direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic21, C84⋊2C4, C21⋊2C42, C12⋊2Dic7, C28⋊2Dic3, C22.3D42, C6.7(C4×D7), C3⋊2(C4×Dic7), C7⋊2(C4×Dic3), C14.7(C4×S3), (C2×C84).9C2, (C2×C28).7S3, C2.2(C4×D21), (C2×C4).6D21, (C2×C12).7D7, C42.16(C2×C4), (C2×C6).21D14, (C2×C14).21D6, C6.8(C2×Dic7), C14.8(C2×Dic3), C2.2(C2×Dic21), (C2×C42).22C22, (C2×Dic21).9C2, SmallGroup(336,97)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C4×Dic21 |
Generators and relations for C4×Dic21
G = < a,b,c | a4=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 256 in 60 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C7, C2×C4, C2×C4, Dic3, C12, C2×C6, C14, C14, C42, C21, C2×Dic3, C2×C12, Dic7, C28, C2×C14, C42, C42, C4×Dic3, C2×Dic7, C2×C28, Dic21, C84, C2×C42, C4×Dic7, C2×Dic21, C2×C84, C4×Dic21
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, D7, C42, C4×S3, C2×Dic3, Dic7, D14, D21, C4×Dic3, C4×D7, C2×Dic7, Dic21, D42, C4×Dic7, C4×D21, C2×Dic21, C4×Dic21
►Regular action on 336 points
Generators in S336
(1 133 308 249)(2 134 309 250)(3 135 310 251)(4 136 311 252)(5 137 312 211)(6 138 313 212)(7 139 314 213)(8 140 315 214)(9 141 316 215)(10 142 317 216)(11 143 318 217)(12 144 319 218)(13 145 320 219)(14 146 321 220)(15 147 322 221)(16 148 323 222)(17 149 324 223)(18 150 325 224)(19 151 326 225)(20 152 327 226)(21 153 328 227)(22 154 329 228)(23 155 330 229)(24 156 331 230)(25 157 332 231)(26 158 333 232)(27 159 334 233)(28 160 335 234)(29 161 336 235)(30 162 295 236)(31 163 296 237)(32 164 297 238)(33 165 298 239)(34 166 299 240)(35 167 300 241)(36 168 301 242)(37 127 302 243)(38 128 303 244)(39 129 304 245)(40 130 305 246)(41 131 306 247)(42 132 307 248)(43 260 102 197)(44 261 103 198)(45 262 104 199)(46 263 105 200)(47 264 106 201)(48 265 107 202)(49 266 108 203)(50 267 109 204)(51 268 110 205)(52 269 111 206)(53 270 112 207)(54 271 113 208)(55 272 114 209)(56 273 115 210)(57 274 116 169)(58 275 117 170)(59 276 118 171)(60 277 119 172)(61 278 120 173)(62 279 121 174)(63 280 122 175)(64 281 123 176)(65 282 124 177)(66 283 125 178)(67 284 126 179)(68 285 85 180)(69 286 86 181)(70 287 87 182)(71 288 88 183)(72 289 89 184)(73 290 90 185)(74 291 91 186)(75 292 92 187)(76 293 93 188)(77 294 94 189)(78 253 95 190)(79 254 96 191)(80 255 97 192)(81 256 98 193)(82 257 99 194)(83 258 100 195)(84 259 101 196)
(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)
(1 279 22 258)(2 278 23 257)(3 277 24 256)(4 276 25 255)(5 275 26 254)(6 274 27 253)(7 273 28 294)(8 272 29 293)(9 271 30 292)(10 270 31 291)(11 269 32 290)(12 268 33 289)(13 267 34 288)(14 266 35 287)(15 265 36 286)(16 264 37 285)(17 263 38 284)(18 262 39 283)(19 261 40 282)(20 260 41 281)(21 259 42 280)(43 247 64 226)(44 246 65 225)(45 245 66 224)(46 244 67 223)(47 243 68 222)(48 242 69 221)(49 241 70 220)(50 240 71 219)(51 239 72 218)(52 238 73 217)(53 237 74 216)(54 236 75 215)(55 235 76 214)(56 234 77 213)(57 233 78 212)(58 232 79 211)(59 231 80 252)(60 230 81 251)(61 229 82 250)(62 228 83 249)(63 227 84 248)(85 148 106 127)(86 147 107 168)(87 146 108 167)(88 145 109 166)(89 144 110 165)(90 143 111 164)(91 142 112 163)(92 141 113 162)(93 140 114 161)(94 139 115 160)(95 138 116 159)(96 137 117 158)(97 136 118 157)(98 135 119 156)(99 134 120 155)(100 133 121 154)(101 132 122 153)(102 131 123 152)(103 130 124 151)(104 129 125 150)(105 128 126 149)(169 334 190 313)(170 333 191 312)(171 332 192 311)(172 331 193 310)(173 330 194 309)(174 329 195 308)(175 328 196 307)(176 327 197 306)(177 326 198 305)(178 325 199 304)(179 324 200 303)(180 323 201 302)(181 322 202 301)(182 321 203 300)(183 320 204 299)(184 319 205 298)(185 318 206 297)(186 317 207 296)(187 316 208 295)(188 315 209 336)(189 314 210 335)
G:=sub<Sym(336)| (1,133,308,249)(2,134,309,250)(3,135,310,251)(4,136,311,252)(5,137,312,211)(6,138,313,212)(7,139,314,213)(8,140,315,214)(9,141,316,215)(10,142,317,216)(11,143,318,217)(12,144,319,218)(13,145,320,219)(14,146,321,220)(15,147,322,221)(16,148,323,222)(17,149,324,223)(18,150,325,224)(19,151,326,225)(20,152,327,226)(21,153,328,227)(22,154,329,228)(23,155,330,229)(24,156,331,230)(25,157,332,231)(26,158,333,232)(27,159,334,233)(28,160,335,234)(29,161,336,235)(30,162,295,236)(31,163,296,237)(32,164,297,238)(33,165,298,239)(34,166,299,240)(35,167,300,241)(36,168,301,242)(37,127,302,243)(38,128,303,244)(39,129,304,245)(40,130,305,246)(41,131,306,247)(42,132,307,248)(43,260,102,197)(44,261,103,198)(45,262,104,199)(46,263,105,200)(47,264,106,201)(48,265,107,202)(49,266,108,203)(50,267,109,204)(51,268,110,205)(52,269,111,206)(53,270,112,207)(54,271,113,208)(55,272,114,209)(56,273,115,210)(57,274,116,169)(58,275,117,170)(59,276,118,171)(60,277,119,172)(61,278,120,173)(62,279,121,174)(63,280,122,175)(64,281,123,176)(65,282,124,177)(66,283,125,178)(67,284,126,179)(68,285,85,180)(69,286,86,181)(70,287,87,182)(71,288,88,183)(72,289,89,184)(73,290,90,185)(74,291,91,186)(75,292,92,187)(76,293,93,188)(77,294,94,189)(78,253,95,190)(79,254,96,191)(80,255,97,192)(81,256,98,193)(82,257,99,194)(83,258,100,195)(84,259,101,196), (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), (1,279,22,258)(2,278,23,257)(3,277,24,256)(4,276,25,255)(5,275,26,254)(6,274,27,253)(7,273,28,294)(8,272,29,293)(9,271,30,292)(10,270,31,291)(11,269,32,290)(12,268,33,289)(13,267,34,288)(14,266,35,287)(15,265,36,286)(16,264,37,285)(17,263,38,284)(18,262,39,283)(19,261,40,282)(20,260,41,281)(21,259,42,280)(43,247,64,226)(44,246,65,225)(45,245,66,224)(46,244,67,223)(47,243,68,222)(48,242,69,221)(49,241,70,220)(50,240,71,219)(51,239,72,218)(52,238,73,217)(53,237,74,216)(54,236,75,215)(55,235,76,214)(56,234,77,213)(57,233,78,212)(58,232,79,211)(59,231,80,252)(60,230,81,251)(61,229,82,250)(62,228,83,249)(63,227,84,248)(85,148,106,127)(86,147,107,168)(87,146,108,167)(88,145,109,166)(89,144,110,165)(90,143,111,164)(91,142,112,163)(92,141,113,162)(93,140,114,161)(94,139,115,160)(95,138,116,159)(96,137,117,158)(97,136,118,157)(98,135,119,156)(99,134,120,155)(100,133,121,154)(101,132,122,153)(102,131,123,152)(103,130,124,151)(104,129,125,150)(105,128,126,149)(169,334,190,313)(170,333,191,312)(171,332,192,311)(172,331,193,310)(173,330,194,309)(174,329,195,308)(175,328,196,307)(176,327,197,306)(177,326,198,305)(178,325,199,304)(179,324,200,303)(180,323,201,302)(181,322,202,301)(182,321,203,300)(183,320,204,299)(184,319,205,298)(185,318,206,297)(186,317,207,296)(187,316,208,295)(188,315,209,336)(189,314,210,335)>;
G:=Group( (1,133,308,249)(2,134,309,250)(3,135,310,251)(4,136,311,252)(5,137,312,211)(6,138,313,212)(7,139,314,213)(8,140,315,214)(9,141,316,215)(10,142,317,216)(11,143,318,217)(12,144,319,218)(13,145,320,219)(14,146,321,220)(15,147,322,221)(16,148,323,222)(17,149,324,223)(18,150,325,224)(19,151,326,225)(20,152,327,226)(21,153,328,227)(22,154,329,228)(23,155,330,229)(24,156,331,230)(25,157,332,231)(26,158,333,232)(27,159,334,233)(28,160,335,234)(29,161,336,235)(30,162,295,236)(31,163,296,237)(32,164,297,238)(33,165,298,239)(34,166,299,240)(35,167,300,241)(36,168,301,242)(37,127,302,243)(38,128,303,244)(39,129,304,245)(40,130,305,246)(41,131,306,247)(42,132,307,248)(43,260,102,197)(44,261,103,198)(45,262,104,199)(46,263,105,200)(47,264,106,201)(48,265,107,202)(49,266,108,203)(50,267,109,204)(51,268,110,205)(52,269,111,206)(53,270,112,207)(54,271,113,208)(55,272,114,209)(56,273,115,210)(57,274,116,169)(58,275,117,170)(59,276,118,171)(60,277,119,172)(61,278,120,173)(62,279,121,174)(63,280,122,175)(64,281,123,176)(65,282,124,177)(66,283,125,178)(67,284,126,179)(68,285,85,180)(69,286,86,181)(70,287,87,182)(71,288,88,183)(72,289,89,184)(73,290,90,185)(74,291,91,186)(75,292,92,187)(76,293,93,188)(77,294,94,189)(78,253,95,190)(79,254,96,191)(80,255,97,192)(81,256,98,193)(82,257,99,194)(83,258,100,195)(84,259,101,196), (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), (1,279,22,258)(2,278,23,257)(3,277,24,256)(4,276,25,255)(5,275,26,254)(6,274,27,253)(7,273,28,294)(8,272,29,293)(9,271,30,292)(10,270,31,291)(11,269,32,290)(12,268,33,289)(13,267,34,288)(14,266,35,287)(15,265,36,286)(16,264,37,285)(17,263,38,284)(18,262,39,283)(19,261,40,282)(20,260,41,281)(21,259,42,280)(43,247,64,226)(44,246,65,225)(45,245,66,224)(46,244,67,223)(47,243,68,222)(48,242,69,221)(49,241,70,220)(50,240,71,219)(51,239,72,218)(52,238,73,217)(53,237,74,216)(54,236,75,215)(55,235,76,214)(56,234,77,213)(57,233,78,212)(58,232,79,211)(59,231,80,252)(60,230,81,251)(61,229,82,250)(62,228,83,249)(63,227,84,248)(85,148,106,127)(86,147,107,168)(87,146,108,167)(88,145,109,166)(89,144,110,165)(90,143,111,164)(91,142,112,163)(92,141,113,162)(93,140,114,161)(94,139,115,160)(95,138,116,159)(96,137,117,158)(97,136,118,157)(98,135,119,156)(99,134,120,155)(100,133,121,154)(101,132,122,153)(102,131,123,152)(103,130,124,151)(104,129,125,150)(105,128,126,149)(169,334,190,313)(170,333,191,312)(171,332,192,311)(172,331,193,310)(173,330,194,309)(174,329,195,308)(175,328,196,307)(176,327,197,306)(177,326,198,305)(178,325,199,304)(179,324,200,303)(180,323,201,302)(181,322,202,301)(182,321,203,300)(183,320,204,299)(184,319,205,298)(185,318,206,297)(186,317,207,296)(187,316,208,295)(188,315,209,336)(189,314,210,335) );
G=PermutationGroup([[(1,133,308,249),(2,134,309,250),(3,135,310,251),(4,136,311,252),(5,137,312,211),(6,138,313,212),(7,139,314,213),(8,140,315,214),(9,141,316,215),(10,142,317,216),(11,143,318,217),(12,144,319,218),(13,145,320,219),(14,146,321,220),(15,147,322,221),(16,148,323,222),(17,149,324,223),(18,150,325,224),(19,151,326,225),(20,152,327,226),(21,153,328,227),(22,154,329,228),(23,155,330,229),(24,156,331,230),(25,157,332,231),(26,158,333,232),(27,159,334,233),(28,160,335,234),(29,161,336,235),(30,162,295,236),(31,163,296,237),(32,164,297,238),(33,165,298,239),(34,166,299,240),(35,167,300,241),(36,168,301,242),(37,127,302,243),(38,128,303,244),(39,129,304,245),(40,130,305,246),(41,131,306,247),(42,132,307,248),(43,260,102,197),(44,261,103,198),(45,262,104,199),(46,263,105,200),(47,264,106,201),(48,265,107,202),(49,266,108,203),(50,267,109,204),(51,268,110,205),(52,269,111,206),(53,270,112,207),(54,271,113,208),(55,272,114,209),(56,273,115,210),(57,274,116,169),(58,275,117,170),(59,276,118,171),(60,277,119,172),(61,278,120,173),(62,279,121,174),(63,280,122,175),(64,281,123,176),(65,282,124,177),(66,283,125,178),(67,284,126,179),(68,285,85,180),(69,286,86,181),(70,287,87,182),(71,288,88,183),(72,289,89,184),(73,290,90,185),(74,291,91,186),(75,292,92,187),(76,293,93,188),(77,294,94,189),(78,253,95,190),(79,254,96,191),(80,255,97,192),(81,256,98,193),(82,257,99,194),(83,258,100,195),(84,259,101,196)], [(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)], [(1,279,22,258),(2,278,23,257),(3,277,24,256),(4,276,25,255),(5,275,26,254),(6,274,27,253),(7,273,28,294),(8,272,29,293),(9,271,30,292),(10,270,31,291),(11,269,32,290),(12,268,33,289),(13,267,34,288),(14,266,35,287),(15,265,36,286),(16,264,37,285),(17,263,38,284),(18,262,39,283),(19,261,40,282),(20,260,41,281),(21,259,42,280),(43,247,64,226),(44,246,65,225),(45,245,66,224),(46,244,67,223),(47,243,68,222),(48,242,69,221),(49,241,70,220),(50,240,71,219),(51,239,72,218),(52,238,73,217),(53,237,74,216),(54,236,75,215),(55,235,76,214),(56,234,77,213),(57,233,78,212),(58,232,79,211),(59,231,80,252),(60,230,81,251),(61,229,82,250),(62,228,83,249),(63,227,84,248),(85,148,106,127),(86,147,107,168),(87,146,108,167),(88,145,109,166),(89,144,110,165),(90,143,111,164),(91,142,112,163),(92,141,113,162),(93,140,114,161),(94,139,115,160),(95,138,116,159),(96,137,117,158),(97,136,118,157),(98,135,119,156),(99,134,120,155),(100,133,121,154),(101,132,122,153),(102,131,123,152),(103,130,124,151),(104,129,125,150),(105,128,126,149),(169,334,190,313),(170,333,191,312),(171,332,192,311),(172,331,193,310),(173,330,194,309),(174,329,195,308),(175,328,196,307),(176,327,197,306),(177,326,198,305),(178,325,199,304),(179,324,200,303),(180,323,201,302),(181,322,202,301),(182,321,203,300),(183,320,204,299),(184,319,205,298),(185,318,206,297),(186,317,207,296),(187,316,208,295),(188,315,209,336),(189,314,210,335)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 21 | ··· | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | D7 | C4×S3 | Dic7 | D14 | D21 | C4×D7 | Dic21 | D42 | C4×D21 |
| kernel | C4×Dic21 | C2×Dic21 | C2×C84 | Dic21 | C84 | C2×C28 | C28 | C2×C14 | C2×C12 | C14 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 3 | 4 | 6 | 3 | 6 | 12 | 12 | 6 | 24 |
Matrix representation of C4×Dic21 ►in GL5(𝔽337)
| 189 | 0 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 | 0 |
| 0 | 0 | 189 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 336 |
| 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 336 | 303 | 0 | 0 |
| 0 | 0 | 0 | 137 | 100 |
| 0 | 0 | 0 | 137 | 223 |
| 148 | 0 | 0 | 0 | 0 |
| 0 | 195 | 275 | 0 | 0 |
| 0 | 48 | 142 | 0 | 0 |
| 0 | 0 | 0 | 101 | 273 |
| 0 | 0 | 0 | 33 | 236 |
G:=sub<GL(5,GF(337))| [189,0,0,0,0,0,189,0,0,0,0,0,189,0,0,0,0,0,336,0,0,0,0,0,336],[336,0,0,0,0,0,0,336,0,0,0,1,303,0,0,0,0,0,137,137,0,0,0,100,223],[148,0,0,0,0,0,195,48,0,0,0,275,142,0,0,0,0,0,101,33,0,0,0,273,236] >;
C4×Dic21 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{21} % in TeX
G:=Group("C4xDic21"); // GroupNames label
G:=SmallGroup(336,97);
// by ID
G=gap.SmallGroup(336,97);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,55,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^4=b^42=1,c^2=b^21,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations