metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C84⋊1C4, C4⋊Dic21, C6.4D28, C2.1D84, C42.5Q8, C28⋊1Dic3, C12⋊1Dic7, C42.22D4, C14.4D12, C6.5Dic14, C14.5Dic6, C2.2Dic42, C22.5D42, C21⋊5(C4⋊C4), (C2×C84).5C2, (C2×C28).3S3, C3⋊2(C4⋊Dic7), C7⋊2(C4⋊Dic3), (C2×C4).3D21, (C2×C12).3D7, C42.31(C2×C4), (C2×C6).23D14, (C2×C14).23D6, C6.9(C2×Dic7), C14.9(C2×Dic3), C2.4(C2×Dic21), (C2×C42).24C22, (C2×Dic21).2C2, SmallGroup(336,99)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C84⋊C4
G = < a,b | a84=b4=1, bab-1=a-1 >
Smallest permutation representation of C84⋊C4
►Regular action on 336 points
Generators in S336
(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)
(1 149 264 215)(2 148 265 214)(3 147 266 213)(4 146 267 212)(5 145 268 211)(6 144 269 210)(7 143 270 209)(8 142 271 208)(9 141 272 207)(10 140 273 206)(11 139 274 205)(12 138 275 204)(13 137 276 203)(14 136 277 202)(15 135 278 201)(16 134 279 200)(17 133 280 199)(18 132 281 198)(19 131 282 197)(20 130 283 196)(21 129 284 195)(22 128 285 194)(23 127 286 193)(24 126 287 192)(25 125 288 191)(26 124 289 190)(27 123 290 189)(28 122 291 188)(29 121 292 187)(30 120 293 186)(31 119 294 185)(32 118 295 184)(33 117 296 183)(34 116 297 182)(35 115 298 181)(36 114 299 180)(37 113 300 179)(38 112 301 178)(39 111 302 177)(40 110 303 176)(41 109 304 175)(42 108 305 174)(43 107 306 173)(44 106 307 172)(45 105 308 171)(46 104 309 170)(47 103 310 169)(48 102 311 252)(49 101 312 251)(50 100 313 250)(51 99 314 249)(52 98 315 248)(53 97 316 247)(54 96 317 246)(55 95 318 245)(56 94 319 244)(57 93 320 243)(58 92 321 242)(59 91 322 241)(60 90 323 240)(61 89 324 239)(62 88 325 238)(63 87 326 237)(64 86 327 236)(65 85 328 235)(66 168 329 234)(67 167 330 233)(68 166 331 232)(69 165 332 231)(70 164 333 230)(71 163 334 229)(72 162 335 228)(73 161 336 227)(74 160 253 226)(75 159 254 225)(76 158 255 224)(77 157 256 223)(78 156 257 222)(79 155 258 221)(80 154 259 220)(81 153 260 219)(82 152 261 218)(83 151 262 217)(84 150 263 216)
G:=sub<Sym(336)| (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), (1,149,264,215)(2,148,265,214)(3,147,266,213)(4,146,267,212)(5,145,268,211)(6,144,269,210)(7,143,270,209)(8,142,271,208)(9,141,272,207)(10,140,273,206)(11,139,274,205)(12,138,275,204)(13,137,276,203)(14,136,277,202)(15,135,278,201)(16,134,279,200)(17,133,280,199)(18,132,281,198)(19,131,282,197)(20,130,283,196)(21,129,284,195)(22,128,285,194)(23,127,286,193)(24,126,287,192)(25,125,288,191)(26,124,289,190)(27,123,290,189)(28,122,291,188)(29,121,292,187)(30,120,293,186)(31,119,294,185)(32,118,295,184)(33,117,296,183)(34,116,297,182)(35,115,298,181)(36,114,299,180)(37,113,300,179)(38,112,301,178)(39,111,302,177)(40,110,303,176)(41,109,304,175)(42,108,305,174)(43,107,306,173)(44,106,307,172)(45,105,308,171)(46,104,309,170)(47,103,310,169)(48,102,311,252)(49,101,312,251)(50,100,313,250)(51,99,314,249)(52,98,315,248)(53,97,316,247)(54,96,317,246)(55,95,318,245)(56,94,319,244)(57,93,320,243)(58,92,321,242)(59,91,322,241)(60,90,323,240)(61,89,324,239)(62,88,325,238)(63,87,326,237)(64,86,327,236)(65,85,328,235)(66,168,329,234)(67,167,330,233)(68,166,331,232)(69,165,332,231)(70,164,333,230)(71,163,334,229)(72,162,335,228)(73,161,336,227)(74,160,253,226)(75,159,254,225)(76,158,255,224)(77,157,256,223)(78,156,257,222)(79,155,258,221)(80,154,259,220)(81,153,260,219)(82,152,261,218)(83,151,262,217)(84,150,263,216)>;
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), (1,149,264,215)(2,148,265,214)(3,147,266,213)(4,146,267,212)(5,145,268,211)(6,144,269,210)(7,143,270,209)(8,142,271,208)(9,141,272,207)(10,140,273,206)(11,139,274,205)(12,138,275,204)(13,137,276,203)(14,136,277,202)(15,135,278,201)(16,134,279,200)(17,133,280,199)(18,132,281,198)(19,131,282,197)(20,130,283,196)(21,129,284,195)(22,128,285,194)(23,127,286,193)(24,126,287,192)(25,125,288,191)(26,124,289,190)(27,123,290,189)(28,122,291,188)(29,121,292,187)(30,120,293,186)(31,119,294,185)(32,118,295,184)(33,117,296,183)(34,116,297,182)(35,115,298,181)(36,114,299,180)(37,113,300,179)(38,112,301,178)(39,111,302,177)(40,110,303,176)(41,109,304,175)(42,108,305,174)(43,107,306,173)(44,106,307,172)(45,105,308,171)(46,104,309,170)(47,103,310,169)(48,102,311,252)(49,101,312,251)(50,100,313,250)(51,99,314,249)(52,98,315,248)(53,97,316,247)(54,96,317,246)(55,95,318,245)(56,94,319,244)(57,93,320,243)(58,92,321,242)(59,91,322,241)(60,90,323,240)(61,89,324,239)(62,88,325,238)(63,87,326,237)(64,86,327,236)(65,85,328,235)(66,168,329,234)(67,167,330,233)(68,166,331,232)(69,165,332,231)(70,164,333,230)(71,163,334,229)(72,162,335,228)(73,161,336,227)(74,160,253,226)(75,159,254,225)(76,158,255,224)(77,157,256,223)(78,156,257,222)(79,155,258,221)(80,154,259,220)(81,153,260,219)(82,152,261,218)(83,151,262,217)(84,150,263,216) );
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)], [(1,149,264,215),(2,148,265,214),(3,147,266,213),(4,146,267,212),(5,145,268,211),(6,144,269,210),(7,143,270,209),(8,142,271,208),(9,141,272,207),(10,140,273,206),(11,139,274,205),(12,138,275,204),(13,137,276,203),(14,136,277,202),(15,135,278,201),(16,134,279,200),(17,133,280,199),(18,132,281,198),(19,131,282,197),(20,130,283,196),(21,129,284,195),(22,128,285,194),(23,127,286,193),(24,126,287,192),(25,125,288,191),(26,124,289,190),(27,123,290,189),(28,122,291,188),(29,121,292,187),(30,120,293,186),(31,119,294,185),(32,118,295,184),(33,117,296,183),(34,116,297,182),(35,115,298,181),(36,114,299,180),(37,113,300,179),(38,112,301,178),(39,111,302,177),(40,110,303,176),(41,109,304,175),(42,108,305,174),(43,107,306,173),(44,106,307,172),(45,105,308,171),(46,104,309,170),(47,103,310,169),(48,102,311,252),(49,101,312,251),(50,100,313,250),(51,99,314,249),(52,98,315,248),(53,97,316,247),(54,96,317,246),(55,95,318,245),(56,94,319,244),(57,93,320,243),(58,92,321,242),(59,91,322,241),(60,90,323,240),(61,89,324,239),(62,88,325,238),(63,87,326,237),(64,86,327,236),(65,85,328,235),(66,168,329,234),(67,167,330,233),(68,166,331,232),(69,165,332,231),(70,164,333,230),(71,163,334,229),(72,162,335,228),(73,161,336,227),(74,160,253,226),(75,159,254,225),(76,158,255,224),(77,157,256,223),(78,156,257,222),(79,155,258,221),(80,154,259,220),(81,153,260,219),(82,152,261,218),(83,151,262,217),(84,150,263,216)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 42 | 42 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | + | - | + | - | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D7 | Dic6 | D12 | Dic7 | D14 | D21 | Dic14 | D28 | Dic21 | D42 | Dic42 | D84 |
| kernel | C84⋊C4 | C2×Dic21 | C2×C84 | C84 | C2×C28 | C42 | C42 | C28 | C2×C14 | C2×C12 | C14 | C14 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 6 | 12 | 6 | 12 | 12 |
Matrix representation of C84⋊C4 ►in GL3(𝔽337) generated by
| 1 | 0 | 0 |
| 0 | 3 | 277 |
| 0 | 260 | 80 |
| 148 | 0 | 0 |
| 0 | 286 | 172 |
| 0 | 75 | 51 |
G:=sub<GL(3,GF(337))| [1,0,0,0,3,260,0,277,80],[148,0,0,0,286,75,0,172,51] >;
C84⋊C4 in GAP, Magma, Sage, TeX
C_{84}\rtimes C_4 % in TeX
G:=Group("C84:C4"); // GroupNames label
G:=SmallGroup(336,99);
// by ID
G=gap.SmallGroup(336,99);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,55,964,10373]);
// Polycyclic
G:=Group<a,b|a^84=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export