direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×Dic9, C72⋊3C4, C18.3C42, C24.8Dic3, C9⋊C8⋊5C4, C9⋊2(C4×C8), C6.8(S3×C8), C2.2(C8×D9), C3.(C8×Dic3), C18.3(C2×C8), C4.20(C4×D9), (C2×C8).10D9, C12.69(C4×S3), (C2×C72).11C2, (C2×C24).25S3, C36.25(C2×C4), (C2×C4).90D18, C6.8(C4×Dic3), C2.2(C4×Dic9), C22.8(C4×D9), (C2×C12).404D6, (C2×Dic9).7C4, C4.11(C2×Dic9), (C4×Dic9).10C2, C12.51(C2×Dic3), (C2×C36).102C22, (C2×C9⋊C8).12C2, (C2×C18).9(C2×C4), (C2×C6).34(C4×S3), SmallGroup(288,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C8×Dic9 |
Generators and relations for C8×Dic9
G = < a,b,c | a8=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 176 in 66 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, C9, Dic3, C12, C2×C6, C42, C2×C8, C2×C8, C18, C18, C3⋊C8, C24, C2×Dic3, C2×C12, C4×C8, Dic9, C36, C2×C18, C2×C3⋊C8, C4×Dic3, C2×C24, C9⋊C8, C72, C2×Dic9, C2×C36, C8×Dic3, C2×C9⋊C8, C4×Dic9, C2×C72, C8×Dic9
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, D9, C4×S3, C2×Dic3, C4×C8, Dic9, D18, S3×C8, C4×Dic3, C4×D9, C2×Dic9, C8×Dic3, C8×D9, C4×Dic9, C8×Dic9
►Regular action on 288 points
Generators in S288
(1 288 133 160 269 45 200 30)(2 271 134 161 270 46 201 31)(3 272 135 162 253 47 202 32)(4 273 136 145 254 48 203 33)(5 274 137 146 255 49 204 34)(6 275 138 147 256 50 205 35)(7 276 139 148 257 51 206 36)(8 277 140 149 258 52 207 19)(9 278 141 150 259 53 208 20)(10 279 142 151 260 54 209 21)(11 280 143 152 261 37 210 22)(12 281 144 153 262 38 211 23)(13 282 127 154 263 39 212 24)(14 283 128 155 264 40 213 25)(15 284 129 156 265 41 214 26)(16 285 130 157 266 42 215 27)(17 286 131 158 267 43 216 28)(18 287 132 159 268 44 199 29)(55 121 80 197 99 163 230 252)(56 122 81 198 100 164 231 235)(57 123 82 181 101 165 232 236)(58 124 83 182 102 166 233 237)(59 125 84 183 103 167 234 238)(60 126 85 184 104 168 217 239)(61 109 86 185 105 169 218 240)(62 110 87 186 106 170 219 241)(63 111 88 187 107 171 220 242)(64 112 89 188 108 172 221 243)(65 113 90 189 91 173 222 244)(66 114 73 190 92 174 223 245)(67 115 74 191 93 175 224 246)(68 116 75 192 94 176 225 247)(69 117 76 193 95 177 226 248)(70 118 77 194 96 178 227 249)(71 119 78 195 97 179 228 250)(72 120 79 196 98 180 229 251)
(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)
(1 230 10 221)(2 229 11 220)(3 228 12 219)(4 227 13 218)(5 226 14 217)(6 225 15 234)(7 224 16 233)(8 223 17 232)(9 222 18 231)(19 174 28 165)(20 173 29 164)(21 172 30 163)(22 171 31 180)(23 170 32 179)(24 169 33 178)(25 168 34 177)(26 167 35 176)(27 166 36 175)(37 187 46 196)(38 186 47 195)(39 185 48 194)(40 184 49 193)(41 183 50 192)(42 182 51 191)(43 181 52 190)(44 198 53 189)(45 197 54 188)(55 142 64 133)(56 141 65 132)(57 140 66 131)(58 139 67 130)(59 138 68 129)(60 137 69 128)(61 136 70 127)(62 135 71 144)(63 134 72 143)(73 267 82 258)(74 266 83 257)(75 265 84 256)(76 264 85 255)(77 263 86 254)(78 262 87 253)(79 261 88 270)(80 260 89 269)(81 259 90 268)(91 199 100 208)(92 216 101 207)(93 215 102 206)(94 214 103 205)(95 213 104 204)(96 212 105 203)(97 211 106 202)(98 210 107 201)(99 209 108 200)(109 145 118 154)(110 162 119 153)(111 161 120 152)(112 160 121 151)(113 159 122 150)(114 158 123 149)(115 157 124 148)(116 156 125 147)(117 155 126 146)(235 278 244 287)(236 277 245 286)(237 276 246 285)(238 275 247 284)(239 274 248 283)(240 273 249 282)(241 272 250 281)(242 271 251 280)(243 288 252 279)
G:=sub<Sym(288)| (1,288,133,160,269,45,200,30)(2,271,134,161,270,46,201,31)(3,272,135,162,253,47,202,32)(4,273,136,145,254,48,203,33)(5,274,137,146,255,49,204,34)(6,275,138,147,256,50,205,35)(7,276,139,148,257,51,206,36)(8,277,140,149,258,52,207,19)(9,278,141,150,259,53,208,20)(10,279,142,151,260,54,209,21)(11,280,143,152,261,37,210,22)(12,281,144,153,262,38,211,23)(13,282,127,154,263,39,212,24)(14,283,128,155,264,40,213,25)(15,284,129,156,265,41,214,26)(16,285,130,157,266,42,215,27)(17,286,131,158,267,43,216,28)(18,287,132,159,268,44,199,29)(55,121,80,197,99,163,230,252)(56,122,81,198,100,164,231,235)(57,123,82,181,101,165,232,236)(58,124,83,182,102,166,233,237)(59,125,84,183,103,167,234,238)(60,126,85,184,104,168,217,239)(61,109,86,185,105,169,218,240)(62,110,87,186,106,170,219,241)(63,111,88,187,107,171,220,242)(64,112,89,188,108,172,221,243)(65,113,90,189,91,173,222,244)(66,114,73,190,92,174,223,245)(67,115,74,191,93,175,224,246)(68,116,75,192,94,176,225,247)(69,117,76,193,95,177,226,248)(70,118,77,194,96,178,227,249)(71,119,78,195,97,179,228,250)(72,120,79,196,98,180,229,251), (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), (1,230,10,221)(2,229,11,220)(3,228,12,219)(4,227,13,218)(5,226,14,217)(6,225,15,234)(7,224,16,233)(8,223,17,232)(9,222,18,231)(19,174,28,165)(20,173,29,164)(21,172,30,163)(22,171,31,180)(23,170,32,179)(24,169,33,178)(25,168,34,177)(26,167,35,176)(27,166,36,175)(37,187,46,196)(38,186,47,195)(39,185,48,194)(40,184,49,193)(41,183,50,192)(42,182,51,191)(43,181,52,190)(44,198,53,189)(45,197,54,188)(55,142,64,133)(56,141,65,132)(57,140,66,131)(58,139,67,130)(59,138,68,129)(60,137,69,128)(61,136,70,127)(62,135,71,144)(63,134,72,143)(73,267,82,258)(74,266,83,257)(75,265,84,256)(76,264,85,255)(77,263,86,254)(78,262,87,253)(79,261,88,270)(80,260,89,269)(81,259,90,268)(91,199,100,208)(92,216,101,207)(93,215,102,206)(94,214,103,205)(95,213,104,204)(96,212,105,203)(97,211,106,202)(98,210,107,201)(99,209,108,200)(109,145,118,154)(110,162,119,153)(111,161,120,152)(112,160,121,151)(113,159,122,150)(114,158,123,149)(115,157,124,148)(116,156,125,147)(117,155,126,146)(235,278,244,287)(236,277,245,286)(237,276,246,285)(238,275,247,284)(239,274,248,283)(240,273,249,282)(241,272,250,281)(242,271,251,280)(243,288,252,279)>;
G:=Group( (1,288,133,160,269,45,200,30)(2,271,134,161,270,46,201,31)(3,272,135,162,253,47,202,32)(4,273,136,145,254,48,203,33)(5,274,137,146,255,49,204,34)(6,275,138,147,256,50,205,35)(7,276,139,148,257,51,206,36)(8,277,140,149,258,52,207,19)(9,278,141,150,259,53,208,20)(10,279,142,151,260,54,209,21)(11,280,143,152,261,37,210,22)(12,281,144,153,262,38,211,23)(13,282,127,154,263,39,212,24)(14,283,128,155,264,40,213,25)(15,284,129,156,265,41,214,26)(16,285,130,157,266,42,215,27)(17,286,131,158,267,43,216,28)(18,287,132,159,268,44,199,29)(55,121,80,197,99,163,230,252)(56,122,81,198,100,164,231,235)(57,123,82,181,101,165,232,236)(58,124,83,182,102,166,233,237)(59,125,84,183,103,167,234,238)(60,126,85,184,104,168,217,239)(61,109,86,185,105,169,218,240)(62,110,87,186,106,170,219,241)(63,111,88,187,107,171,220,242)(64,112,89,188,108,172,221,243)(65,113,90,189,91,173,222,244)(66,114,73,190,92,174,223,245)(67,115,74,191,93,175,224,246)(68,116,75,192,94,176,225,247)(69,117,76,193,95,177,226,248)(70,118,77,194,96,178,227,249)(71,119,78,195,97,179,228,250)(72,120,79,196,98,180,229,251), (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), (1,230,10,221)(2,229,11,220)(3,228,12,219)(4,227,13,218)(5,226,14,217)(6,225,15,234)(7,224,16,233)(8,223,17,232)(9,222,18,231)(19,174,28,165)(20,173,29,164)(21,172,30,163)(22,171,31,180)(23,170,32,179)(24,169,33,178)(25,168,34,177)(26,167,35,176)(27,166,36,175)(37,187,46,196)(38,186,47,195)(39,185,48,194)(40,184,49,193)(41,183,50,192)(42,182,51,191)(43,181,52,190)(44,198,53,189)(45,197,54,188)(55,142,64,133)(56,141,65,132)(57,140,66,131)(58,139,67,130)(59,138,68,129)(60,137,69,128)(61,136,70,127)(62,135,71,144)(63,134,72,143)(73,267,82,258)(74,266,83,257)(75,265,84,256)(76,264,85,255)(77,263,86,254)(78,262,87,253)(79,261,88,270)(80,260,89,269)(81,259,90,268)(91,199,100,208)(92,216,101,207)(93,215,102,206)(94,214,103,205)(95,213,104,204)(96,212,105,203)(97,211,106,202)(98,210,107,201)(99,209,108,200)(109,145,118,154)(110,162,119,153)(111,161,120,152)(112,160,121,151)(113,159,122,150)(114,158,123,149)(115,157,124,148)(116,156,125,147)(117,155,126,146)(235,278,244,287)(236,277,245,286)(237,276,246,285)(238,275,247,284)(239,274,248,283)(240,273,249,282)(241,272,250,281)(242,271,251,280)(243,288,252,279) );
G=PermutationGroup([[(1,288,133,160,269,45,200,30),(2,271,134,161,270,46,201,31),(3,272,135,162,253,47,202,32),(4,273,136,145,254,48,203,33),(5,274,137,146,255,49,204,34),(6,275,138,147,256,50,205,35),(7,276,139,148,257,51,206,36),(8,277,140,149,258,52,207,19),(9,278,141,150,259,53,208,20),(10,279,142,151,260,54,209,21),(11,280,143,152,261,37,210,22),(12,281,144,153,262,38,211,23),(13,282,127,154,263,39,212,24),(14,283,128,155,264,40,213,25),(15,284,129,156,265,41,214,26),(16,285,130,157,266,42,215,27),(17,286,131,158,267,43,216,28),(18,287,132,159,268,44,199,29),(55,121,80,197,99,163,230,252),(56,122,81,198,100,164,231,235),(57,123,82,181,101,165,232,236),(58,124,83,182,102,166,233,237),(59,125,84,183,103,167,234,238),(60,126,85,184,104,168,217,239),(61,109,86,185,105,169,218,240),(62,110,87,186,106,170,219,241),(63,111,88,187,107,171,220,242),(64,112,89,188,108,172,221,243),(65,113,90,189,91,173,222,244),(66,114,73,190,92,174,223,245),(67,115,74,191,93,175,224,246),(68,116,75,192,94,176,225,247),(69,117,76,193,95,177,226,248),(70,118,77,194,96,178,227,249),(71,119,78,195,97,179,228,250),(72,120,79,196,98,180,229,251)], [(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)], [(1,230,10,221),(2,229,11,220),(3,228,12,219),(4,227,13,218),(5,226,14,217),(6,225,15,234),(7,224,16,233),(8,223,17,232),(9,222,18,231),(19,174,28,165),(20,173,29,164),(21,172,30,163),(22,171,31,180),(23,170,32,179),(24,169,33,178),(25,168,34,177),(26,167,35,176),(27,166,36,175),(37,187,46,196),(38,186,47,195),(39,185,48,194),(40,184,49,193),(41,183,50,192),(42,182,51,191),(43,181,52,190),(44,198,53,189),(45,197,54,188),(55,142,64,133),(56,141,65,132),(57,140,66,131),(58,139,67,130),(59,138,68,129),(60,137,69,128),(61,136,70,127),(62,135,71,144),(63,134,72,143),(73,267,82,258),(74,266,83,257),(75,265,84,256),(76,264,85,255),(77,263,86,254),(78,262,87,253),(79,261,88,270),(80,260,89,269),(81,259,90,268),(91,199,100,208),(92,216,101,207),(93,215,102,206),(94,214,103,205),(95,213,104,204),(96,212,105,203),(97,211,106,202),(98,210,107,201),(99,209,108,200),(109,145,118,154),(110,162,119,153),(111,161,120,152),(112,160,121,151),(113,159,122,150),(114,158,123,149),(115,157,124,148),(116,156,125,147),(117,155,126,146),(235,278,244,287),(236,277,245,286),(237,276,246,285),(238,275,247,284),(239,274,248,283),(240,273,249,282),(241,272,250,281),(242,271,251,280),(243,288,252,279)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 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 | C4 | C4 | C4 | C8 | S3 | Dic3 | D6 | D9 | C4×S3 | C4×S3 | Dic9 | D18 | S3×C8 | C4×D9 | C4×D9 | C8×D9 |
| kernel | C8×Dic9 | C2×C9⋊C8 | C4×Dic9 | C2×C72 | C9⋊C8 | C72 | C2×Dic9 | Dic9 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 1 | 2 | 1 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C8×Dic9 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 63 |
| 72 | 0 | 0 |
| 0 | 70 | 42 |
| 0 | 31 | 28 |
| 27 | 0 | 0 |
| 0 | 61 | 51 |
| 0 | 63 | 12 |
G:=sub<GL(3,GF(73))| [1,0,0,0,63,0,0,0,63],[72,0,0,0,70,31,0,42,28],[27,0,0,0,61,63,0,51,12] >;
C8×Dic9 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_9 % in TeX
G:=Group("C8xDic9"); // GroupNames label
G:=SmallGroup(288,21);
// by ID
G=gap.SmallGroup(288,21);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,64,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^8=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations