metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic88, C16.D11, C11⋊1Q32, C22.3D8, C4.3D44, C2.5D88, C176.1C2, C44.26D4, C8.15D22, C88.16C22, Dic44.1C2, SmallGroup(352,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic88
G = < a,b | a176=1, b2=a88, bab-1=a-1 >
Smallest permutation representation of Dic88
►Regular action on 352 points
Generators in S352
(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)(177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352)
(1 244 89 332)(2 243 90 331)(3 242 91 330)(4 241 92 329)(5 240 93 328)(6 239 94 327)(7 238 95 326)(8 237 96 325)(9 236 97 324)(10 235 98 323)(11 234 99 322)(12 233 100 321)(13 232 101 320)(14 231 102 319)(15 230 103 318)(16 229 104 317)(17 228 105 316)(18 227 106 315)(19 226 107 314)(20 225 108 313)(21 224 109 312)(22 223 110 311)(23 222 111 310)(24 221 112 309)(25 220 113 308)(26 219 114 307)(27 218 115 306)(28 217 116 305)(29 216 117 304)(30 215 118 303)(31 214 119 302)(32 213 120 301)(33 212 121 300)(34 211 122 299)(35 210 123 298)(36 209 124 297)(37 208 125 296)(38 207 126 295)(39 206 127 294)(40 205 128 293)(41 204 129 292)(42 203 130 291)(43 202 131 290)(44 201 132 289)(45 200 133 288)(46 199 134 287)(47 198 135 286)(48 197 136 285)(49 196 137 284)(50 195 138 283)(51 194 139 282)(52 193 140 281)(53 192 141 280)(54 191 142 279)(55 190 143 278)(56 189 144 277)(57 188 145 276)(58 187 146 275)(59 186 147 274)(60 185 148 273)(61 184 149 272)(62 183 150 271)(63 182 151 270)(64 181 152 269)(65 180 153 268)(66 179 154 267)(67 178 155 266)(68 177 156 265)(69 352 157 264)(70 351 158 263)(71 350 159 262)(72 349 160 261)(73 348 161 260)(74 347 162 259)(75 346 163 258)(76 345 164 257)(77 344 165 256)(78 343 166 255)(79 342 167 254)(80 341 168 253)(81 340 169 252)(82 339 170 251)(83 338 171 250)(84 337 172 249)(85 336 173 248)(86 335 174 247)(87 334 175 246)(88 333 176 245)
G:=sub<Sym(352)| (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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352), (1,244,89,332)(2,243,90,331)(3,242,91,330)(4,241,92,329)(5,240,93,328)(6,239,94,327)(7,238,95,326)(8,237,96,325)(9,236,97,324)(10,235,98,323)(11,234,99,322)(12,233,100,321)(13,232,101,320)(14,231,102,319)(15,230,103,318)(16,229,104,317)(17,228,105,316)(18,227,106,315)(19,226,107,314)(20,225,108,313)(21,224,109,312)(22,223,110,311)(23,222,111,310)(24,221,112,309)(25,220,113,308)(26,219,114,307)(27,218,115,306)(28,217,116,305)(29,216,117,304)(30,215,118,303)(31,214,119,302)(32,213,120,301)(33,212,121,300)(34,211,122,299)(35,210,123,298)(36,209,124,297)(37,208,125,296)(38,207,126,295)(39,206,127,294)(40,205,128,293)(41,204,129,292)(42,203,130,291)(43,202,131,290)(44,201,132,289)(45,200,133,288)(46,199,134,287)(47,198,135,286)(48,197,136,285)(49,196,137,284)(50,195,138,283)(51,194,139,282)(52,193,140,281)(53,192,141,280)(54,191,142,279)(55,190,143,278)(56,189,144,277)(57,188,145,276)(58,187,146,275)(59,186,147,274)(60,185,148,273)(61,184,149,272)(62,183,150,271)(63,182,151,270)(64,181,152,269)(65,180,153,268)(66,179,154,267)(67,178,155,266)(68,177,156,265)(69,352,157,264)(70,351,158,263)(71,350,159,262)(72,349,160,261)(73,348,161,260)(74,347,162,259)(75,346,163,258)(76,345,164,257)(77,344,165,256)(78,343,166,255)(79,342,167,254)(80,341,168,253)(81,340,169,252)(82,339,170,251)(83,338,171,250)(84,337,172,249)(85,336,173,248)(86,335,174,247)(87,334,175,246)(88,333,176,245)>;
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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352), (1,244,89,332)(2,243,90,331)(3,242,91,330)(4,241,92,329)(5,240,93,328)(6,239,94,327)(7,238,95,326)(8,237,96,325)(9,236,97,324)(10,235,98,323)(11,234,99,322)(12,233,100,321)(13,232,101,320)(14,231,102,319)(15,230,103,318)(16,229,104,317)(17,228,105,316)(18,227,106,315)(19,226,107,314)(20,225,108,313)(21,224,109,312)(22,223,110,311)(23,222,111,310)(24,221,112,309)(25,220,113,308)(26,219,114,307)(27,218,115,306)(28,217,116,305)(29,216,117,304)(30,215,118,303)(31,214,119,302)(32,213,120,301)(33,212,121,300)(34,211,122,299)(35,210,123,298)(36,209,124,297)(37,208,125,296)(38,207,126,295)(39,206,127,294)(40,205,128,293)(41,204,129,292)(42,203,130,291)(43,202,131,290)(44,201,132,289)(45,200,133,288)(46,199,134,287)(47,198,135,286)(48,197,136,285)(49,196,137,284)(50,195,138,283)(51,194,139,282)(52,193,140,281)(53,192,141,280)(54,191,142,279)(55,190,143,278)(56,189,144,277)(57,188,145,276)(58,187,146,275)(59,186,147,274)(60,185,148,273)(61,184,149,272)(62,183,150,271)(63,182,151,270)(64,181,152,269)(65,180,153,268)(66,179,154,267)(67,178,155,266)(68,177,156,265)(69,352,157,264)(70,351,158,263)(71,350,159,262)(72,349,160,261)(73,348,161,260)(74,347,162,259)(75,346,163,258)(76,345,164,257)(77,344,165,256)(78,343,166,255)(79,342,167,254)(80,341,168,253)(81,340,169,252)(82,339,170,251)(83,338,171,250)(84,337,172,249)(85,336,173,248)(86,335,174,247)(87,334,175,246)(88,333,176,245) );
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),(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)], [(1,244,89,332),(2,243,90,331),(3,242,91,330),(4,241,92,329),(5,240,93,328),(6,239,94,327),(7,238,95,326),(8,237,96,325),(9,236,97,324),(10,235,98,323),(11,234,99,322),(12,233,100,321),(13,232,101,320),(14,231,102,319),(15,230,103,318),(16,229,104,317),(17,228,105,316),(18,227,106,315),(19,226,107,314),(20,225,108,313),(21,224,109,312),(22,223,110,311),(23,222,111,310),(24,221,112,309),(25,220,113,308),(26,219,114,307),(27,218,115,306),(28,217,116,305),(29,216,117,304),(30,215,118,303),(31,214,119,302),(32,213,120,301),(33,212,121,300),(34,211,122,299),(35,210,123,298),(36,209,124,297),(37,208,125,296),(38,207,126,295),(39,206,127,294),(40,205,128,293),(41,204,129,292),(42,203,130,291),(43,202,131,290),(44,201,132,289),(45,200,133,288),(46,199,134,287),(47,198,135,286),(48,197,136,285),(49,196,137,284),(50,195,138,283),(51,194,139,282),(52,193,140,281),(53,192,141,280),(54,191,142,279),(55,190,143,278),(56,189,144,277),(57,188,145,276),(58,187,146,275),(59,186,147,274),(60,185,148,273),(61,184,149,272),(62,183,150,271),(63,182,151,270),(64,181,152,269),(65,180,153,268),(66,179,154,267),(67,178,155,266),(68,177,156,265),(69,352,157,264),(70,351,158,263),(71,350,159,262),(72,349,160,261),(73,348,161,260),(74,347,162,259),(75,346,163,258),(76,345,164,257),(77,344,165,256),(78,343,166,255),(79,342,167,254),(80,341,168,253),(81,340,169,252),(82,339,170,251),(83,338,171,250),(84,337,172,249),(85,336,173,248),(86,335,174,247),(87,334,175,246),(88,333,176,245)]])
91 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 11A | ··· | 11E | 16A | 16B | 16C | 16D | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 2 | 88 | 88 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | D8 | D11 | Q32 | D22 | D44 | D88 | Dic88 |
| kernel | Dic88 | C176 | Dic44 | C44 | C22 | C16 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 5 | 4 | 5 | 10 | 20 | 40 |
Matrix representation of Dic88 ►in GL2(𝔽353) generated by
| 222 | 199 |
| 4 | 2 |
| 254 | 38 |
| 318 | 99 |
G:=sub<GL(2,GF(353))| [222,4,199,2],[254,318,38,99] >;
Dic88 in GAP, Magma, Sage, TeX
{\rm Dic}_{88} % in TeX
G:=Group("Dic88"); // GroupNames label
G:=SmallGroup(352,7);
// by ID
G=gap.SmallGroup(352,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,73,79,218,122,579,69,11525]);
// Polycyclic
G:=Group<a,b|a^176=1,b^2=a^88,b*a*b^-1=a^-1>;
// generators/relations
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