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G = C9×Q8⋊C4order 288 = 25·32

Direct product of C9 and Q8⋊C4

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

Aliases: C9×Q8⋊C4, Q82C36, 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

C1C4 — C9×Q8⋊C4
C1C2C6C2×C6C2×C12C2×C36C9×C4⋊C4 — C9×Q8⋊C4
C1C2C4 — C9×Q8⋊C4
C1C2×C18C2×C36 — C9×Q8⋊C4

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 [×3], C3, C4 [×2], C4 [×3], C22, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C9, C12 [×2], C12 [×3], C2×C6, C4⋊C4, C2×C8, C2×Q8, C18 [×3], C24, C2×C12, C2×C12 [×2], C3×Q8 [×2], C3×Q8, Q8⋊C4, C36 [×2], C36 [×3], C2×C18, C3×C4⋊C4, C2×C24, C6×Q8, C72, C2×C36, C2×C36 [×2], Q8×C9 [×2], Q8×C9, C3×Q8⋊C4, C9×C4⋊C4, C2×C72, Q8×C18, C9×Q8⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, SD16, Q16, C18 [×3], C2×C12, C3×D4 [×2], Q8⋊C4, C36 [×2], C2×C18, C3×C22⋊C4, C3×SD16, C3×Q16, C2×C36, D4×C9 [×2], C3×Q8⋊C4, C9×C22⋊C4, C9×SD16, C9×Q16, C9×Q8⋊C4

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

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

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

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

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

dim111111111111111222222222222
type++++++-
imageC1C2C2C2C3C4C6C6C6C9C12C18C18C18C36D4D4SD16Q16C3×D4C3×D4C3×SD16C3×Q16D4×C9D4×C9C9×SD16C9×Q16
kernelC9×Q8⋊C4C9×C4⋊C4C2×C72Q8×C18C3×Q8⋊C4Q8×C9C3×C4⋊C4C2×C24C6×Q8Q8⋊C4C3×Q8C4⋊C4C2×C8C2×Q8Q8C36C2×C18C18C18C12C2×C6C6C6C4C22C2C2
# reps111124222686662411222244661212

Matrix representation of C9×Q8⋊C4 in GL3(𝔽73) generated by

100
0550
0055
,
100
001
0720
,
7200
01732
03256
,
4600
05421
02119
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

׿
×
𝔽