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G = Dic78order 312 = 23·3·13

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic78, C4.D39, C392Q8, C52.1S3, C2.3D78, C6.8D26, C26.8D6, C32Dic26, C132Dic6, C156.1C2, C12.1D13, C78.8C22, Dic39.1C2, SmallGroup(312,37)

Series: Derived Chief Lower central Upper central

C1C78 — Dic78
C1C13C39C78Dic39 — Dic78
C39C78 — Dic78
C1C2C4

Generators and relations for Dic78
 G = < a,b | a156=1, b2=a78, bab-1=a-1 >

39C4
39C4
39Q8
13Dic3
13Dic3
3Dic13
3Dic13
13Dic6
3Dic26

Smallest permutation representation of Dic78
Regular action on 312 points
Generators in S312
(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)
(1 195 79 273)(2 194 80 272)(3 193 81 271)(4 192 82 270)(5 191 83 269)(6 190 84 268)(7 189 85 267)(8 188 86 266)(9 187 87 265)(10 186 88 264)(11 185 89 263)(12 184 90 262)(13 183 91 261)(14 182 92 260)(15 181 93 259)(16 180 94 258)(17 179 95 257)(18 178 96 256)(19 177 97 255)(20 176 98 254)(21 175 99 253)(22 174 100 252)(23 173 101 251)(24 172 102 250)(25 171 103 249)(26 170 104 248)(27 169 105 247)(28 168 106 246)(29 167 107 245)(30 166 108 244)(31 165 109 243)(32 164 110 242)(33 163 111 241)(34 162 112 240)(35 161 113 239)(36 160 114 238)(37 159 115 237)(38 158 116 236)(39 157 117 235)(40 312 118 234)(41 311 119 233)(42 310 120 232)(43 309 121 231)(44 308 122 230)(45 307 123 229)(46 306 124 228)(47 305 125 227)(48 304 126 226)(49 303 127 225)(50 302 128 224)(51 301 129 223)(52 300 130 222)(53 299 131 221)(54 298 132 220)(55 297 133 219)(56 296 134 218)(57 295 135 217)(58 294 136 216)(59 293 137 215)(60 292 138 214)(61 291 139 213)(62 290 140 212)(63 289 141 211)(64 288 142 210)(65 287 143 209)(66 286 144 208)(67 285 145 207)(68 284 146 206)(69 283 147 205)(70 282 148 204)(71 281 149 203)(72 280 150 202)(73 279 151 201)(74 278 152 200)(75 277 153 199)(76 276 154 198)(77 275 155 197)(78 274 156 196)

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

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

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

81 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order1234446121213···1326···2639···3952···5278···78156···156
size112278782222···22···22···22···22···22···2

81 irreducible representations

dim1112222222222
type++++-+-+++-+-
imageC1C2C2S3Q8D6Dic6D13D26D39Dic26D78Dic78
kernelDic78Dic39C156C52C39C26C13C12C6C4C3C2C1
# reps12111126612121224

Matrix representation of Dic78 in GL2(𝔽157) generated by

11998
5942
,
150103
307
G:=sub<GL(2,GF(157))| [119,59,98,42],[150,30,103,7] >;

Dic78 in GAP, Magma, Sage, TeX

{\rm Dic}_{78}
% in TeX

G:=Group("Dic78");
// GroupNames label

G:=SmallGroup(312,37);
// by ID

G=gap.SmallGroup(312,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,20,61,26,323,7204]);
// Polycyclic

G:=Group<a,b|a^156=1,b^2=a^78,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic78 in TeX

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