Copied to
clipboard

G = Dic74order 296 = 23·37

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic74, C37⋊Q8, C4.D37, C2.3D74, C148.1C2, C74.1C22, Dic37.1C2, SmallGroup(296,4)

Series: Derived Chief Lower central Upper central

C1C74 — Dic74
C1C37C74Dic37 — Dic74
C37C74 — Dic74
C1C2C4

Generators and relations for Dic74
 G = < a,b | a148=1, b2=a74, bab-1=a-1 >

37C4
37C4
37Q8

Smallest permutation representation of Dic74
Regular action on 296 points
Generators in S296
(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)
(1 171 75 245)(2 170 76 244)(3 169 77 243)(4 168 78 242)(5 167 79 241)(6 166 80 240)(7 165 81 239)(8 164 82 238)(9 163 83 237)(10 162 84 236)(11 161 85 235)(12 160 86 234)(13 159 87 233)(14 158 88 232)(15 157 89 231)(16 156 90 230)(17 155 91 229)(18 154 92 228)(19 153 93 227)(20 152 94 226)(21 151 95 225)(22 150 96 224)(23 149 97 223)(24 296 98 222)(25 295 99 221)(26 294 100 220)(27 293 101 219)(28 292 102 218)(29 291 103 217)(30 290 104 216)(31 289 105 215)(32 288 106 214)(33 287 107 213)(34 286 108 212)(35 285 109 211)(36 284 110 210)(37 283 111 209)(38 282 112 208)(39 281 113 207)(40 280 114 206)(41 279 115 205)(42 278 116 204)(43 277 117 203)(44 276 118 202)(45 275 119 201)(46 274 120 200)(47 273 121 199)(48 272 122 198)(49 271 123 197)(50 270 124 196)(51 269 125 195)(52 268 126 194)(53 267 127 193)(54 266 128 192)(55 265 129 191)(56 264 130 190)(57 263 131 189)(58 262 132 188)(59 261 133 187)(60 260 134 186)(61 259 135 185)(62 258 136 184)(63 257 137 183)(64 256 138 182)(65 255 139 181)(66 254 140 180)(67 253 141 179)(68 252 142 178)(69 251 143 177)(70 250 144 176)(71 249 145 175)(72 248 146 174)(73 247 147 173)(74 246 148 172)

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

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

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

77 conjugacy classes

class 1  2 4A4B4C37A···37R74A···74R148A···148AJ
order1244437···3774···74148···148
size11274742···22···22···2

77 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D37D74Dic74
kernelDic74Dic37C148C37C4C2C1
# reps1211181836

Matrix representation of Dic74 in GL2(𝔽149) generated by

29102
4778
,
6858
4181
G:=sub<GL(2,GF(149))| [29,47,102,78],[68,41,58,81] >;

Dic74 in GAP, Magma, Sage, TeX

{\rm Dic}_{74}
% in TeX

G:=Group("Dic74");
// GroupNames label

G:=SmallGroup(296,4);
// by ID

G=gap.SmallGroup(296,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-37,16,49,21,4611]);
// Polycyclic

G:=Group<a,b|a^148=1,b^2=a^74,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic74 in TeX

׿
×
𝔽