○D4 [×2], D9, C22×S3, C42.C2, D18 [×3], C4○D12, D42S3, S3×Q8, C22×D9, Dic3.Q8, D365C2, D42D9, Q8×D9, Dic9.Q8

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

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

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

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

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12223444444444466699912···1218···1836···36
size1111222441818181836362222224···42···24···4

54 irreducible representations

dim11111222222224444
type++++++-+++----
imageC1C2C2C2C2S3Q8D6C4○D4D9D18C4○D12D365C2D42S3S3×Q8D42D9Q8×D9
kernelDic9.Q8C4×Dic9Dic9⋊C4C4⋊Dic9C9×C4⋊C4C3×C4⋊C4Dic9C2×C12C18C4⋊C4C2×C4C6C2C6C6C2C2
# reps114111234394121133

Matrix representation of Dic9.Q8 in GL4(𝔽37) generated by

202600
113100
0010
0001
,
8900
12900
00360
00036
,
71400
233000
0001
00360
,
262000
311100
003223
00235
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

׿
×
𝔽