direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×Q8⋊C4, Q8⋊2C36, C36.61D4, C18.6Q16, C18.10SD16, (Q8×C9)⋊4C4, C4⋊C4.1C18, C4.2(C2×C36), (C2×C72).3C2, (C2×C8).1C18, (C2×C24).3C6, C4.12(D4×C9), C2.1(C9×Q16), C6.6(C3×Q16), C36.31(C2×C4), (C2×C18).47D4, C12.70(C3×D4), (Q8×C18).7C2, (C2×Q8).4C18, (C3×Q8).6C12, (C6×Q8).15C6, C2.2(C9×SD16), C22.9(D4×C9), C12.31(C2×C12), C6.10(C3×SD16), C18.25(C22⋊C4), (C2×C36).114C22, (C9×C4⋊C4).8C2, (C3×C4⋊C4).9C6, C3.(C3×Q8⋊C4), (C2×C6).56(C3×D4), C2.7(C9×C22⋊C4), (C3×Q8⋊C4).C3, (C2×C4).13(C2×C18), C6.25(C3×C22⋊C4), (C2×C12).134(C2×C6), SmallGroup(288,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×Q8⋊C4
G = < a,b,c,d | a9=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 90 in 63 conjugacy classes, 42 normal (36 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C9, C12, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C18, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C36, C36, C2×C18, C3×C4⋊C4, C2×C24, C6×Q8, C72, C2×C36, C2×C36, Q8×C9, Q8×C9, C3×Q8⋊C4, C9×C4⋊C4, C2×C72, Q8×C18, C9×Q8⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, SD16, Q16, C18, C2×C12, C3×D4, Q8⋊C4, C36, C2×C18, C3×C22⋊C4, C3×SD16, C3×Q16, C2×C36, D4×C9, C3×Q8⋊C4, C9×C22⋊C4, C9×SD16, C9×Q16, C9×Q8⋊C4
►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 59 77 107)(2 60 78 108)(3 61 79 100)(4 62 80 101)(5 63 81 102)(6 55 73 103)(7 56 74 104)(8 57 75 105)(9 58 76 106)(10 82 38 72)(11 83 39 64)(12 84 40 65)(13 85 41 66)(14 86 42 67)(15 87 43 68)(16 88 44 69)(17 89 45 70)(18 90 37 71)(19 36 237 259)(20 28 238 260)(21 29 239 261)(22 30 240 253)(23 31 241 254)(24 32 242 255)(25 33 243 256)(26 34 235 257)(27 35 236 258)(46 265 286 276)(47 266 287 277)(48 267 288 278)(49 268 280 279)(50 269 281 271)(51 270 282 272)(52 262 283 273)(53 263 284 274)(54 264 285 275)(91 179 149 113)(92 180 150 114)(93 172 151 115)(94 173 152 116)(95 174 153 117)(96 175 145 109)(97 176 146 110)(98 177 147 111)(99 178 148 112)(118 158 129 139)(119 159 130 140)(120 160 131 141)(121 161 132 142)(122 162 133 143)(123 154 134 144)(124 155 135 136)(125 156 127 137)(126 157 128 138)(163 251 221 185)(164 252 222 186)(165 244 223 187)(166 245 224 188)(167 246 225 189)(168 247 217 181)(169 248 218 182)(170 249 219 183)(171 250 220 184)(190 230 201 211)(191 231 202 212)(192 232 203 213)(193 233 204 214)(194 234 205 215)(195 226 206 216)(196 227 207 208)(197 228 199 209)(198 229 200 210)
(1 118 77 129)(2 119 78 130)(3 120 79 131)(4 121 80 132)(5 122 81 133)(6 123 73 134)(7 124 74 135)(8 125 75 127)(9 126 76 128)(10 145 38 96)(11 146 39 97)(12 147 40 98)(13 148 41 99)(14 149 42 91)(15 150 43 92)(16 151 44 93)(17 152 45 94)(18 153 37 95)(19 213 237 232)(20 214 238 233)(21 215 239 234)(22 216 240 226)(23 208 241 227)(24 209 242 228)(25 210 243 229)(26 211 235 230)(27 212 236 231)(28 204 260 193)(29 205 261 194)(30 206 253 195)(31 207 254 196)(32 199 255 197)(33 200 256 198)(34 201 257 190)(35 202 258 191)(36 203 259 192)(46 224 286 166)(47 225 287 167)(48 217 288 168)(49 218 280 169)(50 219 281 170)(51 220 282 171)(52 221 283 163)(53 222 284 164)(54 223 285 165)(55 144 103 154)(56 136 104 155)(57 137 105 156)(58 138 106 157)(59 139 107 158)(60 140 108 159)(61 141 100 160)(62 142 101 161)(63 143 102 162)(64 110 83 176)(65 111 84 177)(66 112 85 178)(67 113 86 179)(68 114 87 180)(69 115 88 172)(70 116 89 173)(71 117 90 174)(72 109 82 175)(181 267 247 278)(182 268 248 279)(183 269 249 271)(184 270 250 272)(185 262 251 273)(186 263 252 274)(187 264 244 275)(188 265 245 276)(189 266 246 277)
(1 201 14 163)(2 202 15 164)(3 203 16 165)(4 204 17 166)(5 205 18 167)(6 206 10 168)(7 207 11 169)(8 199 12 170)(9 200 13 171)(19 151 264 120)(20 152 265 121)(21 153 266 122)(22 145 267 123)(23 146 268 124)(24 147 269 125)(25 148 270 126)(26 149 262 118)(27 150 263 119)(28 173 286 142)(29 174 287 143)(30 175 288 144)(31 176 280 136)(32 177 281 137)(33 178 282 138)(34 179 283 139)(35 180 284 140)(36 172 285 141)(37 225 81 194)(38 217 73 195)(39 218 74 196)(40 219 75 197)(41 220 76 198)(42 221 77 190)(43 222 78 191)(44 223 79 192)(45 224 80 193)(46 161 260 116)(47 162 261 117)(48 154 253 109)(49 155 254 110)(50 156 255 111)(51 157 256 112)(52 158 257 113)(53 159 258 114)(54 160 259 115)(55 226 82 181)(56 227 83 182)(57 228 84 183)(58 229 85 184)(59 230 86 185)(60 231 87 186)(61 232 88 187)(62 233 89 188)(63 234 90 189)(64 248 104 208)(65 249 105 209)(66 250 106 210)(67 251 107 211)(68 252 108 212)(69 244 100 213)(70 245 101 214)(71 246 102 215)(72 247 103 216)(91 273 129 235)(92 274 130 236)(93 275 131 237)(94 276 132 238)(95 277 133 239)(96 278 134 240)(97 279 135 241)(98 271 127 242)(99 272 128 243)
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), (1,59,77,107)(2,60,78,108)(3,61,79,100)(4,62,80,101)(5,63,81,102)(6,55,73,103)(7,56,74,104)(8,57,75,105)(9,58,76,106)(10,82,38,72)(11,83,39,64)(12,84,40,65)(13,85,41,66)(14,86,42,67)(15,87,43,68)(16,88,44,69)(17,89,45,70)(18,90,37,71)(19,36,237,259)(20,28,238,260)(21,29,239,261)(22,30,240,253)(23,31,241,254)(24,32,242,255)(25,33,243,256)(26,34,235,257)(27,35,236,258)(46,265,286,276)(47,266,287,277)(48,267,288,278)(49,268,280,279)(50,269,281,271)(51,270,282,272)(52,262,283,273)(53,263,284,274)(54,264,285,275)(91,179,149,113)(92,180,150,114)(93,172,151,115)(94,173,152,116)(95,174,153,117)(96,175,145,109)(97,176,146,110)(98,177,147,111)(99,178,148,112)(118,158,129,139)(119,159,130,140)(120,160,131,141)(121,161,132,142)(122,162,133,143)(123,154,134,144)(124,155,135,136)(125,156,127,137)(126,157,128,138)(163,251,221,185)(164,252,222,186)(165,244,223,187)(166,245,224,188)(167,246,225,189)(168,247,217,181)(169,248,218,182)(170,249,219,183)(171,250,220,184)(190,230,201,211)(191,231,202,212)(192,232,203,213)(193,233,204,214)(194,234,205,215)(195,226,206,216)(196,227,207,208)(197,228,199,209)(198,229,200,210), (1,118,77,129)(2,119,78,130)(3,120,79,131)(4,121,80,132)(5,122,81,133)(6,123,73,134)(7,124,74,135)(8,125,75,127)(9,126,76,128)(10,145,38,96)(11,146,39,97)(12,147,40,98)(13,148,41,99)(14,149,42,91)(15,150,43,92)(16,151,44,93)(17,152,45,94)(18,153,37,95)(19,213,237,232)(20,214,238,233)(21,215,239,234)(22,216,240,226)(23,208,241,227)(24,209,242,228)(25,210,243,229)(26,211,235,230)(27,212,236,231)(28,204,260,193)(29,205,261,194)(30,206,253,195)(31,207,254,196)(32,199,255,197)(33,200,256,198)(34,201,257,190)(35,202,258,191)(36,203,259,192)(46,224,286,166)(47,225,287,167)(48,217,288,168)(49,218,280,169)(50,219,281,170)(51,220,282,171)(52,221,283,163)(53,222,284,164)(54,223,285,165)(55,144,103,154)(56,136,104,155)(57,137,105,156)(58,138,106,157)(59,139,107,158)(60,140,108,159)(61,141,100,160)(62,142,101,161)(63,143,102,162)(64,110,83,176)(65,111,84,177)(66,112,85,178)(67,113,86,179)(68,114,87,180)(69,115,88,172)(70,116,89,173)(71,117,90,174)(72,109,82,175)(181,267,247,278)(182,268,248,279)(183,269,249,271)(184,270,250,272)(185,262,251,273)(186,263,252,274)(187,264,244,275)(188,265,245,276)(189,266,246,277), (1,201,14,163)(2,202,15,164)(3,203,16,165)(4,204,17,166)(5,205,18,167)(6,206,10,168)(7,207,11,169)(8,199,12,170)(9,200,13,171)(19,151,264,120)(20,152,265,121)(21,153,266,122)(22,145,267,123)(23,146,268,124)(24,147,269,125)(25,148,270,126)(26,149,262,118)(27,150,263,119)(28,173,286,142)(29,174,287,143)(30,175,288,144)(31,176,280,136)(32,177,281,137)(33,178,282,138)(34,179,283,139)(35,180,284,140)(36,172,285,141)(37,225,81,194)(38,217,73,195)(39,218,74,196)(40,219,75,197)(41,220,76,198)(42,221,77,190)(43,222,78,191)(44,223,79,192)(45,224,80,193)(46,161,260,116)(47,162,261,117)(48,154,253,109)(49,155,254,110)(50,156,255,111)(51,157,256,112)(52,158,257,113)(53,159,258,114)(54,160,259,115)(55,226,82,181)(56,227,83,182)(57,228,84,183)(58,229,85,184)(59,230,86,185)(60,231,87,186)(61,232,88,187)(62,233,89,188)(63,234,90,189)(64,248,104,208)(65,249,105,209)(66,250,106,210)(67,251,107,211)(68,252,108,212)(69,244,100,213)(70,245,101,214)(71,246,102,215)(72,247,103,216)(91,273,129,235)(92,274,130,236)(93,275,131,237)(94,276,132,238)(95,277,133,239)(96,278,134,240)(97,279,135,241)(98,271,127,242)(99,272,128,243)>;
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), (1,59,77,107)(2,60,78,108)(3,61,79,100)(4,62,80,101)(5,63,81,102)(6,55,73,103)(7,56,74,104)(8,57,75,105)(9,58,76,106)(10,82,38,72)(11,83,39,64)(12,84,40,65)(13,85,41,66)(14,86,42,67)(15,87,43,68)(16,88,44,69)(17,89,45,70)(18,90,37,71)(19,36,237,259)(20,28,238,260)(21,29,239,261)(22,30,240,253)(23,31,241,254)(24,32,242,255)(25,33,243,256)(26,34,235,257)(27,35,236,258)(46,265,286,276)(47,266,287,277)(48,267,288,278)(49,268,280,279)(50,269,281,271)(51,270,282,272)(52,262,283,273)(53,263,284,274)(54,264,285,275)(91,179,149,113)(92,180,150,114)(93,172,151,115)(94,173,152,116)(95,174,153,117)(96,175,145,109)(97,176,146,110)(98,177,147,111)(99,178,148,112)(118,158,129,139)(119,159,130,140)(120,160,131,141)(121,161,132,142)(122,162,133,143)(123,154,134,144)(124,155,135,136)(125,156,127,137)(126,157,128,138)(163,251,221,185)(164,252,222,186)(165,244,223,187)(166,245,224,188)(167,246,225,189)(168,247,217,181)(169,248,218,182)(170,249,219,183)(171,250,220,184)(190,230,201,211)(191,231,202,212)(192,232,203,213)(193,233,204,214)(194,234,205,215)(195,226,206,216)(196,227,207,208)(197,228,199,209)(198,229,200,210), (1,118,77,129)(2,119,78,130)(3,120,79,131)(4,121,80,132)(5,122,81,133)(6,123,73,134)(7,124,74,135)(8,125,75,127)(9,126,76,128)(10,145,38,96)(11,146,39,97)(12,147,40,98)(13,148,41,99)(14,149,42,91)(15,150,43,92)(16,151,44,93)(17,152,45,94)(18,153,37,95)(19,213,237,232)(20,214,238,233)(21,215,239,234)(22,216,240,226)(23,208,241,227)(24,209,242,228)(25,210,243,229)(26,211,235,230)(27,212,236,231)(28,204,260,193)(29,205,261,194)(30,206,253,195)(31,207,254,196)(32,199,255,197)(33,200,256,198)(34,201,257,190)(35,202,258,191)(36,203,259,192)(46,224,286,166)(47,225,287,167)(48,217,288,168)(49,218,280,169)(50,219,281,170)(51,220,282,171)(52,221,283,163)(53,222,284,164)(54,223,285,165)(55,144,103,154)(56,136,104,155)(57,137,105,156)(58,138,106,157)(59,139,107,158)(60,140,108,159)(61,141,100,160)(62,142,101,161)(63,143,102,162)(64,110,83,176)(65,111,84,177)(66,112,85,178)(67,113,86,179)(68,114,87,180)(69,115,88,172)(70,116,89,173)(71,117,90,174)(72,109,82,175)(181,267,247,278)(182,268,248,279)(183,269,249,271)(184,270,250,272)(185,262,251,273)(186,263,252,274)(187,264,244,275)(188,265,245,276)(189,266,246,277), (1,201,14,163)(2,202,15,164)(3,203,16,165)(4,204,17,166)(5,205,18,167)(6,206,10,168)(7,207,11,169)(8,199,12,170)(9,200,13,171)(19,151,264,120)(20,152,265,121)(21,153,266,122)(22,145,267,123)(23,146,268,124)(24,147,269,125)(25,148,270,126)(26,149,262,118)(27,150,263,119)(28,173,286,142)(29,174,287,143)(30,175,288,144)(31,176,280,136)(32,177,281,137)(33,178,282,138)(34,179,283,139)(35,180,284,140)(36,172,285,141)(37,225,81,194)(38,217,73,195)(39,218,74,196)(40,219,75,197)(41,220,76,198)(42,221,77,190)(43,222,78,191)(44,223,79,192)(45,224,80,193)(46,161,260,116)(47,162,261,117)(48,154,253,109)(49,155,254,110)(50,156,255,111)(51,157,256,112)(52,158,257,113)(53,159,258,114)(54,160,259,115)(55,226,82,181)(56,227,83,182)(57,228,84,183)(58,229,85,184)(59,230,86,185)(60,231,87,186)(61,232,88,187)(62,233,89,188)(63,234,90,189)(64,248,104,208)(65,249,105,209)(66,250,106,210)(67,251,107,211)(68,252,108,212)(69,244,100,213)(70,245,101,214)(71,246,102,215)(72,247,103,216)(91,273,129,235)(92,274,130,236)(93,275,131,237)(94,276,132,238)(95,277,133,239)(96,278,134,240)(97,279,135,241)(98,271,127,242)(99,272,128,243) );
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)], [(1,59,77,107),(2,60,78,108),(3,61,79,100),(4,62,80,101),(5,63,81,102),(6,55,73,103),(7,56,74,104),(8,57,75,105),(9,58,76,106),(10,82,38,72),(11,83,39,64),(12,84,40,65),(13,85,41,66),(14,86,42,67),(15,87,43,68),(16,88,44,69),(17,89,45,70),(18,90,37,71),(19,36,237,259),(20,28,238,260),(21,29,239,261),(22,30,240,253),(23,31,241,254),(24,32,242,255),(25,33,243,256),(26,34,235,257),(27,35,236,258),(46,265,286,276),(47,266,287,277),(48,267,288,278),(49,268,280,279),(50,269,281,271),(51,270,282,272),(52,262,283,273),(53,263,284,274),(54,264,285,275),(91,179,149,113),(92,180,150,114),(93,172,151,115),(94,173,152,116),(95,174,153,117),(96,175,145,109),(97,176,146,110),(98,177,147,111),(99,178,148,112),(118,158,129,139),(119,159,130,140),(120,160,131,141),(121,161,132,142),(122,162,133,143),(123,154,134,144),(124,155,135,136),(125,156,127,137),(126,157,128,138),(163,251,221,185),(164,252,222,186),(165,244,223,187),(166,245,224,188),(167,246,225,189),(168,247,217,181),(169,248,218,182),(170,249,219,183),(171,250,220,184),(190,230,201,211),(191,231,202,212),(192,232,203,213),(193,233,204,214),(194,234,205,215),(195,226,206,216),(196,227,207,208),(197,228,199,209),(198,229,200,210)], [(1,118,77,129),(2,119,78,130),(3,120,79,131),(4,121,80,132),(5,122,81,133),(6,123,73,134),(7,124,74,135),(8,125,75,127),(9,126,76,128),(10,145,38,96),(11,146,39,97),(12,147,40,98),(13,148,41,99),(14,149,42,91),(15,150,43,92),(16,151,44,93),(17,152,45,94),(18,153,37,95),(19,213,237,232),(20,214,238,233),(21,215,239,234),(22,216,240,226),(23,208,241,227),(24,209,242,228),(25,210,243,229),(26,211,235,230),(27,212,236,231),(28,204,260,193),(29,205,261,194),(30,206,253,195),(31,207,254,196),(32,199,255,197),(33,200,256,198),(34,201,257,190),(35,202,258,191),(36,203,259,192),(46,224,286,166),(47,225,287,167),(48,217,288,168),(49,218,280,169),(50,219,281,170),(51,220,282,171),(52,221,283,163),(53,222,284,164),(54,223,285,165),(55,144,103,154),(56,136,104,155),(57,137,105,156),(58,138,106,157),(59,139,107,158),(60,140,108,159),(61,141,100,160),(62,142,101,161),(63,143,102,162),(64,110,83,176),(65,111,84,177),(66,112,85,178),(67,113,86,179),(68,114,87,180),(69,115,88,172),(70,116,89,173),(71,117,90,174),(72,109,82,175),(181,267,247,278),(182,268,248,279),(183,269,249,271),(184,270,250,272),(185,262,251,273),(186,263,252,274),(187,264,244,275),(188,265,245,276),(189,266,246,277)], [(1,201,14,163),(2,202,15,164),(3,203,16,165),(4,204,17,166),(5,205,18,167),(6,206,10,168),(7,207,11,169),(8,199,12,170),(9,200,13,171),(19,151,264,120),(20,152,265,121),(21,153,266,122),(22,145,267,123),(23,146,268,124),(24,147,269,125),(25,148,270,126),(26,149,262,118),(27,150,263,119),(28,173,286,142),(29,174,287,143),(30,175,288,144),(31,176,280,136),(32,177,281,137),(33,178,282,138),(34,179,283,139),(35,180,284,140),(36,172,285,141),(37,225,81,194),(38,217,73,195),(39,218,74,196),(40,219,75,197),(41,220,76,198),(42,221,77,190),(43,222,78,191),(44,223,79,192),(45,224,80,193),(46,161,260,116),(47,162,261,117),(48,154,253,109),(49,155,254,110),(50,156,255,111),(51,157,256,112),(52,158,257,113),(53,159,258,114),(54,160,259,115),(55,226,82,181),(56,227,83,182),(57,228,84,183),(58,229,85,184),(59,230,86,185),(60,231,87,186),(61,232,88,187),(62,233,89,188),(63,234,90,189),(64,248,104,208),(65,249,105,209),(66,250,106,210),(67,251,107,211),(68,252,108,212),(69,244,100,213),(70,245,101,214),(71,246,102,215),(72,247,103,216),(91,273,129,235),(92,274,130,236),(93,275,131,237),(94,276,132,238),(95,277,133,239),(96,278,134,240),(97,279,135,241),(98,271,127,242),(99,272,128,243)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 18A | ··· | 18R | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36AJ | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C36 | D4 | D4 | SD16 | Q16 | C3×D4 | C3×D4 | C3×SD16 | C3×Q16 | D4×C9 | D4×C9 | C9×SD16 | C9×Q16 |
| kernel | C9×Q8⋊C4 | C9×C4⋊C4 | C2×C72 | Q8×C18 | C3×Q8⋊C4 | Q8×C9 | C3×C4⋊C4 | C2×C24 | C6×Q8 | Q8⋊C4 | C3×Q8 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C36 | C2×C18 | C18 | C18 | C12 | C2×C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 |
Matrix representation of C9×Q8⋊C4 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 0 | 55 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 72 | 0 |
| 72 | 0 | 0 |
| 0 | 17 | 32 |
| 0 | 32 | 56 |
| 46 | 0 | 0 |
| 0 | 54 | 21 |
| 0 | 21 | 19 |
G:=sub<GL(3,GF(73))| [1,0,0,0,55,0,0,0,55],[1,0,0,0,0,72,0,1,0],[72,0,0,0,17,32,0,32,56],[46,0,0,0,54,21,0,21,19] >;
C9×Q8⋊C4 in GAP, Magma, Sage, TeX
C_9\times Q_8\rtimes C_4
% in TeX
G:=Group("C9xQ8:C4"); // GroupNames label
G:=SmallGroup(288,53);
// by ID
G=gap.SmallGroup(288,53);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,1016,268,4371,2194,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations