direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic11, C55⋊4C4, C11⋊3C20, C22.3C10, C110.2C2, C10.2D11, C2.(C5×D11), SmallGroup(220,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C5×Dic11 |
Generators and relations for C5×Dic11
G = < a,b,c | a5=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic11
►Regular action on 220 points
Generators in S220
(1 110 71 66 26)(2 89 72 45 27)(3 90 73 46 28)(4 91 74 47 29)(5 92 75 48 30)(6 93 76 49 31)(7 94 77 50 32)(8 95 78 51 33)(9 96 79 52 34)(10 97 80 53 35)(11 98 81 54 36)(12 99 82 55 37)(13 100 83 56 38)(14 101 84 57 39)(15 102 85 58 40)(16 103 86 59 41)(17 104 87 60 42)(18 105 88 61 43)(19 106 67 62 44)(20 107 68 63 23)(21 108 69 64 24)(22 109 70 65 25)(111 210 188 155 144)(112 211 189 156 145)(113 212 190 157 146)(114 213 191 158 147)(115 214 192 159 148)(116 215 193 160 149)(117 216 194 161 150)(118 217 195 162 151)(119 218 196 163 152)(120 219 197 164 153)(121 220 198 165 154)(122 199 177 166 133)(123 200 178 167 134)(124 201 179 168 135)(125 202 180 169 136)(126 203 181 170 137)(127 204 182 171 138)(128 205 183 172 139)(129 206 184 173 140)(130 207 185 174 141)(131 208 186 175 142)(132 209 187 176 143)
(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)
(1 122 12 111)(2 121 13 132)(3 120 14 131)(4 119 15 130)(5 118 16 129)(6 117 17 128)(7 116 18 127)(8 115 19 126)(9 114 20 125)(10 113 21 124)(11 112 22 123)(23 136 34 147)(24 135 35 146)(25 134 36 145)(26 133 37 144)(27 154 38 143)(28 153 39 142)(29 152 40 141)(30 151 41 140)(31 150 42 139)(32 149 43 138)(33 148 44 137)(45 165 56 176)(46 164 57 175)(47 163 58 174)(48 162 59 173)(49 161 60 172)(50 160 61 171)(51 159 62 170)(52 158 63 169)(53 157 64 168)(54 156 65 167)(55 155 66 166)(67 181 78 192)(68 180 79 191)(69 179 80 190)(70 178 81 189)(71 177 82 188)(72 198 83 187)(73 197 84 186)(74 196 85 185)(75 195 86 184)(76 194 87 183)(77 193 88 182)(89 220 100 209)(90 219 101 208)(91 218 102 207)(92 217 103 206)(93 216 104 205)(94 215 105 204)(95 214 106 203)(96 213 107 202)(97 212 108 201)(98 211 109 200)(99 210 110 199)
G:=sub<Sym(220)| (1,110,71,66,26)(2,89,72,45,27)(3,90,73,46,28)(4,91,74,47,29)(5,92,75,48,30)(6,93,76,49,31)(7,94,77,50,32)(8,95,78,51,33)(9,96,79,52,34)(10,97,80,53,35)(11,98,81,54,36)(12,99,82,55,37)(13,100,83,56,38)(14,101,84,57,39)(15,102,85,58,40)(16,103,86,59,41)(17,104,87,60,42)(18,105,88,61,43)(19,106,67,62,44)(20,107,68,63,23)(21,108,69,64,24)(22,109,70,65,25)(111,210,188,155,144)(112,211,189,156,145)(113,212,190,157,146)(114,213,191,158,147)(115,214,192,159,148)(116,215,193,160,149)(117,216,194,161,150)(118,217,195,162,151)(119,218,196,163,152)(120,219,197,164,153)(121,220,198,165,154)(122,199,177,166,133)(123,200,178,167,134)(124,201,179,168,135)(125,202,180,169,136)(126,203,181,170,137)(127,204,182,171,138)(128,205,183,172,139)(129,206,184,173,140)(130,207,185,174,141)(131,208,186,175,142)(132,209,187,176,143), (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), (1,122,12,111)(2,121,13,132)(3,120,14,131)(4,119,15,130)(5,118,16,129)(6,117,17,128)(7,116,18,127)(8,115,19,126)(9,114,20,125)(10,113,21,124)(11,112,22,123)(23,136,34,147)(24,135,35,146)(25,134,36,145)(26,133,37,144)(27,154,38,143)(28,153,39,142)(29,152,40,141)(30,151,41,140)(31,150,42,139)(32,149,43,138)(33,148,44,137)(45,165,56,176)(46,164,57,175)(47,163,58,174)(48,162,59,173)(49,161,60,172)(50,160,61,171)(51,159,62,170)(52,158,63,169)(53,157,64,168)(54,156,65,167)(55,155,66,166)(67,181,78,192)(68,180,79,191)(69,179,80,190)(70,178,81,189)(71,177,82,188)(72,198,83,187)(73,197,84,186)(74,196,85,185)(75,195,86,184)(76,194,87,183)(77,193,88,182)(89,220,100,209)(90,219,101,208)(91,218,102,207)(92,217,103,206)(93,216,104,205)(94,215,105,204)(95,214,106,203)(96,213,107,202)(97,212,108,201)(98,211,109,200)(99,210,110,199)>;
G:=Group( (1,110,71,66,26)(2,89,72,45,27)(3,90,73,46,28)(4,91,74,47,29)(5,92,75,48,30)(6,93,76,49,31)(7,94,77,50,32)(8,95,78,51,33)(9,96,79,52,34)(10,97,80,53,35)(11,98,81,54,36)(12,99,82,55,37)(13,100,83,56,38)(14,101,84,57,39)(15,102,85,58,40)(16,103,86,59,41)(17,104,87,60,42)(18,105,88,61,43)(19,106,67,62,44)(20,107,68,63,23)(21,108,69,64,24)(22,109,70,65,25)(111,210,188,155,144)(112,211,189,156,145)(113,212,190,157,146)(114,213,191,158,147)(115,214,192,159,148)(116,215,193,160,149)(117,216,194,161,150)(118,217,195,162,151)(119,218,196,163,152)(120,219,197,164,153)(121,220,198,165,154)(122,199,177,166,133)(123,200,178,167,134)(124,201,179,168,135)(125,202,180,169,136)(126,203,181,170,137)(127,204,182,171,138)(128,205,183,172,139)(129,206,184,173,140)(130,207,185,174,141)(131,208,186,175,142)(132,209,187,176,143), (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), (1,122,12,111)(2,121,13,132)(3,120,14,131)(4,119,15,130)(5,118,16,129)(6,117,17,128)(7,116,18,127)(8,115,19,126)(9,114,20,125)(10,113,21,124)(11,112,22,123)(23,136,34,147)(24,135,35,146)(25,134,36,145)(26,133,37,144)(27,154,38,143)(28,153,39,142)(29,152,40,141)(30,151,41,140)(31,150,42,139)(32,149,43,138)(33,148,44,137)(45,165,56,176)(46,164,57,175)(47,163,58,174)(48,162,59,173)(49,161,60,172)(50,160,61,171)(51,159,62,170)(52,158,63,169)(53,157,64,168)(54,156,65,167)(55,155,66,166)(67,181,78,192)(68,180,79,191)(69,179,80,190)(70,178,81,189)(71,177,82,188)(72,198,83,187)(73,197,84,186)(74,196,85,185)(75,195,86,184)(76,194,87,183)(77,193,88,182)(89,220,100,209)(90,219,101,208)(91,218,102,207)(92,217,103,206)(93,216,104,205)(94,215,105,204)(95,214,106,203)(96,213,107,202)(97,212,108,201)(98,211,109,200)(99,210,110,199) );
G=PermutationGroup([[(1,110,71,66,26),(2,89,72,45,27),(3,90,73,46,28),(4,91,74,47,29),(5,92,75,48,30),(6,93,76,49,31),(7,94,77,50,32),(8,95,78,51,33),(9,96,79,52,34),(10,97,80,53,35),(11,98,81,54,36),(12,99,82,55,37),(13,100,83,56,38),(14,101,84,57,39),(15,102,85,58,40),(16,103,86,59,41),(17,104,87,60,42),(18,105,88,61,43),(19,106,67,62,44),(20,107,68,63,23),(21,108,69,64,24),(22,109,70,65,25),(111,210,188,155,144),(112,211,189,156,145),(113,212,190,157,146),(114,213,191,158,147),(115,214,192,159,148),(116,215,193,160,149),(117,216,194,161,150),(118,217,195,162,151),(119,218,196,163,152),(120,219,197,164,153),(121,220,198,165,154),(122,199,177,166,133),(123,200,178,167,134),(124,201,179,168,135),(125,202,180,169,136),(126,203,181,170,137),(127,204,182,171,138),(128,205,183,172,139),(129,206,184,173,140),(130,207,185,174,141),(131,208,186,175,142),(132,209,187,176,143)], [(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)], [(1,122,12,111),(2,121,13,132),(3,120,14,131),(4,119,15,130),(5,118,16,129),(6,117,17,128),(7,116,18,127),(8,115,19,126),(9,114,20,125),(10,113,21,124),(11,112,22,123),(23,136,34,147),(24,135,35,146),(25,134,36,145),(26,133,37,144),(27,154,38,143),(28,153,39,142),(29,152,40,141),(30,151,41,140),(31,150,42,139),(32,149,43,138),(33,148,44,137),(45,165,56,176),(46,164,57,175),(47,163,58,174),(48,162,59,173),(49,161,60,172),(50,160,61,171),(51,159,62,170),(52,158,63,169),(53,157,64,168),(54,156,65,167),(55,155,66,166),(67,181,78,192),(68,180,79,191),(69,179,80,190),(70,178,81,189),(71,177,82,188),(72,198,83,187),(73,197,84,186),(74,196,85,185),(75,195,86,184),(76,194,87,183),(77,193,88,182),(89,220,100,209),(90,219,101,208),(91,218,102,207),(92,217,103,206),(93,216,104,205),(94,215,105,204),(95,214,106,203),(96,213,107,202),(97,212,108,201),(98,211,109,200),(99,210,110,199)]])
C5×Dic11 is a maximal subgroup of
D55⋊2C4 C11⋊D20 C55⋊Q8 C20×D11
70 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11E | 20A | ··· | 20H | 22A | ··· | 22E | 55A | ··· | 55T | 110A | ··· | 110T |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 20 | ··· | 20 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D11 | Dic11 | C5×D11 | C5×Dic11 |
| kernel | C5×Dic11 | C110 | C55 | Dic11 | C22 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 5 | 5 | 20 | 20 |
Matrix representation of C5×Dic11 ►in GL3(𝔽661) generated by
| 1 | 0 | 0 |
| 0 | 471 | 0 |
| 0 | 0 | 471 |
| 660 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 660 | 585 |
| 555 | 0 | 0 |
| 0 | 511 | 116 |
| 0 | 279 | 150 |
G:=sub<GL(3,GF(661))| [1,0,0,0,471,0,0,0,471],[660,0,0,0,0,660,0,1,585],[555,0,0,0,511,279,0,116,150] >;
C5×Dic11 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{11} % in TeX
G:=Group("C5xDic11"); // GroupNames label
G:=SmallGroup(220,4);
// by ID
G=gap.SmallGroup(220,4);
# by ID
G:=PCGroup([4,-2,-5,-2,-11,40,3203]);
// Polycyclic
G:=Group<a,b,c|a^5=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export