direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×Q16, C8.C26, Q8.C26, C104.3C2, C26.16D4, C52.19C22, C4.3(C2×C26), C2.5(D4×C13), (Q8×C13).2C2, SmallGroup(208,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×Q16
G = < a,b,c | a13=b8=1, c2=b4, ab=ba, 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 21 186 145 101 52 57 70)(2 22 187 146 102 40 58 71)(3 23 188 147 103 41 59 72)(4 24 189 148 104 42 60 73)(5 25 190 149 92 43 61 74)(6 26 191 150 93 44 62 75)(7 14 192 151 94 45 63 76)(8 15 193 152 95 46 64 77)(9 16 194 153 96 47 65 78)(10 17 195 154 97 48 53 66)(11 18 183 155 98 49 54 67)(12 19 184 156 99 50 55 68)(13 20 185 144 100 51 56 69)(27 198 172 129 169 139 115 86)(28 199 173 130 157 140 116 87)(29 200 174 118 158 141 117 88)(30 201 175 119 159 142 105 89)(31 202 176 120 160 143 106 90)(32 203 177 121 161 131 107 91)(33 204 178 122 162 132 108 79)(34 205 179 123 163 133 109 80)(35 206 180 124 164 134 110 81)(36 207 181 125 165 135 111 82)(37 208 182 126 166 136 112 83)(38 196 170 127 167 137 113 84)(39 197 171 128 168 138 114 85)
(1 141 101 200)(2 142 102 201)(3 143 103 202)(4 131 104 203)(5 132 92 204)(6 133 93 205)(7 134 94 206)(8 135 95 207)(9 136 96 208)(10 137 97 196)(11 138 98 197)(12 139 99 198)(13 140 100 199)(14 164 45 35)(15 165 46 36)(16 166 47 37)(17 167 48 38)(18 168 49 39)(19 169 50 27)(20 157 51 28)(21 158 52 29)(22 159 40 30)(23 160 41 31)(24 161 42 32)(25 162 43 33)(26 163 44 34)(53 84 195 127)(54 85 183 128)(55 86 184 129)(56 87 185 130)(57 88 186 118)(58 89 187 119)(59 90 188 120)(60 91 189 121)(61 79 190 122)(62 80 191 123)(63 81 192 124)(64 82 193 125)(65 83 194 126)(66 113 154 170)(67 114 155 171)(68 115 156 172)(69 116 144 173)(70 117 145 174)(71 105 146 175)(72 106 147 176)(73 107 148 177)(74 108 149 178)(75 109 150 179)(76 110 151 180)(77 111 152 181)(78 112 153 182)
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), (1,21,186,145,101,52,57,70)(2,22,187,146,102,40,58,71)(3,23,188,147,103,41,59,72)(4,24,189,148,104,42,60,73)(5,25,190,149,92,43,61,74)(6,26,191,150,93,44,62,75)(7,14,192,151,94,45,63,76)(8,15,193,152,95,46,64,77)(9,16,194,153,96,47,65,78)(10,17,195,154,97,48,53,66)(11,18,183,155,98,49,54,67)(12,19,184,156,99,50,55,68)(13,20,185,144,100,51,56,69)(27,198,172,129,169,139,115,86)(28,199,173,130,157,140,116,87)(29,200,174,118,158,141,117,88)(30,201,175,119,159,142,105,89)(31,202,176,120,160,143,106,90)(32,203,177,121,161,131,107,91)(33,204,178,122,162,132,108,79)(34,205,179,123,163,133,109,80)(35,206,180,124,164,134,110,81)(36,207,181,125,165,135,111,82)(37,208,182,126,166,136,112,83)(38,196,170,127,167,137,113,84)(39,197,171,128,168,138,114,85), (1,141,101,200)(2,142,102,201)(3,143,103,202)(4,131,104,203)(5,132,92,204)(6,133,93,205)(7,134,94,206)(8,135,95,207)(9,136,96,208)(10,137,97,196)(11,138,98,197)(12,139,99,198)(13,140,100,199)(14,164,45,35)(15,165,46,36)(16,166,47,37)(17,167,48,38)(18,168,49,39)(19,169,50,27)(20,157,51,28)(21,158,52,29)(22,159,40,30)(23,160,41,31)(24,161,42,32)(25,162,43,33)(26,163,44,34)(53,84,195,127)(54,85,183,128)(55,86,184,129)(56,87,185,130)(57,88,186,118)(58,89,187,119)(59,90,188,120)(60,91,189,121)(61,79,190,122)(62,80,191,123)(63,81,192,124)(64,82,193,125)(65,83,194,126)(66,113,154,170)(67,114,155,171)(68,115,156,172)(69,116,144,173)(70,117,145,174)(71,105,146,175)(72,106,147,176)(73,107,148,177)(74,108,149,178)(75,109,150,179)(76,110,151,180)(77,111,152,181)(78,112,153,182)>;
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), (1,21,186,145,101,52,57,70)(2,22,187,146,102,40,58,71)(3,23,188,147,103,41,59,72)(4,24,189,148,104,42,60,73)(5,25,190,149,92,43,61,74)(6,26,191,150,93,44,62,75)(7,14,192,151,94,45,63,76)(8,15,193,152,95,46,64,77)(9,16,194,153,96,47,65,78)(10,17,195,154,97,48,53,66)(11,18,183,155,98,49,54,67)(12,19,184,156,99,50,55,68)(13,20,185,144,100,51,56,69)(27,198,172,129,169,139,115,86)(28,199,173,130,157,140,116,87)(29,200,174,118,158,141,117,88)(30,201,175,119,159,142,105,89)(31,202,176,120,160,143,106,90)(32,203,177,121,161,131,107,91)(33,204,178,122,162,132,108,79)(34,205,179,123,163,133,109,80)(35,206,180,124,164,134,110,81)(36,207,181,125,165,135,111,82)(37,208,182,126,166,136,112,83)(38,196,170,127,167,137,113,84)(39,197,171,128,168,138,114,85), (1,141,101,200)(2,142,102,201)(3,143,103,202)(4,131,104,203)(5,132,92,204)(6,133,93,205)(7,134,94,206)(8,135,95,207)(9,136,96,208)(10,137,97,196)(11,138,98,197)(12,139,99,198)(13,140,100,199)(14,164,45,35)(15,165,46,36)(16,166,47,37)(17,167,48,38)(18,168,49,39)(19,169,50,27)(20,157,51,28)(21,158,52,29)(22,159,40,30)(23,160,41,31)(24,161,42,32)(25,162,43,33)(26,163,44,34)(53,84,195,127)(54,85,183,128)(55,86,184,129)(56,87,185,130)(57,88,186,118)(58,89,187,119)(59,90,188,120)(60,91,189,121)(61,79,190,122)(62,80,191,123)(63,81,192,124)(64,82,193,125)(65,83,194,126)(66,113,154,170)(67,114,155,171)(68,115,156,172)(69,116,144,173)(70,117,145,174)(71,105,146,175)(72,106,147,176)(73,107,148,177)(74,108,149,178)(75,109,150,179)(76,110,151,180)(77,111,152,181)(78,112,153,182) );
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)], [(1,21,186,145,101,52,57,70),(2,22,187,146,102,40,58,71),(3,23,188,147,103,41,59,72),(4,24,189,148,104,42,60,73),(5,25,190,149,92,43,61,74),(6,26,191,150,93,44,62,75),(7,14,192,151,94,45,63,76),(8,15,193,152,95,46,64,77),(9,16,194,153,96,47,65,78),(10,17,195,154,97,48,53,66),(11,18,183,155,98,49,54,67),(12,19,184,156,99,50,55,68),(13,20,185,144,100,51,56,69),(27,198,172,129,169,139,115,86),(28,199,173,130,157,140,116,87),(29,200,174,118,158,141,117,88),(30,201,175,119,159,142,105,89),(31,202,176,120,160,143,106,90),(32,203,177,121,161,131,107,91),(33,204,178,122,162,132,108,79),(34,205,179,123,163,133,109,80),(35,206,180,124,164,134,110,81),(36,207,181,125,165,135,111,82),(37,208,182,126,166,136,112,83),(38,196,170,127,167,137,113,84),(39,197,171,128,168,138,114,85)], [(1,141,101,200),(2,142,102,201),(3,143,103,202),(4,131,104,203),(5,132,92,204),(6,133,93,205),(7,134,94,206),(8,135,95,207),(9,136,96,208),(10,137,97,196),(11,138,98,197),(12,139,99,198),(13,140,100,199),(14,164,45,35),(15,165,46,36),(16,166,47,37),(17,167,48,38),(18,168,49,39),(19,169,50,27),(20,157,51,28),(21,158,52,29),(22,159,40,30),(23,160,41,31),(24,161,42,32),(25,162,43,33),(26,163,44,34),(53,84,195,127),(54,85,183,128),(55,86,184,129),(56,87,185,130),(57,88,186,118),(58,89,187,119),(59,90,188,120),(60,91,189,121),(61,79,190,122),(62,80,191,123),(63,81,192,124),(64,82,193,125),(65,83,194,126),(66,113,154,170),(67,114,155,171),(68,115,156,172),(69,116,144,173),(70,117,145,174),(71,105,146,175),(72,106,147,176),(73,107,148,177),(74,108,149,178),(75,109,150,179),(76,110,151,180),(77,111,152,181),(78,112,153,182)]])
C13×Q16 is a maximal subgroup of
C8.6D26 C13⋊Q32 Q16⋊D13 D104⋊C2
91 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13L | 26A | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52AJ | 104A | ··· | 104X |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D4 | Q16 | D4×C13 | C13×Q16 |
| kernel | C13×Q16 | C104 | Q8×C13 | Q16 | C8 | Q8 | C26 | C13 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 2 | 12 | 24 |
Matrix representation of C13×Q16 ►in GL2(𝔽313) generated by
| 294 | 0 |
| 0 | 294 |
| 0 | 120 |
| 253 | 120 |
| 284 | 118 |
| 30 | 29 |
G:=sub<GL(2,GF(313))| [294,0,0,294],[0,253,120,120],[284,30,118,29] >;
C13×Q16 in GAP, Magma, Sage, TeX
C_{13}\times Q_{16} % in TeX
G:=Group("C13xQ16"); // GroupNames label
G:=SmallGroup(208,27);
// by ID
G=gap.SmallGroup(208,27);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-2,520,541,526,3123,1568,58]);
// Polycyclic
G:=Group<a,b,c|a^13=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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