direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×Dic5, C55⋊5C4, C5⋊2C44, C10.C22, C22.2D5, C110.3C2, C2.(D5×C11), SmallGroup(220,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C11×Dic5 |
Generators and relations for C11×Dic5
G = < a,b,c | a11=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×Dic5
►Regular action on 220 points
Generators in S220
(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)
(1 139 154 208 182 176 165 128 193 116)(2 140 144 209 183 166 155 129 194 117)(3 141 145 199 184 167 156 130 195 118)(4 142 146 200 185 168 157 131 196 119)(5 143 147 201 186 169 158 132 197 120)(6 133 148 202 187 170 159 122 198 121)(7 134 149 203 177 171 160 123 188 111)(8 135 150 204 178 172 161 124 189 112)(9 136 151 205 179 173 162 125 190 113)(10 137 152 206 180 174 163 126 191 114)(11 138 153 207 181 175 164 127 192 115)(12 39 105 215 92 78 23 50 61 67)(13 40 106 216 93 79 24 51 62 68)(14 41 107 217 94 80 25 52 63 69)(15 42 108 218 95 81 26 53 64 70)(16 43 109 219 96 82 27 54 65 71)(17 44 110 220 97 83 28 55 66 72)(18 34 100 210 98 84 29 45 56 73)(19 35 101 211 99 85 30 46 57 74)(20 36 102 212 89 86 31 47 58 75)(21 37 103 213 90 87 32 48 59 76)(22 38 104 214 91 88 33 49 60 77)
(1 56 176 210)(2 57 166 211)(3 58 167 212)(4 59 168 213)(5 60 169 214)(6 61 170 215)(7 62 171 216)(8 63 172 217)(9 64 173 218)(10 65 174 219)(11 66 175 220)(12 202 78 198)(13 203 79 188)(14 204 80 189)(15 205 81 190)(16 206 82 191)(17 207 83 192)(18 208 84 193)(19 209 85 194)(20 199 86 195)(21 200 87 196)(22 201 88 197)(23 122 39 148)(24 123 40 149)(25 124 41 150)(26 125 42 151)(27 126 43 152)(28 127 44 153)(29 128 34 154)(30 129 35 144)(31 130 36 145)(32 131 37 146)(33 132 38 147)(45 165 100 139)(46 155 101 140)(47 156 102 141)(48 157 103 142)(49 158 104 143)(50 159 105 133)(51 160 106 134)(52 161 107 135)(53 162 108 136)(54 163 109 137)(55 164 110 138)(67 187 92 121)(68 177 93 111)(69 178 94 112)(70 179 95 113)(71 180 96 114)(72 181 97 115)(73 182 98 116)(74 183 99 117)(75 184 89 118)(76 185 90 119)(77 186 91 120)
G:=sub<Sym(220)| (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), (1,139,154,208,182,176,165,128,193,116)(2,140,144,209,183,166,155,129,194,117)(3,141,145,199,184,167,156,130,195,118)(4,142,146,200,185,168,157,131,196,119)(5,143,147,201,186,169,158,132,197,120)(6,133,148,202,187,170,159,122,198,121)(7,134,149,203,177,171,160,123,188,111)(8,135,150,204,178,172,161,124,189,112)(9,136,151,205,179,173,162,125,190,113)(10,137,152,206,180,174,163,126,191,114)(11,138,153,207,181,175,164,127,192,115)(12,39,105,215,92,78,23,50,61,67)(13,40,106,216,93,79,24,51,62,68)(14,41,107,217,94,80,25,52,63,69)(15,42,108,218,95,81,26,53,64,70)(16,43,109,219,96,82,27,54,65,71)(17,44,110,220,97,83,28,55,66,72)(18,34,100,210,98,84,29,45,56,73)(19,35,101,211,99,85,30,46,57,74)(20,36,102,212,89,86,31,47,58,75)(21,37,103,213,90,87,32,48,59,76)(22,38,104,214,91,88,33,49,60,77), (1,56,176,210)(2,57,166,211)(3,58,167,212)(4,59,168,213)(5,60,169,214)(6,61,170,215)(7,62,171,216)(8,63,172,217)(9,64,173,218)(10,65,174,219)(11,66,175,220)(12,202,78,198)(13,203,79,188)(14,204,80,189)(15,205,81,190)(16,206,82,191)(17,207,83,192)(18,208,84,193)(19,209,85,194)(20,199,86,195)(21,200,87,196)(22,201,88,197)(23,122,39,148)(24,123,40,149)(25,124,41,150)(26,125,42,151)(27,126,43,152)(28,127,44,153)(29,128,34,154)(30,129,35,144)(31,130,36,145)(32,131,37,146)(33,132,38,147)(45,165,100,139)(46,155,101,140)(47,156,102,141)(48,157,103,142)(49,158,104,143)(50,159,105,133)(51,160,106,134)(52,161,107,135)(53,162,108,136)(54,163,109,137)(55,164,110,138)(67,187,92,121)(68,177,93,111)(69,178,94,112)(70,179,95,113)(71,180,96,114)(72,181,97,115)(73,182,98,116)(74,183,99,117)(75,184,89,118)(76,185,90,119)(77,186,91,120)>;
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), (1,139,154,208,182,176,165,128,193,116)(2,140,144,209,183,166,155,129,194,117)(3,141,145,199,184,167,156,130,195,118)(4,142,146,200,185,168,157,131,196,119)(5,143,147,201,186,169,158,132,197,120)(6,133,148,202,187,170,159,122,198,121)(7,134,149,203,177,171,160,123,188,111)(8,135,150,204,178,172,161,124,189,112)(9,136,151,205,179,173,162,125,190,113)(10,137,152,206,180,174,163,126,191,114)(11,138,153,207,181,175,164,127,192,115)(12,39,105,215,92,78,23,50,61,67)(13,40,106,216,93,79,24,51,62,68)(14,41,107,217,94,80,25,52,63,69)(15,42,108,218,95,81,26,53,64,70)(16,43,109,219,96,82,27,54,65,71)(17,44,110,220,97,83,28,55,66,72)(18,34,100,210,98,84,29,45,56,73)(19,35,101,211,99,85,30,46,57,74)(20,36,102,212,89,86,31,47,58,75)(21,37,103,213,90,87,32,48,59,76)(22,38,104,214,91,88,33,49,60,77), (1,56,176,210)(2,57,166,211)(3,58,167,212)(4,59,168,213)(5,60,169,214)(6,61,170,215)(7,62,171,216)(8,63,172,217)(9,64,173,218)(10,65,174,219)(11,66,175,220)(12,202,78,198)(13,203,79,188)(14,204,80,189)(15,205,81,190)(16,206,82,191)(17,207,83,192)(18,208,84,193)(19,209,85,194)(20,199,86,195)(21,200,87,196)(22,201,88,197)(23,122,39,148)(24,123,40,149)(25,124,41,150)(26,125,42,151)(27,126,43,152)(28,127,44,153)(29,128,34,154)(30,129,35,144)(31,130,36,145)(32,131,37,146)(33,132,38,147)(45,165,100,139)(46,155,101,140)(47,156,102,141)(48,157,103,142)(49,158,104,143)(50,159,105,133)(51,160,106,134)(52,161,107,135)(53,162,108,136)(54,163,109,137)(55,164,110,138)(67,187,92,121)(68,177,93,111)(69,178,94,112)(70,179,95,113)(71,180,96,114)(72,181,97,115)(73,182,98,116)(74,183,99,117)(75,184,89,118)(76,185,90,119)(77,186,91,120) );
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)], [(1,139,154,208,182,176,165,128,193,116),(2,140,144,209,183,166,155,129,194,117),(3,141,145,199,184,167,156,130,195,118),(4,142,146,200,185,168,157,131,196,119),(5,143,147,201,186,169,158,132,197,120),(6,133,148,202,187,170,159,122,198,121),(7,134,149,203,177,171,160,123,188,111),(8,135,150,204,178,172,161,124,189,112),(9,136,151,205,179,173,162,125,190,113),(10,137,152,206,180,174,163,126,191,114),(11,138,153,207,181,175,164,127,192,115),(12,39,105,215,92,78,23,50,61,67),(13,40,106,216,93,79,24,51,62,68),(14,41,107,217,94,80,25,52,63,69),(15,42,108,218,95,81,26,53,64,70),(16,43,109,219,96,82,27,54,65,71),(17,44,110,220,97,83,28,55,66,72),(18,34,100,210,98,84,29,45,56,73),(19,35,101,211,99,85,30,46,57,74),(20,36,102,212,89,86,31,47,58,75),(21,37,103,213,90,87,32,48,59,76),(22,38,104,214,91,88,33,49,60,77)], [(1,56,176,210),(2,57,166,211),(3,58,167,212),(4,59,168,213),(5,60,169,214),(6,61,170,215),(7,62,171,216),(8,63,172,217),(9,64,173,218),(10,65,174,219),(11,66,175,220),(12,202,78,198),(13,203,79,188),(14,204,80,189),(15,205,81,190),(16,206,82,191),(17,207,83,192),(18,208,84,193),(19,209,85,194),(20,199,86,195),(21,200,87,196),(22,201,88,197),(23,122,39,148),(24,123,40,149),(25,124,41,150),(26,125,42,151),(27,126,43,152),(28,127,44,153),(29,128,34,154),(30,129,35,144),(31,130,36,145),(32,131,37,146),(33,132,38,147),(45,165,100,139),(46,155,101,140),(47,156,102,141),(48,157,103,142),(49,158,104,143),(50,159,105,133),(51,160,106,134),(52,161,107,135),(53,162,108,136),(54,163,109,137),(55,164,110,138),(67,187,92,121),(68,177,93,111),(69,178,94,112),(70,179,95,113),(71,180,96,114),(72,181,97,115),(73,182,98,116),(74,183,99,117),(75,184,89,118),(76,185,90,119),(77,186,91,120)]])
C11×Dic5 is a maximal subgroup of
C55⋊C8 D55⋊2C4 C5⋊D44 C55⋊Q8 D5×C44
88 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 11A | ··· | 11J | 22A | ··· | 22J | 44A | ··· | 44T | 55A | ··· | 55T | 110A | ··· | 110T |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C11 | C22 | C44 | D5 | Dic5 | D5×C11 | C11×Dic5 |
| kernel | C11×Dic5 | C110 | C55 | Dic5 | C10 | C5 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 2 | 2 | 20 | 20 |
Matrix representation of C11×Dic5 ►in GL2(𝔽661) generated by
| 81 | 0 |
| 0 | 81 |
| 1 | 660 |
| 59 | 603 |
| 0 | 568 |
| 462 | 0 |
G:=sub<GL(2,GF(661))| [81,0,0,81],[1,59,660,603],[0,462,568,0] >;
C11×Dic5 in GAP, Magma, Sage, TeX
C_{11}\times {\rm Dic}_5 % in TeX
G:=Group("C11xDic5"); // GroupNames label
G:=SmallGroup(220,3);
// by ID
G=gap.SmallGroup(220,3);
# by ID
G:=PCGroup([4,-2,-11,-2,-5,88,2819]);
// Polycyclic
G:=Group<a,b,c|a^11=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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