Copied to
clipboard

G = C5×Dic14order 280 = 23·5·7

Direct product of C5 and Dic14

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×Dic14, C353Q8, C20.3D7, C140.3C2, C28.1C10, Dic7.C10, C10.13D14, C70.13C22, C7⋊(C5×Q8), C4.(C5×D7), C2.3(C10×D7), C14.1(C2×C10), (C5×Dic7).2C2, SmallGroup(280,14)

Series: Derived Chief Lower central Upper central

C1C14 — C5×Dic14
C1C7C14C70C5×Dic7 — C5×Dic14
C7C14 — C5×Dic14
C1C10C20

Generators and relations for C5×Dic14
 G = < a,b,c | a5=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C4
7Q8
7C20
7C20
7C5×Q8

Smallest permutation representation of C5×Dic14
Regular action on 280 points
Generators in S280
(1 249 145 280 74)(2 250 146 253 75)(3 251 147 254 76)(4 252 148 255 77)(5 225 149 256 78)(6 226 150 257 79)(7 227 151 258 80)(8 228 152 259 81)(9 229 153 260 82)(10 230 154 261 83)(11 231 155 262 84)(12 232 156 263 57)(13 233 157 264 58)(14 234 158 265 59)(15 235 159 266 60)(16 236 160 267 61)(17 237 161 268 62)(18 238 162 269 63)(19 239 163 270 64)(20 240 164 271 65)(21 241 165 272 66)(22 242 166 273 67)(23 243 167 274 68)(24 244 168 275 69)(25 245 141 276 70)(26 246 142 277 71)(27 247 143 278 72)(28 248 144 279 73)(29 94 138 197 185)(30 95 139 198 186)(31 96 140 199 187)(32 97 113 200 188)(33 98 114 201 189)(34 99 115 202 190)(35 100 116 203 191)(36 101 117 204 192)(37 102 118 205 193)(38 103 119 206 194)(39 104 120 207 195)(40 105 121 208 196)(41 106 122 209 169)(42 107 123 210 170)(43 108 124 211 171)(44 109 125 212 172)(45 110 126 213 173)(46 111 127 214 174)(47 112 128 215 175)(48 85 129 216 176)(49 86 130 217 177)(50 87 131 218 178)(51 88 132 219 179)(52 89 133 220 180)(53 90 134 221 181)(54 91 135 222 182)(55 92 136 223 183)(56 93 137 224 184)
(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)
(1 106 15 92)(2 105 16 91)(3 104 17 90)(4 103 18 89)(5 102 19 88)(6 101 20 87)(7 100 21 86)(8 99 22 85)(9 98 23 112)(10 97 24 111)(11 96 25 110)(12 95 26 109)(13 94 27 108)(14 93 28 107)(29 72 43 58)(30 71 44 57)(31 70 45 84)(32 69 46 83)(33 68 47 82)(34 67 48 81)(35 66 49 80)(36 65 50 79)(37 64 51 78)(38 63 52 77)(39 62 53 76)(40 61 54 75)(41 60 55 74)(42 59 56 73)(113 244 127 230)(114 243 128 229)(115 242 129 228)(116 241 130 227)(117 240 131 226)(118 239 132 225)(119 238 133 252)(120 237 134 251)(121 236 135 250)(122 235 136 249)(123 234 137 248)(124 233 138 247)(125 232 139 246)(126 231 140 245)(141 213 155 199)(142 212 156 198)(143 211 157 197)(144 210 158 224)(145 209 159 223)(146 208 160 222)(147 207 161 221)(148 206 162 220)(149 205 163 219)(150 204 164 218)(151 203 165 217)(152 202 166 216)(153 201 167 215)(154 200 168 214)(169 266 183 280)(170 265 184 279)(171 264 185 278)(172 263 186 277)(173 262 187 276)(174 261 188 275)(175 260 189 274)(176 259 190 273)(177 258 191 272)(178 257 192 271)(179 256 193 270)(180 255 194 269)(181 254 195 268)(182 253 196 267)

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

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

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

85 conjugacy classes

class 1  2 4A4B4C5A5B5C5D7A7B7C10A10B10C10D14A14B14C20A20B20C20D20E···20L28A···28F35A···35L70A···70L140A···140X
order124445555777101010101414142020202020···2028···2835···3570···70140···140
size112141411112221111222222214···142···22···22···22···2

85 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C5C10C10Q8D7D14C5×Q8Dic14C5×D7C10×D7C5×Dic14
kernelC5×Dic14C5×Dic7C140Dic14Dic7C28C35C20C10C7C5C4C2C1
# reps12148413346121224

Matrix representation of C5×Dic14 in GL2(𝔽281) generated by

2320
0232
,
32112
16991
,
20576
4676
G:=sub<GL(2,GF(281))| [232,0,0,232],[32,169,112,91],[205,46,76,76] >;

C5×Dic14 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{14}
% in TeX

G:=Group("C5xDic14");
// GroupNames label

G:=SmallGroup(280,14);
// by ID

G=gap.SmallGroup(280,14);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-7,100,221,106,6004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic14 in TeX

׿
×
𝔽