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G = C4×C84  order 336 = 24·3·7

Abelian group of type [4,84]

direct product, abelian, monomial, 2-elementary

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, C3, C4, C22, C6, C7, C2×C4, C12, C2×C6, C14, C42, C21, C2×C12, C28, C2×C14, C42, C4×C12, C2×C28, C84, C2×C42, C4×C28, C2×C84, C4×C84
Quotients: C1, C2, C3, C4, C22, C6, C7, C2×C4, C12, C2×C6, C14, C42, C21, C2×C12, C28, C2×C14, C42, C4×C12, C2×C28, C84, C2×C42, C4×C28, C2×C84, C4×C84

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

(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)| (1,90,242,294)(2,91,243,295)(3,92,244,296)(4,93,245,297)(5,94,246,298)(6,95,247,299)(7,96,248,300)(8,97,249,301)(9,98,250,302)(10,99,251,303)(11,100,252,304)(12,101,169,305)(13,102,170,306)(14,103,171,307)(15,104,172,308)(16,105,173,309)(17,106,174,310)(18,107,175,311)(19,108,176,312)(20,109,177,313)(21,110,178,314)(22,111,179,315)(23,112,180,316)(24,113,181,317)(25,114,182,318)(26,115,183,319)(27,116,184,320)(28,117,185,321)(29,118,186,322)(30,119,187,323)(31,120,188,324)(32,121,189,325)(33,122,190,326)(34,123,191,327)(35,124,192,328)(36,125,193,329)(37,126,194,330)(38,127,195,331)(39,128,196,332)(40,129,197,333)(41,130,198,334)(42,131,199,335)(43,132,200,336)(44,133,201,253)(45,134,202,254)(46,135,203,255)(47,136,204,256)(48,137,205,257)(49,138,206,258)(50,139,207,259)(51,140,208,260)(52,141,209,261)(53,142,210,262)(54,143,211,263)(55,144,212,264)(56,145,213,265)(57,146,214,266)(58,147,215,267)(59,148,216,268)(60,149,217,269)(61,150,218,270)(62,151,219,271)(63,152,220,272)(64,153,221,273)(65,154,222,274)(66,155,223,275)(67,156,224,276)(68,157,225,277)(69,158,226,278)(70,159,227,279)(71,160,228,280)(72,161,229,281)(73,162,230,282)(74,163,231,283)(75,164,232,284)(76,165,233,285)(77,166,234,286)(78,167,235,287)(79,168,236,288)(80,85,237,289)(81,86,238,290)(82,87,239,291)(83,88,240,292)(84,89,241,293), (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( (1,90,242,294)(2,91,243,295)(3,92,244,296)(4,93,245,297)(5,94,246,298)(6,95,247,299)(7,96,248,300)(8,97,249,301)(9,98,250,302)(10,99,251,303)(11,100,252,304)(12,101,169,305)(13,102,170,306)(14,103,171,307)(15,104,172,308)(16,105,173,309)(17,106,174,310)(18,107,175,311)(19,108,176,312)(20,109,177,313)(21,110,178,314)(22,111,179,315)(23,112,180,316)(24,113,181,317)(25,114,182,318)(26,115,183,319)(27,116,184,320)(28,117,185,321)(29,118,186,322)(30,119,187,323)(31,120,188,324)(32,121,189,325)(33,122,190,326)(34,123,191,327)(35,124,192,328)(36,125,193,329)(37,126,194,330)(38,127,195,331)(39,128,196,332)(40,129,197,333)(41,130,198,334)(42,131,199,335)(43,132,200,336)(44,133,201,253)(45,134,202,254)(46,135,203,255)(47,136,204,256)(48,137,205,257)(49,138,206,258)(50,139,207,259)(51,140,208,260)(52,141,209,261)(53,142,210,262)(54,143,211,263)(55,144,212,264)(56,145,213,265)(57,146,214,266)(58,147,215,267)(59,148,216,268)(60,149,217,269)(61,150,218,270)(62,151,219,271)(63,152,220,272)(64,153,221,273)(65,154,222,274)(66,155,223,275)(67,156,224,276)(68,157,225,277)(69,158,226,278)(70,159,227,279)(71,160,228,280)(72,161,229,281)(73,162,230,282)(74,163,231,283)(75,164,232,284)(76,165,233,285)(77,166,234,286)(78,167,235,287)(79,168,236,288)(80,85,237,289)(81,86,238,290)(82,87,239,291)(83,88,240,292)(84,89,241,293), (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([[(1,90,242,294),(2,91,243,295),(3,92,244,296),(4,93,245,297),(5,94,246,298),(6,95,247,299),(7,96,248,300),(8,97,249,301),(9,98,250,302),(10,99,251,303),(11,100,252,304),(12,101,169,305),(13,102,170,306),(14,103,171,307),(15,104,172,308),(16,105,173,309),(17,106,174,310),(18,107,175,311),(19,108,176,312),(20,109,177,313),(21,110,178,314),(22,111,179,315),(23,112,180,316),(24,113,181,317),(25,114,182,318),(26,115,183,319),(27,116,184,320),(28,117,185,321),(29,118,186,322),(30,119,187,323),(31,120,188,324),(32,121,189,325),(33,122,190,326),(34,123,191,327),(35,124,192,328),(36,125,193,329),(37,126,194,330),(38,127,195,331),(39,128,196,332),(40,129,197,333),(41,130,198,334),(42,131,199,335),(43,132,200,336),(44,133,201,253),(45,134,202,254),(46,135,203,255),(47,136,204,256),(48,137,205,257),(49,138,206,258),(50,139,207,259),(51,140,208,260),(52,141,209,261),(53,142,210,262),(54,143,211,263),(55,144,212,264),(56,145,213,265),(57,146,214,266),(58,147,215,267),(59,148,216,268),(60,149,217,269),(61,150,218,270),(62,151,219,271),(63,152,220,272),(64,153,221,273),(65,154,222,274),(66,155,223,275),(67,156,224,276),(68,157,225,277),(69,158,226,278),(70,159,227,279),(71,160,228,280),(72,161,229,281),(73,162,230,282),(74,163,231,283),(75,164,232,284),(76,165,233,285),(77,166,234,286),(78,167,235,287),(79,168,236,288),(80,85,237,289),(81,86,238,290),(82,87,239,291),(83,88,240,292),(84,89,241,293)], [(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 2A2B2C3A3B4A···4L6A···6F7A···7F12A···12X14A···14R21A···21L28A···28BT42A···42AJ84A···84EN
order1222334···46···67···712···1214···1421···2128···2842···4284···84
size1111111···11···11···11···11···11···11···11···11···1

336 irreducible representations

dim111111111111
type++
imageC1C2C3C4C6C7C12C14C21C28C42C84
kernelC4×C84C2×C84C4×C28C84C2×C28C4×C12C28C2×C12C42C12C2×C4C4
# reps13212662418127236144

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

1890
01
,
160
072
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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