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G = C9×C2.D8order 288 = 25·32

Direct product of C9 and C2.D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×C2.D8, C725C4, C81C36, C24.2C12, C18.14D8, C36.10Q8, C18.7Q16, C2.2(C9×D8), C4⋊C4.3C18, C4.2(Q8×C9), C4.7(C2×C36), (C2×C8).3C18, C6.14(C3×D8), C6.7(C3×Q16), C2.2(C9×Q16), C36.44(C2×C4), (C2×C72).13C2, (C2×C24).13C6, (C2×C18).49D4, C18.13(C4⋊C4), C12.10(C3×Q8), C12.50(C2×C12), C22.11(D4×C9), (C2×C36).117C22, C3.(C3×C2.D8), C2.4(C9×C4⋊C4), (C3×C2.D8).C3, C6.13(C3×C4⋊C4), (C9×C4⋊C4).10C2, (C3×C4⋊C4).11C6, (C2×C6).58(C3×D4), (C2×C4).16(C2×C18), (C2×C12).137(C2×C6), SmallGroup(288,57)

Series: Derived Chief Lower central Upper central

C1C4 — C9×C2.D8
C1C2C6C2×C6C2×C12C2×C36C9×C4⋊C4 — C9×C2.D8
C1C2C4 — C9×C2.D8
C1C2×C18C2×C36 — C9×C2.D8

Generators and relations for C9×C2.D8
 G = < a,b,c,d | a9=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

4C4
4C4
2C2×C4
2C2×C4
4C12
4C12
2C2×C12
2C2×C12
4C36
4C36
2C2×C36
2C2×C36

Smallest permutation representation of C9×C2.D8
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 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)(19 266)(20 267)(21 268)(22 269)(23 270)(24 262)(25 263)(26 264)(27 265)(28 286)(29 287)(30 288)(31 280)(32 281)(33 282)(34 283)(35 284)(36 285)(37 81)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 260)(47 261)(48 253)(49 254)(50 255)(51 256)(52 257)(53 258)(54 259)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 104)(65 105)(66 106)(67 107)(68 108)(69 100)(70 101)(71 102)(72 103)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 127)(99 128)(109 154)(110 155)(111 156)(112 157)(113 158)(114 159)(115 160)(116 161)(117 162)(118 149)(119 150)(120 151)(121 152)(122 153)(123 145)(124 146)(125 147)(126 148)(136 176)(137 177)(138 178)(139 179)(140 180)(141 172)(142 173)(143 174)(144 175)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 199)(171 200)(181 226)(182 227)(183 228)(184 229)(185 230)(186 231)(187 232)(188 233)(189 234)(190 221)(191 222)(192 223)(193 224)(194 225)(195 217)(196 218)(197 219)(198 220)(208 248)(209 249)(210 250)(211 251)(212 252)(213 244)(214 245)(215 246)(216 247)(235 273)(236 274)(237 275)(238 276)(239 277)(240 278)(241 279)(242 271)(243 272)
(1 179 107 149 77 113 59 91)(2 180 108 150 78 114 60 92)(3 172 100 151 79 115 61 93)(4 173 101 152 80 116 62 94)(5 174 102 153 81 117 63 95)(6 175 103 145 73 109 55 96)(7 176 104 146 74 110 56 97)(8 177 105 147 75 111 57 98)(9 178 106 148 76 112 58 99)(10 142 70 121 45 161 89 132)(11 143 71 122 37 162 90 133)(12 144 72 123 38 154 82 134)(13 136 64 124 39 155 83 135)(14 137 65 125 40 156 84 127)(15 138 66 126 41 157 85 128)(16 139 67 118 42 158 86 129)(17 140 68 119 43 159 87 130)(18 141 69 120 44 160 88 131)(19 246 29 167 239 189 261 225)(20 247 30 168 240 181 253 217)(21 248 31 169 241 182 254 218)(22 249 32 170 242 183 255 219)(23 250 33 171 243 184 256 220)(24 251 34 163 235 185 257 221)(25 252 35 164 236 186 258 222)(26 244 36 165 237 187 259 223)(27 245 28 166 238 188 260 224)(46 193 265 214 286 204 276 233)(47 194 266 215 287 205 277 234)(48 195 267 216 288 206 278 226)(49 196 268 208 280 207 279 227)(50 197 269 209 281 199 271 228)(51 198 270 210 282 200 272 229)(52 190 262 211 283 201 273 230)(53 191 263 212 284 202 274 231)(54 192 264 213 285 203 275 232)
(1 283 16 34)(2 284 17 35)(3 285 18 36)(4 286 10 28)(5 287 11 29)(6 288 12 30)(7 280 13 31)(8 281 14 32)(9 282 15 33)(19 102 266 71)(20 103 267 72)(21 104 268 64)(22 105 269 65)(23 106 270 66)(24 107 262 67)(25 108 263 68)(26 100 264 69)(27 101 265 70)(37 261 81 47)(38 253 73 48)(39 254 74 49)(40 255 75 50)(41 256 76 51)(42 257 77 52)(43 258 78 53)(44 259 79 54)(45 260 80 46)(55 278 82 240)(56 279 83 241)(57 271 84 242)(58 272 85 243)(59 273 86 235)(60 274 87 236)(61 275 88 237)(62 276 89 238)(63 277 90 239)(91 201 129 163)(92 202 130 164)(93 203 131 165)(94 204 132 166)(95 205 133 167)(96 206 134 168)(97 207 135 169)(98 199 127 170)(99 200 128 171)(109 226 154 181)(110 227 155 182)(111 228 156 183)(112 229 157 184)(113 230 158 185)(114 231 159 186)(115 232 160 187)(116 233 161 188)(117 234 162 189)(118 221 149 190)(119 222 150 191)(120 223 151 192)(121 224 152 193)(122 225 153 194)(123 217 145 195)(124 218 146 196)(125 219 147 197)(126 220 148 198)(136 248 176 208)(137 249 177 209)(138 250 178 210)(139 251 179 211)(140 252 180 212)(141 244 172 213)(142 245 173 214)(143 246 174 215)(144 247 175 216)

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

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

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

126 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F6A···6F8A8B8C8D9A···9F12A12B12C12D12E···12L18A···18R24A···24H36A···36L36M···36AJ72A···72X
order1222334444446···688889···91212121212···1218···1824···2436···3636···3672···72
size1111112244441···122221···122224···41···12···22···24···42···2

126 irreducible representations

dim111111111111222222222222
type+++-++-
imageC1C2C2C3C4C6C6C9C12C18C18C36Q8D4D8Q16C3×Q8C3×D4C3×D8C3×Q16Q8×C9D4×C9C9×D8C9×Q16
kernelC9×C2.D8C9×C4⋊C4C2×C72C3×C2.D8C72C3×C4⋊C4C2×C24C2.D8C24C4⋊C4C2×C8C8C36C2×C18C18C18C12C2×C6C6C6C4C22C2C2
# reps1212442681262411222244661212

Matrix representation of C9×C2.D8 in GL4(𝔽73) generated by

1000
0100
0040
0004
,
72000
07200
0010
0001
,
722900
10100
005716
005757
,
173100
665600
003059
005943
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,10,0,0,29,1,0,0,0,0,57,57,0,0,16,57],[17,66,0,0,31,56,0,0,0,0,30,59,0,0,59,43] >;

C9×C2.D8 in GAP, Magma, Sage, TeX

C_9\times C_2.D_8
% in TeX

G:=Group("C9xC2.D8");
// GroupNames label

G:=SmallGroup(288,57);
// by ID

G=gap.SmallGroup(288,57);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,1100,268,4371,360]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C9×C2.D8 in TeX

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