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## G = C4×C84order 336 = 24·3·7

### Abelian group of type [4,84]

Aliases: C4×C84, SmallGroup(336,106)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C84
 Chief series C1 — C2 — C22 — C2×C14 — C2×C42 — C2×C84 — C4×C84
 Lower central C1 — C4×C84
 Upper central C1 — C4×C84

Generators and relations for C4×C84
G = < a,b | a4=b84=1, ab=ba >

Subgroups: 60, all normal (12 characteristic)
C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C7, C2×C4 [×3], C12 [×6], C2×C6, C14 [×3], C42, C21, C2×C12 [×3], C28 [×6], C2×C14, C42 [×3], C4×C12, C2×C28 [×3], C84 [×6], C2×C42, C4×C28, C2×C84 [×3], C4×C84
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C7, C2×C4 [×3], C12 [×6], C2×C6, C14 [×3], C42, C21, C2×C12 [×3], C28 [×6], C2×C14, C42 [×3], C4×C12, C2×C28 [×3], C84 [×6], C2×C42, C4×C28, C2×C84 [×3], C4×C84

Smallest permutation representation of C4×C84
Regular action on 336 points
Generators in S336
(1 152 277 247)(2 153 278 248)(3 154 279 249)(4 155 280 250)(5 156 281 251)(6 157 282 252)(7 158 283 169)(8 159 284 170)(9 160 285 171)(10 161 286 172)(11 162 287 173)(12 163 288 174)(13 164 289 175)(14 165 290 176)(15 166 291 177)(16 167 292 178)(17 168 293 179)(18 85 294 180)(19 86 295 181)(20 87 296 182)(21 88 297 183)(22 89 298 184)(23 90 299 185)(24 91 300 186)(25 92 301 187)(26 93 302 188)(27 94 303 189)(28 95 304 190)(29 96 305 191)(30 97 306 192)(31 98 307 193)(32 99 308 194)(33 100 309 195)(34 101 310 196)(35 102 311 197)(36 103 312 198)(37 104 313 199)(38 105 314 200)(39 106 315 201)(40 107 316 202)(41 108 317 203)(42 109 318 204)(43 110 319 205)(44 111 320 206)(45 112 321 207)(46 113 322 208)(47 114 323 209)(48 115 324 210)(49 116 325 211)(50 117 326 212)(51 118 327 213)(52 119 328 214)(53 120 329 215)(54 121 330 216)(55 122 331 217)(56 123 332 218)(57 124 333 219)(58 125 334 220)(59 126 335 221)(60 127 336 222)(61 128 253 223)(62 129 254 224)(63 130 255 225)(64 131 256 226)(65 132 257 227)(66 133 258 228)(67 134 259 229)(68 135 260 230)(69 136 261 231)(70 137 262 232)(71 138 263 233)(72 139 264 234)(73 140 265 235)(74 141 266 236)(75 142 267 237)(76 143 268 238)(77 144 269 239)(78 145 270 240)(79 146 271 241)(80 147 272 242)(81 148 273 243)(82 149 274 244)(83 150 275 245)(84 151 276 246)
(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)

G:=sub<Sym(336)| (1,152,277,247)(2,153,278,248)(3,154,279,249)(4,155,280,250)(5,156,281,251)(6,157,282,252)(7,158,283,169)(8,159,284,170)(9,160,285,171)(10,161,286,172)(11,162,287,173)(12,163,288,174)(13,164,289,175)(14,165,290,176)(15,166,291,177)(16,167,292,178)(17,168,293,179)(18,85,294,180)(19,86,295,181)(20,87,296,182)(21,88,297,183)(22,89,298,184)(23,90,299,185)(24,91,300,186)(25,92,301,187)(26,93,302,188)(27,94,303,189)(28,95,304,190)(29,96,305,191)(30,97,306,192)(31,98,307,193)(32,99,308,194)(33,100,309,195)(34,101,310,196)(35,102,311,197)(36,103,312,198)(37,104,313,199)(38,105,314,200)(39,106,315,201)(40,107,316,202)(41,108,317,203)(42,109,318,204)(43,110,319,205)(44,111,320,206)(45,112,321,207)(46,113,322,208)(47,114,323,209)(48,115,324,210)(49,116,325,211)(50,117,326,212)(51,118,327,213)(52,119,328,214)(53,120,329,215)(54,121,330,216)(55,122,331,217)(56,123,332,218)(57,124,333,219)(58,125,334,220)(59,126,335,221)(60,127,336,222)(61,128,253,223)(62,129,254,224)(63,130,255,225)(64,131,256,226)(65,132,257,227)(66,133,258,228)(67,134,259,229)(68,135,260,230)(69,136,261,231)(70,137,262,232)(71,138,263,233)(72,139,264,234)(73,140,265,235)(74,141,266,236)(75,142,267,237)(76,143,268,238)(77,144,269,239)(78,145,270,240)(79,146,271,241)(80,147,272,242)(81,148,273,243)(82,149,274,244)(83,150,275,245)(84,151,276,246), (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)>;

G:=Group( (1,152,277,247)(2,153,278,248)(3,154,279,249)(4,155,280,250)(5,156,281,251)(6,157,282,252)(7,158,283,169)(8,159,284,170)(9,160,285,171)(10,161,286,172)(11,162,287,173)(12,163,288,174)(13,164,289,175)(14,165,290,176)(15,166,291,177)(16,167,292,178)(17,168,293,179)(18,85,294,180)(19,86,295,181)(20,87,296,182)(21,88,297,183)(22,89,298,184)(23,90,299,185)(24,91,300,186)(25,92,301,187)(26,93,302,188)(27,94,303,189)(28,95,304,190)(29,96,305,191)(30,97,306,192)(31,98,307,193)(32,99,308,194)(33,100,309,195)(34,101,310,196)(35,102,311,197)(36,103,312,198)(37,104,313,199)(38,105,314,200)(39,106,315,201)(40,107,316,202)(41,108,317,203)(42,109,318,204)(43,110,319,205)(44,111,320,206)(45,112,321,207)(46,113,322,208)(47,114,323,209)(48,115,324,210)(49,116,325,211)(50,117,326,212)(51,118,327,213)(52,119,328,214)(53,120,329,215)(54,121,330,216)(55,122,331,217)(56,123,332,218)(57,124,333,219)(58,125,334,220)(59,126,335,221)(60,127,336,222)(61,128,253,223)(62,129,254,224)(63,130,255,225)(64,131,256,226)(65,132,257,227)(66,133,258,228)(67,134,259,229)(68,135,260,230)(69,136,261,231)(70,137,262,232)(71,138,263,233)(72,139,264,234)(73,140,265,235)(74,141,266,236)(75,142,267,237)(76,143,268,238)(77,144,269,239)(78,145,270,240)(79,146,271,241)(80,147,272,242)(81,148,273,243)(82,149,274,244)(83,150,275,245)(84,151,276,246), (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) );

G=PermutationGroup([(1,152,277,247),(2,153,278,248),(3,154,279,249),(4,155,280,250),(5,156,281,251),(6,157,282,252),(7,158,283,169),(8,159,284,170),(9,160,285,171),(10,161,286,172),(11,162,287,173),(12,163,288,174),(13,164,289,175),(14,165,290,176),(15,166,291,177),(16,167,292,178),(17,168,293,179),(18,85,294,180),(19,86,295,181),(20,87,296,182),(21,88,297,183),(22,89,298,184),(23,90,299,185),(24,91,300,186),(25,92,301,187),(26,93,302,188),(27,94,303,189),(28,95,304,190),(29,96,305,191),(30,97,306,192),(31,98,307,193),(32,99,308,194),(33,100,309,195),(34,101,310,196),(35,102,311,197),(36,103,312,198),(37,104,313,199),(38,105,314,200),(39,106,315,201),(40,107,316,202),(41,108,317,203),(42,109,318,204),(43,110,319,205),(44,111,320,206),(45,112,321,207),(46,113,322,208),(47,114,323,209),(48,115,324,210),(49,116,325,211),(50,117,326,212),(51,118,327,213),(52,119,328,214),(53,120,329,215),(54,121,330,216),(55,122,331,217),(56,123,332,218),(57,124,333,219),(58,125,334,220),(59,126,335,221),(60,127,336,222),(61,128,253,223),(62,129,254,224),(63,130,255,225),(64,131,256,226),(65,132,257,227),(66,133,258,228),(67,134,259,229),(68,135,260,230),(69,136,261,231),(70,137,262,232),(71,138,263,233),(72,139,264,234),(73,140,265,235),(74,141,266,236),(75,142,267,237),(76,143,268,238),(77,144,269,239),(78,145,270,240),(79,146,271,241),(80,147,272,242),(81,148,273,243),(82,149,274,244),(83,150,275,245),(84,151,276,246)], [(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)])

336 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4L 6A ··· 6F 7A ··· 7F 12A ··· 12X 14A ··· 14R 21A ··· 21L 28A ··· 28BT 42A ··· 42AJ 84A ··· 84EN order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 7 ··· 7 12 ··· 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

336 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C7 C12 C14 C21 C28 C42 C84 kernel C4×C84 C2×C84 C4×C28 C84 C2×C28 C4×C12 C28 C2×C12 C42 C12 C2×C4 C4 # reps 1 3 2 12 6 6 24 18 12 72 36 144

Matrix representation of C4×C84 in GL2(𝔽337) generated by

 189 0 0 1
,
 16 0 0 72
G:=sub<GL(2,GF(337))| [189,0,0,1],[16,0,0,72] >;

C4×C84 in GAP, Magma, Sage, TeX

C_4\times C_{84}
% in TeX

G:=Group("C4xC84");
// GroupNames label

G:=SmallGroup(336,106);
// by ID

G=gap.SmallGroup(336,106);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,-2,504,1015]);
// Polycyclic

G:=Group<a,b|a^4=b^84=1,a*b=b*a>;
// generators/relations

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