direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4⋊Q8, C36⋊4Q8, C36.41D4, C42.5C18, C4⋊(Q8×C9), C4.5(D4×C9), C4⋊C4.5C18, C6.73(C6×D4), C6.22(C6×Q8), C2.5(Q8×C18), (C4×C12).21C6, (C4×C36).11C2, C2.10(D4×C18), C12.41(C3×D4), C18.73(C2×D4), (C2×Q8).5C18, (Q8×C18).8C2, (C6×Q8).18C6, C18.22(C2×Q8), C12.12(C3×Q8), (C2×C18).83C23, (C2×C36).125C22, C22.18(C22×C18), C3.(C3×C4⋊Q8), (C3×C4⋊Q8).C3, (C3×C4⋊C4).17C6, (C9×C4⋊C4).12C2, (C2×C4).23(C2×C18), (C2×C12).141(C2×C6), (C2×C6).88(C22×C6), SmallGroup(288,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4⋊Q8
G = < a,b,c,d | a9=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 126 in 102 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C9, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C18, C18, C2×C12, C2×C12, C3×Q8, C4⋊Q8, C36, C36, C2×C18, C4×C12, C3×C4⋊C4, C6×Q8, C2×C36, C2×C36, Q8×C9, C3×C4⋊Q8, C4×C36, C9×C4⋊C4, Q8×C18, C9×C4⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C9, C2×C6, C2×D4, C2×Q8, C18, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C2×C18, C6×D4, C6×Q8, D4×C9, Q8×C9, C22×C18, C3×C4⋊Q8, D4×C18, Q8×C18, C9×C4⋊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)
(1 129 46 142)(2 130 47 143)(3 131 48 144)(4 132 49 136)(5 133 50 137)(6 134 51 138)(7 135 52 139)(8 127 53 140)(9 128 54 141)(10 171 23 175)(11 163 24 176)(12 164 25 177)(13 165 26 178)(14 166 27 179)(15 167 19 180)(16 168 20 172)(17 169 21 173)(18 170 22 174)(28 157 37 148)(29 158 38 149)(30 159 39 150)(31 160 40 151)(32 161 41 152)(33 162 42 153)(34 154 43 145)(35 155 44 146)(36 156 45 147)(55 104 68 91)(56 105 69 92)(57 106 70 93)(58 107 71 94)(59 108 72 95)(60 100 64 96)(61 101 65 97)(62 102 66 98)(63 103 67 99)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)(181 286 190 273)(182 287 191 274)(183 288 192 275)(184 280 193 276)(185 281 194 277)(186 282 195 278)(187 283 196 279)(188 284 197 271)(189 285 198 272)(199 235 212 248)(200 236 213 249)(201 237 214 250)(202 238 215 251)(203 239 216 252)(204 240 208 244)(205 241 209 245)(206 242 210 246)(207 243 211 247)(217 253 226 262)(218 254 227 263)(219 255 228 264)(220 256 229 265)(221 257 230 266)(222 258 231 267)(223 259 232 268)(224 260 233 269)(225 261 234 270)
(1 93 43 109)(2 94 44 110)(3 95 45 111)(4 96 37 112)(5 97 38 113)(6 98 39 114)(7 99 40 115)(8 91 41 116)(9 92 42 117)(10 207 279 223)(11 199 271 224)(12 200 272 225)(13 201 273 217)(14 202 274 218)(15 203 275 219)(16 204 276 220)(17 205 277 221)(18 206 278 222)(19 216 288 228)(20 208 280 229)(21 209 281 230)(22 210 282 231)(23 211 283 232)(24 212 284 233)(25 213 285 234)(26 214 286 226)(27 215 287 227)(28 121 49 100)(29 122 50 101)(30 123 51 102)(31 124 52 103)(32 125 53 104)(33 126 54 105)(34 118 46 106)(35 119 47 107)(36 120 48 108)(55 152 80 127)(56 153 81 128)(57 145 73 129)(58 146 74 130)(59 147 75 131)(60 148 76 132)(61 149 77 133)(62 150 78 134)(63 151 79 135)(64 157 85 136)(65 158 86 137)(66 159 87 138)(67 160 88 139)(68 161 89 140)(69 162 90 141)(70 154 82 142)(71 155 83 143)(72 156 84 144)(163 235 188 260)(164 236 189 261)(165 237 181 253)(166 238 182 254)(167 239 183 255)(168 240 184 256)(169 241 185 257)(170 242 186 258)(171 243 187 259)(172 244 193 265)(173 245 194 266)(174 246 195 267)(175 247 196 268)(176 248 197 269)(177 249 198 270)(178 250 190 262)(179 251 191 263)(180 252 192 264)
(1 165 43 181)(2 166 44 182)(3 167 45 183)(4 168 37 184)(5 169 38 185)(6 170 39 186)(7 171 40 187)(8 163 41 188)(9 164 42 189)(10 151 279 135)(11 152 271 127)(12 153 272 128)(13 145 273 129)(14 146 274 130)(15 147 275 131)(16 148 276 132)(17 149 277 133)(18 150 278 134)(19 156 288 144)(20 157 280 136)(21 158 281 137)(22 159 282 138)(23 160 283 139)(24 161 284 140)(25 162 285 141)(26 154 286 142)(27 155 287 143)(28 193 49 172)(29 194 50 173)(30 195 51 174)(31 196 52 175)(32 197 53 176)(33 198 54 177)(34 190 46 178)(35 191 47 179)(36 192 48 180)(55 224 80 199)(56 225 81 200)(57 217 73 201)(58 218 74 202)(59 219 75 203)(60 220 76 204)(61 221 77 205)(62 222 78 206)(63 223 79 207)(64 229 85 208)(65 230 86 209)(66 231 87 210)(67 232 88 211)(68 233 89 212)(69 234 90 213)(70 226 82 214)(71 227 83 215)(72 228 84 216)(91 260 116 235)(92 261 117 236)(93 253 109 237)(94 254 110 238)(95 255 111 239)(96 256 112 240)(97 257 113 241)(98 258 114 242)(99 259 115 243)(100 265 121 244)(101 266 122 245)(102 267 123 246)(103 268 124 247)(104 269 125 248)(105 270 126 249)(106 262 118 250)(107 263 119 251)(108 264 120 252)
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), (1,129,46,142)(2,130,47,143)(3,131,48,144)(4,132,49,136)(5,133,50,137)(6,134,51,138)(7,135,52,139)(8,127,53,140)(9,128,54,141)(10,171,23,175)(11,163,24,176)(12,164,25,177)(13,165,26,178)(14,166,27,179)(15,167,19,180)(16,168,20,172)(17,169,21,173)(18,170,22,174)(28,157,37,148)(29,158,38,149)(30,159,39,150)(31,160,40,151)(32,161,41,152)(33,162,42,153)(34,154,43,145)(35,155,44,146)(36,156,45,147)(55,104,68,91)(56,105,69,92)(57,106,70,93)(58,107,71,94)(59,108,72,95)(60,100,64,96)(61,101,65,97)(62,102,66,98)(63,103,67,99)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(181,286,190,273)(182,287,191,274)(183,288,192,275)(184,280,193,276)(185,281,194,277)(186,282,195,278)(187,283,196,279)(188,284,197,271)(189,285,198,272)(199,235,212,248)(200,236,213,249)(201,237,214,250)(202,238,215,251)(203,239,216,252)(204,240,208,244)(205,241,209,245)(206,242,210,246)(207,243,211,247)(217,253,226,262)(218,254,227,263)(219,255,228,264)(220,256,229,265)(221,257,230,266)(222,258,231,267)(223,259,232,268)(224,260,233,269)(225,261,234,270), (1,93,43,109)(2,94,44,110)(3,95,45,111)(4,96,37,112)(5,97,38,113)(6,98,39,114)(7,99,40,115)(8,91,41,116)(9,92,42,117)(10,207,279,223)(11,199,271,224)(12,200,272,225)(13,201,273,217)(14,202,274,218)(15,203,275,219)(16,204,276,220)(17,205,277,221)(18,206,278,222)(19,216,288,228)(20,208,280,229)(21,209,281,230)(22,210,282,231)(23,211,283,232)(24,212,284,233)(25,213,285,234)(26,214,286,226)(27,215,287,227)(28,121,49,100)(29,122,50,101)(30,123,51,102)(31,124,52,103)(32,125,53,104)(33,126,54,105)(34,118,46,106)(35,119,47,107)(36,120,48,108)(55,152,80,127)(56,153,81,128)(57,145,73,129)(58,146,74,130)(59,147,75,131)(60,148,76,132)(61,149,77,133)(62,150,78,134)(63,151,79,135)(64,157,85,136)(65,158,86,137)(66,159,87,138)(67,160,88,139)(68,161,89,140)(69,162,90,141)(70,154,82,142)(71,155,83,143)(72,156,84,144)(163,235,188,260)(164,236,189,261)(165,237,181,253)(166,238,182,254)(167,239,183,255)(168,240,184,256)(169,241,185,257)(170,242,186,258)(171,243,187,259)(172,244,193,265)(173,245,194,266)(174,246,195,267)(175,247,196,268)(176,248,197,269)(177,249,198,270)(178,250,190,262)(179,251,191,263)(180,252,192,264), (1,165,43,181)(2,166,44,182)(3,167,45,183)(4,168,37,184)(5,169,38,185)(6,170,39,186)(7,171,40,187)(8,163,41,188)(9,164,42,189)(10,151,279,135)(11,152,271,127)(12,153,272,128)(13,145,273,129)(14,146,274,130)(15,147,275,131)(16,148,276,132)(17,149,277,133)(18,150,278,134)(19,156,288,144)(20,157,280,136)(21,158,281,137)(22,159,282,138)(23,160,283,139)(24,161,284,140)(25,162,285,141)(26,154,286,142)(27,155,287,143)(28,193,49,172)(29,194,50,173)(30,195,51,174)(31,196,52,175)(32,197,53,176)(33,198,54,177)(34,190,46,178)(35,191,47,179)(36,192,48,180)(55,224,80,199)(56,225,81,200)(57,217,73,201)(58,218,74,202)(59,219,75,203)(60,220,76,204)(61,221,77,205)(62,222,78,206)(63,223,79,207)(64,229,85,208)(65,230,86,209)(66,231,87,210)(67,232,88,211)(68,233,89,212)(69,234,90,213)(70,226,82,214)(71,227,83,215)(72,228,84,216)(91,260,116,235)(92,261,117,236)(93,253,109,237)(94,254,110,238)(95,255,111,239)(96,256,112,240)(97,257,113,241)(98,258,114,242)(99,259,115,243)(100,265,121,244)(101,266,122,245)(102,267,123,246)(103,268,124,247)(104,269,125,248)(105,270,126,249)(106,262,118,250)(107,263,119,251)(108,264,120,252)>;
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), (1,129,46,142)(2,130,47,143)(3,131,48,144)(4,132,49,136)(5,133,50,137)(6,134,51,138)(7,135,52,139)(8,127,53,140)(9,128,54,141)(10,171,23,175)(11,163,24,176)(12,164,25,177)(13,165,26,178)(14,166,27,179)(15,167,19,180)(16,168,20,172)(17,169,21,173)(18,170,22,174)(28,157,37,148)(29,158,38,149)(30,159,39,150)(31,160,40,151)(32,161,41,152)(33,162,42,153)(34,154,43,145)(35,155,44,146)(36,156,45,147)(55,104,68,91)(56,105,69,92)(57,106,70,93)(58,107,71,94)(59,108,72,95)(60,100,64,96)(61,101,65,97)(62,102,66,98)(63,103,67,99)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(181,286,190,273)(182,287,191,274)(183,288,192,275)(184,280,193,276)(185,281,194,277)(186,282,195,278)(187,283,196,279)(188,284,197,271)(189,285,198,272)(199,235,212,248)(200,236,213,249)(201,237,214,250)(202,238,215,251)(203,239,216,252)(204,240,208,244)(205,241,209,245)(206,242,210,246)(207,243,211,247)(217,253,226,262)(218,254,227,263)(219,255,228,264)(220,256,229,265)(221,257,230,266)(222,258,231,267)(223,259,232,268)(224,260,233,269)(225,261,234,270), (1,93,43,109)(2,94,44,110)(3,95,45,111)(4,96,37,112)(5,97,38,113)(6,98,39,114)(7,99,40,115)(8,91,41,116)(9,92,42,117)(10,207,279,223)(11,199,271,224)(12,200,272,225)(13,201,273,217)(14,202,274,218)(15,203,275,219)(16,204,276,220)(17,205,277,221)(18,206,278,222)(19,216,288,228)(20,208,280,229)(21,209,281,230)(22,210,282,231)(23,211,283,232)(24,212,284,233)(25,213,285,234)(26,214,286,226)(27,215,287,227)(28,121,49,100)(29,122,50,101)(30,123,51,102)(31,124,52,103)(32,125,53,104)(33,126,54,105)(34,118,46,106)(35,119,47,107)(36,120,48,108)(55,152,80,127)(56,153,81,128)(57,145,73,129)(58,146,74,130)(59,147,75,131)(60,148,76,132)(61,149,77,133)(62,150,78,134)(63,151,79,135)(64,157,85,136)(65,158,86,137)(66,159,87,138)(67,160,88,139)(68,161,89,140)(69,162,90,141)(70,154,82,142)(71,155,83,143)(72,156,84,144)(163,235,188,260)(164,236,189,261)(165,237,181,253)(166,238,182,254)(167,239,183,255)(168,240,184,256)(169,241,185,257)(170,242,186,258)(171,243,187,259)(172,244,193,265)(173,245,194,266)(174,246,195,267)(175,247,196,268)(176,248,197,269)(177,249,198,270)(178,250,190,262)(179,251,191,263)(180,252,192,264), (1,165,43,181)(2,166,44,182)(3,167,45,183)(4,168,37,184)(5,169,38,185)(6,170,39,186)(7,171,40,187)(8,163,41,188)(9,164,42,189)(10,151,279,135)(11,152,271,127)(12,153,272,128)(13,145,273,129)(14,146,274,130)(15,147,275,131)(16,148,276,132)(17,149,277,133)(18,150,278,134)(19,156,288,144)(20,157,280,136)(21,158,281,137)(22,159,282,138)(23,160,283,139)(24,161,284,140)(25,162,285,141)(26,154,286,142)(27,155,287,143)(28,193,49,172)(29,194,50,173)(30,195,51,174)(31,196,52,175)(32,197,53,176)(33,198,54,177)(34,190,46,178)(35,191,47,179)(36,192,48,180)(55,224,80,199)(56,225,81,200)(57,217,73,201)(58,218,74,202)(59,219,75,203)(60,220,76,204)(61,221,77,205)(62,222,78,206)(63,223,79,207)(64,229,85,208)(65,230,86,209)(66,231,87,210)(67,232,88,211)(68,233,89,212)(69,234,90,213)(70,226,82,214)(71,227,83,215)(72,228,84,216)(91,260,116,235)(92,261,117,236)(93,253,109,237)(94,254,110,238)(95,255,111,239)(96,256,112,240)(97,257,113,241)(98,258,114,242)(99,259,115,243)(100,265,121,244)(101,266,122,245)(102,267,123,246)(103,268,124,247)(104,269,125,248)(105,270,126,249)(106,262,118,250)(107,263,119,251)(108,264,120,252) );
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)], [(1,129,46,142),(2,130,47,143),(3,131,48,144),(4,132,49,136),(5,133,50,137),(6,134,51,138),(7,135,52,139),(8,127,53,140),(9,128,54,141),(10,171,23,175),(11,163,24,176),(12,164,25,177),(13,165,26,178),(14,166,27,179),(15,167,19,180),(16,168,20,172),(17,169,21,173),(18,170,22,174),(28,157,37,148),(29,158,38,149),(30,159,39,150),(31,160,40,151),(32,161,41,152),(33,162,42,153),(34,154,43,145),(35,155,44,146),(36,156,45,147),(55,104,68,91),(56,105,69,92),(57,106,70,93),(58,107,71,94),(59,108,72,95),(60,100,64,96),(61,101,65,97),(62,102,66,98),(63,103,67,99),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117),(181,286,190,273),(182,287,191,274),(183,288,192,275),(184,280,193,276),(185,281,194,277),(186,282,195,278),(187,283,196,279),(188,284,197,271),(189,285,198,272),(199,235,212,248),(200,236,213,249),(201,237,214,250),(202,238,215,251),(203,239,216,252),(204,240,208,244),(205,241,209,245),(206,242,210,246),(207,243,211,247),(217,253,226,262),(218,254,227,263),(219,255,228,264),(220,256,229,265),(221,257,230,266),(222,258,231,267),(223,259,232,268),(224,260,233,269),(225,261,234,270)], [(1,93,43,109),(2,94,44,110),(3,95,45,111),(4,96,37,112),(5,97,38,113),(6,98,39,114),(7,99,40,115),(8,91,41,116),(9,92,42,117),(10,207,279,223),(11,199,271,224),(12,200,272,225),(13,201,273,217),(14,202,274,218),(15,203,275,219),(16,204,276,220),(17,205,277,221),(18,206,278,222),(19,216,288,228),(20,208,280,229),(21,209,281,230),(22,210,282,231),(23,211,283,232),(24,212,284,233),(25,213,285,234),(26,214,286,226),(27,215,287,227),(28,121,49,100),(29,122,50,101),(30,123,51,102),(31,124,52,103),(32,125,53,104),(33,126,54,105),(34,118,46,106),(35,119,47,107),(36,120,48,108),(55,152,80,127),(56,153,81,128),(57,145,73,129),(58,146,74,130),(59,147,75,131),(60,148,76,132),(61,149,77,133),(62,150,78,134),(63,151,79,135),(64,157,85,136),(65,158,86,137),(66,159,87,138),(67,160,88,139),(68,161,89,140),(69,162,90,141),(70,154,82,142),(71,155,83,143),(72,156,84,144),(163,235,188,260),(164,236,189,261),(165,237,181,253),(166,238,182,254),(167,239,183,255),(168,240,184,256),(169,241,185,257),(170,242,186,258),(171,243,187,259),(172,244,193,265),(173,245,194,266),(174,246,195,267),(175,247,196,268),(176,248,197,269),(177,249,198,270),(178,250,190,262),(179,251,191,263),(180,252,192,264)], [(1,165,43,181),(2,166,44,182),(3,167,45,183),(4,168,37,184),(5,169,38,185),(6,170,39,186),(7,171,40,187),(8,163,41,188),(9,164,42,189),(10,151,279,135),(11,152,271,127),(12,153,272,128),(13,145,273,129),(14,146,274,130),(15,147,275,131),(16,148,276,132),(17,149,277,133),(18,150,278,134),(19,156,288,144),(20,157,280,136),(21,158,281,137),(22,159,282,138),(23,160,283,139),(24,161,284,140),(25,162,285,141),(26,154,286,142),(27,155,287,143),(28,193,49,172),(29,194,50,173),(30,195,51,174),(31,196,52,175),(32,197,53,176),(33,198,54,177),(34,190,46,178),(35,191,47,179),(36,192,48,180),(55,224,80,199),(56,225,81,200),(57,217,73,201),(58,218,74,202),(59,219,75,203),(60,220,76,204),(61,221,77,205),(62,222,78,206),(63,223,79,207),(64,229,85,208),(65,230,86,209),(66,231,87,210),(67,232,88,211),(68,233,89,212),(69,234,90,213),(70,226,82,214),(71,227,83,215),(72,228,84,216),(91,260,116,235),(92,261,117,236),(93,253,109,237),(94,254,110,238),(95,255,111,239),(96,256,112,240),(97,257,113,241),(98,258,114,242),(99,259,115,243),(100,265,121,244),(101,266,122,245),(102,267,123,246),(103,268,124,247),(104,269,125,248),(105,270,126,249),(106,262,118,250),(107,263,119,251),(108,264,120,252)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12L | 12M | ··· | 12T | 18A | ··· | 18R | 36A | ··· | 36AJ | 36AK | ··· | 36BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | Q8 | C3×D4 | C3×Q8 | D4×C9 | Q8×C9 |
| kernel | C9×C4⋊Q8 | C4×C36 | C9×C4⋊C4 | Q8×C18 | C3×C4⋊Q8 | C4×C12 | C3×C4⋊C4 | C6×Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C36 | C36 | C12 | C12 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 6 | 6 | 24 | 12 | 2 | 4 | 4 | 8 | 12 | 24 |
Matrix representation of C9×C4⋊Q8 ►in GL4(𝔽37) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 35 | 0 | 0 |
| 1 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 29 | 6 |
| 22 | 9 | 0 | 0 |
| 8 | 15 | 0 | 0 |
| 0 | 0 | 16 | 13 |
| 0 | 0 | 3 | 21 |
G:=sub<GL(4,GF(37))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[1,1,0,0,35,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,31,29,0,0,0,6],[22,8,0,0,9,15,0,0,0,0,16,3,0,0,13,21] >;
C9×C4⋊Q8 in GAP, Magma, Sage, TeX
C_9\times C_4\rtimes Q_8
% in TeX
G:=Group("C9xC4:Q8"); // GroupNames label
G:=SmallGroup(288,178);
// by ID
G=gap.SmallGroup(288,178);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,365,176,1094,268,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations