metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C13⋊2Q16, Q8.D13, C4.4D26, C26.10D4, C52.4C22, Dic26.2C2, C13⋊2C8.1C2, (Q8×C13).1C2, C2.7(C13⋊D4), SmallGroup(208,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13⋊Q16
G = < a,b,c | a13=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13⋊Q16
►Regular action on 208 points
Generators in S208
(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)
(1 183 31 170 22 196 50 157)(2 195 32 182 23 208 51 169)(3 194 33 181 24 207 52 168)(4 193 34 180 25 206 40 167)(5 192 35 179 26 205 41 166)(6 191 36 178 14 204 42 165)(7 190 37 177 15 203 43 164)(8 189 38 176 16 202 44 163)(9 188 39 175 17 201 45 162)(10 187 27 174 18 200 46 161)(11 186 28 173 19 199 47 160)(12 185 29 172 20 198 48 159)(13 184 30 171 21 197 49 158)(53 131 102 105 67 144 81 118)(54 143 103 117 68 156 82 130)(55 142 104 116 69 155 83 129)(56 141 92 115 70 154 84 128)(57 140 93 114 71 153 85 127)(58 139 94 113 72 152 86 126)(59 138 95 112 73 151 87 125)(60 137 96 111 74 150 88 124)(61 136 97 110 75 149 89 123)(62 135 98 109 76 148 90 122)(63 134 99 108 77 147 91 121)(64 133 100 107 78 146 79 120)(65 132 101 106 66 145 80 119)
(1 67 22 53)(2 68 23 54)(3 69 24 55)(4 70 25 56)(5 71 26 57)(6 72 14 58)(7 73 15 59)(8 74 16 60)(9 75 17 61)(10 76 18 62)(11 77 19 63)(12 78 20 64)(13 66 21 65)(27 98 46 90)(28 99 47 91)(29 100 48 79)(30 101 49 80)(31 102 50 81)(32 103 51 82)(33 104 52 83)(34 92 40 84)(35 93 41 85)(36 94 42 86)(37 95 43 87)(38 96 44 88)(39 97 45 89)(105 196 118 183)(106 197 119 184)(107 198 120 185)(108 199 121 186)(109 200 122 187)(110 201 123 188)(111 202 124 189)(112 203 125 190)(113 204 126 191)(114 205 127 192)(115 206 128 193)(116 207 129 194)(117 208 130 195)(131 157 144 170)(132 158 145 171)(133 159 146 172)(134 160 147 173)(135 161 148 174)(136 162 149 175)(137 163 150 176)(138 164 151 177)(139 165 152 178)(140 166 153 179)(141 167 154 180)(142 168 155 181)(143 169 156 182)
G:=sub<Sym(208)| (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), (1,183,31,170,22,196,50,157)(2,195,32,182,23,208,51,169)(3,194,33,181,24,207,52,168)(4,193,34,180,25,206,40,167)(5,192,35,179,26,205,41,166)(6,191,36,178,14,204,42,165)(7,190,37,177,15,203,43,164)(8,189,38,176,16,202,44,163)(9,188,39,175,17,201,45,162)(10,187,27,174,18,200,46,161)(11,186,28,173,19,199,47,160)(12,185,29,172,20,198,48,159)(13,184,30,171,21,197,49,158)(53,131,102,105,67,144,81,118)(54,143,103,117,68,156,82,130)(55,142,104,116,69,155,83,129)(56,141,92,115,70,154,84,128)(57,140,93,114,71,153,85,127)(58,139,94,113,72,152,86,126)(59,138,95,112,73,151,87,125)(60,137,96,111,74,150,88,124)(61,136,97,110,75,149,89,123)(62,135,98,109,76,148,90,122)(63,134,99,108,77,147,91,121)(64,133,100,107,78,146,79,120)(65,132,101,106,66,145,80,119), (1,67,22,53)(2,68,23,54)(3,69,24,55)(4,70,25,56)(5,71,26,57)(6,72,14,58)(7,73,15,59)(8,74,16,60)(9,75,17,61)(10,76,18,62)(11,77,19,63)(12,78,20,64)(13,66,21,65)(27,98,46,90)(28,99,47,91)(29,100,48,79)(30,101,49,80)(31,102,50,81)(32,103,51,82)(33,104,52,83)(34,92,40,84)(35,93,41,85)(36,94,42,86)(37,95,43,87)(38,96,44,88)(39,97,45,89)(105,196,118,183)(106,197,119,184)(107,198,120,185)(108,199,121,186)(109,200,122,187)(110,201,123,188)(111,202,124,189)(112,203,125,190)(113,204,126,191)(114,205,127,192)(115,206,128,193)(116,207,129,194)(117,208,130,195)(131,157,144,170)(132,158,145,171)(133,159,146,172)(134,160,147,173)(135,161,148,174)(136,162,149,175)(137,163,150,176)(138,164,151,177)(139,165,152,178)(140,166,153,179)(141,167,154,180)(142,168,155,181)(143,169,156,182)>;
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), (1,183,31,170,22,196,50,157)(2,195,32,182,23,208,51,169)(3,194,33,181,24,207,52,168)(4,193,34,180,25,206,40,167)(5,192,35,179,26,205,41,166)(6,191,36,178,14,204,42,165)(7,190,37,177,15,203,43,164)(8,189,38,176,16,202,44,163)(9,188,39,175,17,201,45,162)(10,187,27,174,18,200,46,161)(11,186,28,173,19,199,47,160)(12,185,29,172,20,198,48,159)(13,184,30,171,21,197,49,158)(53,131,102,105,67,144,81,118)(54,143,103,117,68,156,82,130)(55,142,104,116,69,155,83,129)(56,141,92,115,70,154,84,128)(57,140,93,114,71,153,85,127)(58,139,94,113,72,152,86,126)(59,138,95,112,73,151,87,125)(60,137,96,111,74,150,88,124)(61,136,97,110,75,149,89,123)(62,135,98,109,76,148,90,122)(63,134,99,108,77,147,91,121)(64,133,100,107,78,146,79,120)(65,132,101,106,66,145,80,119), (1,67,22,53)(2,68,23,54)(3,69,24,55)(4,70,25,56)(5,71,26,57)(6,72,14,58)(7,73,15,59)(8,74,16,60)(9,75,17,61)(10,76,18,62)(11,77,19,63)(12,78,20,64)(13,66,21,65)(27,98,46,90)(28,99,47,91)(29,100,48,79)(30,101,49,80)(31,102,50,81)(32,103,51,82)(33,104,52,83)(34,92,40,84)(35,93,41,85)(36,94,42,86)(37,95,43,87)(38,96,44,88)(39,97,45,89)(105,196,118,183)(106,197,119,184)(107,198,120,185)(108,199,121,186)(109,200,122,187)(110,201,123,188)(111,202,124,189)(112,203,125,190)(113,204,126,191)(114,205,127,192)(115,206,128,193)(116,207,129,194)(117,208,130,195)(131,157,144,170)(132,158,145,171)(133,159,146,172)(134,160,147,173)(135,161,148,174)(136,162,149,175)(137,163,150,176)(138,164,151,177)(139,165,152,178)(140,166,153,179)(141,167,154,180)(142,168,155,181)(143,169,156,182) );
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)], [(1,183,31,170,22,196,50,157),(2,195,32,182,23,208,51,169),(3,194,33,181,24,207,52,168),(4,193,34,180,25,206,40,167),(5,192,35,179,26,205,41,166),(6,191,36,178,14,204,42,165),(7,190,37,177,15,203,43,164),(8,189,38,176,16,202,44,163),(9,188,39,175,17,201,45,162),(10,187,27,174,18,200,46,161),(11,186,28,173,19,199,47,160),(12,185,29,172,20,198,48,159),(13,184,30,171,21,197,49,158),(53,131,102,105,67,144,81,118),(54,143,103,117,68,156,82,130),(55,142,104,116,69,155,83,129),(56,141,92,115,70,154,84,128),(57,140,93,114,71,153,85,127),(58,139,94,113,72,152,86,126),(59,138,95,112,73,151,87,125),(60,137,96,111,74,150,88,124),(61,136,97,110,75,149,89,123),(62,135,98,109,76,148,90,122),(63,134,99,108,77,147,91,121),(64,133,100,107,78,146,79,120),(65,132,101,106,66,145,80,119)], [(1,67,22,53),(2,68,23,54),(3,69,24,55),(4,70,25,56),(5,71,26,57),(6,72,14,58),(7,73,15,59),(8,74,16,60),(9,75,17,61),(10,76,18,62),(11,77,19,63),(12,78,20,64),(13,66,21,65),(27,98,46,90),(28,99,47,91),(29,100,48,79),(30,101,49,80),(31,102,50,81),(32,103,51,82),(33,104,52,83),(34,92,40,84),(35,93,41,85),(36,94,42,86),(37,95,43,87),(38,96,44,88),(39,97,45,89),(105,196,118,183),(106,197,119,184),(107,198,120,185),(108,199,121,186),(109,200,122,187),(110,201,123,188),(111,202,124,189),(112,203,125,190),(113,204,126,191),(114,205,127,192),(115,206,128,193),(116,207,129,194),(117,208,130,195),(131,157,144,170),(132,158,145,171),(133,159,146,172),(134,160,147,173),(135,161,148,174),(136,162,149,175),(137,163,150,176),(138,164,151,177),(139,165,152,178),(140,166,153,179),(141,167,154,180),(142,168,155,181),(143,169,156,182)]])
C13⋊Q16 is a maximal subgroup of
D4.D26 D26.6D4 Q16×D13 Q16⋊D13 Q8.D26 C52.C23 D4.9D26
C13⋊Q16 is a maximal quotient of C26.D8 C26.Q16 Q8⋊Dic13
37 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52R |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 4 | 52 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | Q16 | D13 | D26 | C13⋊D4 | C13⋊Q16 |
| kernel | C13⋊Q16 | C13⋊2C8 | Dic26 | Q8×C13 | C26 | C13 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 |
Matrix representation of C13⋊Q16 ►in GL4(𝔽313) generated by
| 0 | 1 | 0 | 0 |
| 312 | 293 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 64 | 98 | 0 | 0 |
| 70 | 249 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 118 | 120 |
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 256 | 270 |
| 0 | 0 | 243 | 57 |
G:=sub<GL(4,GF(313))| [0,312,0,0,1,293,0,0,0,0,1,0,0,0,0,1],[64,70,0,0,98,249,0,0,0,0,0,118,0,0,61,120],[312,0,0,0,0,312,0,0,0,0,256,243,0,0,270,57] >;
C13⋊Q16 in GAP, Magma, Sage, TeX
C_{13}\rtimes Q_{16} % in TeX
G:=Group("C13:Q16"); // GroupNames label
G:=SmallGroup(208,18);
// by ID
G=gap.SmallGroup(208,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,46,182,97,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^13=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export