direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C26, C52.42D4, C104⋊13C22, C52.45C23, (C2×C8)⋊5C26, C8⋊3(C2×C26), (C2×Q8)⋊3C26, Q8⋊1(C2×C26), C4.7(D4×C13), (C2×C104)⋊13C2, (Q8×C26)⋊10C2, (C2×D4).6C26, D4.1(C2×C26), (C2×C26).53D4, C26.75(C2×D4), C2.12(D4×C26), (D4×C26).13C2, C4.2(C22×C26), (Q8×C13)⋊9C22, C22.15(D4×C13), (C2×C52).130C22, (D4×C13).11C22, (C2×C4).26(C2×C26), SmallGroup(416,194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C26
G = < a,b,c | a26=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C13, C2×C8, SD16, C2×D4, C2×Q8, C26, C26, C26, C2×SD16, C52, C52, C2×C26, C2×C26, C104, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, Q8×C13, C22×C26, C2×C104, C13×SD16, D4×C26, Q8×C26, SD16×C26
Quotients: C1, C2, C22, D4, C23, C13, SD16, C2×D4, C26, C2×SD16, C2×C26, D4×C13, C22×C26, C13×SD16, D4×C26, SD16×C26
►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 37 162 144 72 191 124 82)(2 38 163 145 73 192 125 83)(3 39 164 146 74 193 126 84)(4 40 165 147 75 194 127 85)(5 41 166 148 76 195 128 86)(6 42 167 149 77 196 129 87)(7 43 168 150 78 197 130 88)(8 44 169 151 53 198 105 89)(9 45 170 152 54 199 106 90)(10 46 171 153 55 200 107 91)(11 47 172 154 56 201 108 92)(12 48 173 155 57 202 109 93)(13 49 174 156 58 203 110 94)(14 50 175 131 59 204 111 95)(15 51 176 132 60 205 112 96)(16 52 177 133 61 206 113 97)(17 27 178 134 62 207 114 98)(18 28 179 135 63 208 115 99)(19 29 180 136 64 183 116 100)(20 30 181 137 65 184 117 101)(21 31 182 138 66 185 118 102)(22 32 157 139 67 186 119 103)(23 33 158 140 68 187 120 104)(24 34 159 141 69 188 121 79)(25 35 160 142 70 189 122 80)(26 36 161 143 71 190 123 81)
(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 156)(50 131)(51 132)(52 133)(79 188)(80 189)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 196)(88 197)(89 198)(90 199)(91 200)(92 201)(93 202)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 183)(101 184)(102 185)(103 186)(104 187)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)(113 177)(114 178)(115 179)(116 180)(117 181)(118 182)(119 157)(120 158)(121 159)(122 160)(123 161)(124 162)(125 163)(126 164)(127 165)(128 166)(129 167)(130 168)
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), (1,37,162,144,72,191,124,82)(2,38,163,145,73,192,125,83)(3,39,164,146,74,193,126,84)(4,40,165,147,75,194,127,85)(5,41,166,148,76,195,128,86)(6,42,167,149,77,196,129,87)(7,43,168,150,78,197,130,88)(8,44,169,151,53,198,105,89)(9,45,170,152,54,199,106,90)(10,46,171,153,55,200,107,91)(11,47,172,154,56,201,108,92)(12,48,173,155,57,202,109,93)(13,49,174,156,58,203,110,94)(14,50,175,131,59,204,111,95)(15,51,176,132,60,205,112,96)(16,52,177,133,61,206,113,97)(17,27,178,134,62,207,114,98)(18,28,179,135,63,208,115,99)(19,29,180,136,64,183,116,100)(20,30,181,137,65,184,117,101)(21,31,182,138,66,185,118,102)(22,32,157,139,67,186,119,103)(23,33,158,140,68,187,120,104)(24,34,159,141,69,188,121,79)(25,35,160,142,70,189,122,80)(26,36,161,143,71,190,123,81), (27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,131)(51,132)(52,133)(79,188)(80,189)(81,190)(82,191)(83,192)(84,193)(85,194)(86,195)(87,196)(88,197)(89,198)(90,199)(91,200)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,183)(101,184)(102,185)(103,186)(104,187)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)>;
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), (1,37,162,144,72,191,124,82)(2,38,163,145,73,192,125,83)(3,39,164,146,74,193,126,84)(4,40,165,147,75,194,127,85)(5,41,166,148,76,195,128,86)(6,42,167,149,77,196,129,87)(7,43,168,150,78,197,130,88)(8,44,169,151,53,198,105,89)(9,45,170,152,54,199,106,90)(10,46,171,153,55,200,107,91)(11,47,172,154,56,201,108,92)(12,48,173,155,57,202,109,93)(13,49,174,156,58,203,110,94)(14,50,175,131,59,204,111,95)(15,51,176,132,60,205,112,96)(16,52,177,133,61,206,113,97)(17,27,178,134,62,207,114,98)(18,28,179,135,63,208,115,99)(19,29,180,136,64,183,116,100)(20,30,181,137,65,184,117,101)(21,31,182,138,66,185,118,102)(22,32,157,139,67,186,119,103)(23,33,158,140,68,187,120,104)(24,34,159,141,69,188,121,79)(25,35,160,142,70,189,122,80)(26,36,161,143,71,190,123,81), (27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,131)(51,132)(52,133)(79,188)(80,189)(81,190)(82,191)(83,192)(84,193)(85,194)(86,195)(87,196)(88,197)(89,198)(90,199)(91,200)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,183)(101,184)(102,185)(103,186)(104,187)(105,169)(106,170)(107,171)(108,172)(109,173)(110,174)(111,175)(112,176)(113,177)(114,178)(115,179)(116,180)(117,181)(118,182)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168) );
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)], [(1,37,162,144,72,191,124,82),(2,38,163,145,73,192,125,83),(3,39,164,146,74,193,126,84),(4,40,165,147,75,194,127,85),(5,41,166,148,76,195,128,86),(6,42,167,149,77,196,129,87),(7,43,168,150,78,197,130,88),(8,44,169,151,53,198,105,89),(9,45,170,152,54,199,106,90),(10,46,171,153,55,200,107,91),(11,47,172,154,56,201,108,92),(12,48,173,155,57,202,109,93),(13,49,174,156,58,203,110,94),(14,50,175,131,59,204,111,95),(15,51,176,132,60,205,112,96),(16,52,177,133,61,206,113,97),(17,27,178,134,62,207,114,98),(18,28,179,135,63,208,115,99),(19,29,180,136,64,183,116,100),(20,30,181,137,65,184,117,101),(21,31,182,138,66,185,118,102),(22,32,157,139,67,186,119,103),(23,33,158,140,68,187,120,104),(24,34,159,141,69,188,121,79),(25,35,160,142,70,189,122,80),(26,36,161,143,71,190,123,81)], [(27,134),(28,135),(29,136),(30,137),(31,138),(32,139),(33,140),(34,141),(35,142),(36,143),(37,144),(38,145),(39,146),(40,147),(41,148),(42,149),(43,150),(44,151),(45,152),(46,153),(47,154),(48,155),(49,156),(50,131),(51,132),(52,133),(79,188),(80,189),(81,190),(82,191),(83,192),(84,193),(85,194),(86,195),(87,196),(88,197),(89,198),(90,199),(91,200),(92,201),(93,202),(94,203),(95,204),(96,205),(97,206),(98,207),(99,208),(100,183),(101,184),(102,185),(103,186),(104,187),(105,169),(106,170),(107,171),(108,172),(109,173),(110,174),(111,175),(112,176),(113,177),(114,178),(115,179),(116,180),(117,181),(118,182),(119,157),(120,158),(121,159),(122,160),(123,161),(124,162),(125,163),(126,164),(127,165),(128,166),(129,167),(130,168)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 52A | ··· | 52X | 52Y | ··· | 52AV | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C26 | D4 | D4 | SD16 | D4×C13 | D4×C13 | C13×SD16 |
| kernel | SD16×C26 | C2×C104 | C13×SD16 | D4×C26 | Q8×C26 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C52 | C2×C26 | C26 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 12 | 12 | 48 | 12 | 12 | 1 | 1 | 4 | 12 | 12 | 48 |
Matrix representation of SD16×C26 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 48 | 0 |
| 0 | 0 | 48 |
| 1 | 0 | 0 |
| 0 | 248 | 65 |
| 0 | 248 | 248 |
| 312 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 312 |
G:=sub<GL(3,GF(313))| [312,0,0,0,48,0,0,0,48],[1,0,0,0,248,248,0,65,248],[312,0,0,0,1,0,0,0,312] >;
SD16×C26 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{26} % in TeX
G:=Group("SD16xC26"); // GroupNames label
G:=SmallGroup(416,194);
// by ID
G=gap.SmallGroup(416,194);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,9364,4690,88]);
// Polycyclic
G:=Group<a,b,c|a^26=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations