metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13⋊2C16, C26.2C8, C52.5C4, C8.2D13, C104.2C2, C4.2Dic13, C2.(C13⋊2C8), SmallGroup(208,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊2C16 |
Generators and relations for C13⋊2C16
G = < a,b | a13=b16=1, bab-1=a-1 >
Smallest permutation representation of C13⋊2C16
►Regular action on 208 points
Generators in S208
(1 80 155 92 204 39 190 129 62 30 100 167 117)(2 118 168 101 31 63 130 191 40 205 93 156 65)(3 66 157 94 206 41 192 131 64 32 102 169 119)(4 120 170 103 17 49 132 177 42 207 95 158 67)(5 68 159 96 208 43 178 133 50 18 104 171 121)(6 122 172 105 19 51 134 179 44 193 81 160 69)(7 70 145 82 194 45 180 135 52 20 106 173 123)(8 124 174 107 21 53 136 181 46 195 83 146 71)(9 72 147 84 196 47 182 137 54 22 108 175 125)(10 126 176 109 23 55 138 183 48 197 85 148 73)(11 74 149 86 198 33 184 139 56 24 110 161 127)(12 128 162 111 25 57 140 185 34 199 87 150 75)(13 76 151 88 200 35 186 141 58 26 112 163 113)(14 114 164 97 27 59 142 187 36 201 89 152 77)(15 78 153 90 202 37 188 143 60 28 98 165 115)(16 116 166 99 29 61 144 189 38 203 91 154 79)
(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)
G:=sub<Sym(208)| (1,80,155,92,204,39,190,129,62,30,100,167,117)(2,118,168,101,31,63,130,191,40,205,93,156,65)(3,66,157,94,206,41,192,131,64,32,102,169,119)(4,120,170,103,17,49,132,177,42,207,95,158,67)(5,68,159,96,208,43,178,133,50,18,104,171,121)(6,122,172,105,19,51,134,179,44,193,81,160,69)(7,70,145,82,194,45,180,135,52,20,106,173,123)(8,124,174,107,21,53,136,181,46,195,83,146,71)(9,72,147,84,196,47,182,137,54,22,108,175,125)(10,126,176,109,23,55,138,183,48,197,85,148,73)(11,74,149,86,198,33,184,139,56,24,110,161,127)(12,128,162,111,25,57,140,185,34,199,87,150,75)(13,76,151,88,200,35,186,141,58,26,112,163,113)(14,114,164,97,27,59,142,187,36,201,89,152,77)(15,78,153,90,202,37,188,143,60,28,98,165,115)(16,116,166,99,29,61,144,189,38,203,91,154,79), (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)>;
G:=Group( (1,80,155,92,204,39,190,129,62,30,100,167,117)(2,118,168,101,31,63,130,191,40,205,93,156,65)(3,66,157,94,206,41,192,131,64,32,102,169,119)(4,120,170,103,17,49,132,177,42,207,95,158,67)(5,68,159,96,208,43,178,133,50,18,104,171,121)(6,122,172,105,19,51,134,179,44,193,81,160,69)(7,70,145,82,194,45,180,135,52,20,106,173,123)(8,124,174,107,21,53,136,181,46,195,83,146,71)(9,72,147,84,196,47,182,137,54,22,108,175,125)(10,126,176,109,23,55,138,183,48,197,85,148,73)(11,74,149,86,198,33,184,139,56,24,110,161,127)(12,128,162,111,25,57,140,185,34,199,87,150,75)(13,76,151,88,200,35,186,141,58,26,112,163,113)(14,114,164,97,27,59,142,187,36,201,89,152,77)(15,78,153,90,202,37,188,143,60,28,98,165,115)(16,116,166,99,29,61,144,189,38,203,91,154,79), (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) );
G=PermutationGroup([[(1,80,155,92,204,39,190,129,62,30,100,167,117),(2,118,168,101,31,63,130,191,40,205,93,156,65),(3,66,157,94,206,41,192,131,64,32,102,169,119),(4,120,170,103,17,49,132,177,42,207,95,158,67),(5,68,159,96,208,43,178,133,50,18,104,171,121),(6,122,172,105,19,51,134,179,44,193,81,160,69),(7,70,145,82,194,45,180,135,52,20,106,173,123),(8,124,174,107,21,53,136,181,46,195,83,146,71),(9,72,147,84,196,47,182,137,54,22,108,175,125),(10,126,176,109,23,55,138,183,48,197,85,148,73),(11,74,149,86,198,33,184,139,56,24,110,161,127),(12,128,162,111,25,57,140,185,34,199,87,150,75),(13,76,151,88,200,35,186,141,58,26,112,163,113),(14,114,164,97,27,59,142,187,36,201,89,152,77),(15,78,153,90,202,37,188,143,60,28,98,165,115),(16,116,166,99,29,61,144,189,38,203,91,154,79)], [(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)]])
C13⋊2C16 is a maximal subgroup of
C13⋊C32 C16×D13 C208⋊C2 C52.4C8 C13⋊D16 D8.D13 C8.6D26 C13⋊Q32
C13⋊2C16 is a maximal quotient of
C13⋊2C32
64 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 16A | ··· | 16H | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 16 | ··· | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D13 | Dic13 | C13⋊2C8 | C13⋊2C16 |
| kernel | C13⋊2C16 | C104 | C52 | C26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 6 | 6 | 12 | 24 |
Matrix representation of C13⋊2C16 ►in GL2(𝔽1249) generated by
| 0 | 1 |
| 1248 | 417 |
| 64 | 1172 |
| 382 | 1185 |
G:=sub<GL(2,GF(1249))| [0,1248,1,417],[64,382,1172,1185] >;
C13⋊2C16 in GAP, Magma, Sage, TeX
C_{13}\rtimes_2C_{16} % in TeX
G:=Group("C13:2C16"); // GroupNames label
G:=SmallGroup(208,1);
// by ID
G=gap.SmallGroup(208,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,10,26,42,4804]);
// Polycyclic
G:=Group<a,b|a^13=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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