metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic26.C4, Dic13.12C23, D4.(C13⋊C4), (D4×C13).C4, C13⋊D4.C4, D13⋊C8⋊2C2, C13⋊1(C8○D4), C52.5(C2×C4), D26.1(C2×C4), C13⋊C8.1C22, C52.C4⋊3C2, C26.7(C22×C4), D4⋊2D13.3C2, C13⋊M4(2)⋊2C2, Dic13.1(C2×C4), (C4×D13).11C22, (C2×Dic13).25C22, (C2×C13⋊C8)⋊4C2, C4.5(C2×C13⋊C4), (C2×C26).(C2×C4), C2.8(C22×C13⋊C4), C22.1(C2×C13⋊C4), SmallGroup(416,205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic26.C4
G = < a,b,c | a52=1, b2=c4=a26, bab-1=a-1, cac-1=a5, bc=cb >
Subgroups: 356 in 62 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, D4, D4, Q8, C13, C2×C8, M4(2), C4○D4, D13, C26, C26, C8○D4, Dic13, Dic13, C52, D26, C2×C26, C13⋊C8, C13⋊C8, Dic26, C4×D13, C2×Dic13, C13⋊D4, D4×C13, D13⋊C8, C52.C4, C2×C13⋊C8, C13⋊M4(2), D4⋊2D13, Dic26.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C8○D4, C13⋊C4, C2×C13⋊C4, C22×C13⋊C4, Dic26.C4
►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)
(1 177 27 203)(2 176 28 202)(3 175 29 201)(4 174 30 200)(5 173 31 199)(6 172 32 198)(7 171 33 197)(8 170 34 196)(9 169 35 195)(10 168 36 194)(11 167 37 193)(12 166 38 192)(13 165 39 191)(14 164 40 190)(15 163 41 189)(16 162 42 188)(17 161 43 187)(18 160 44 186)(19 159 45 185)(20 158 46 184)(21 157 47 183)(22 208 48 182)(23 207 49 181)(24 206 50 180)(25 205 51 179)(26 204 52 178)(53 115 79 141)(54 114 80 140)(55 113 81 139)(56 112 82 138)(57 111 83 137)(58 110 84 136)(59 109 85 135)(60 108 86 134)(61 107 87 133)(62 106 88 132)(63 105 89 131)(64 156 90 130)(65 155 91 129)(66 154 92 128)(67 153 93 127)(68 152 94 126)(69 151 95 125)(70 150 96 124)(71 149 97 123)(72 148 98 122)(73 147 99 121)(74 146 100 120)(75 145 101 119)(76 144 102 118)(77 143 103 117)(78 142 104 116)
(1 58 164 149 27 84 190 123)(2 79 189 154 28 53 163 128)(3 100 162 107 29 74 188 133)(4 69 187 112 30 95 161 138)(5 90 160 117 31 64 186 143)(6 59 185 122 32 85 159 148)(7 80 158 127 33 54 184 153)(8 101 183 132 34 75 157 106)(9 70 208 137 35 96 182 111)(10 91 181 142 36 65 207 116)(11 60 206 147 37 86 180 121)(12 81 179 152 38 55 205 126)(13 102 204 105 39 76 178 131)(14 71 177 110 40 97 203 136)(15 92 202 115 41 66 176 141)(16 61 175 120 42 87 201 146)(17 82 200 125 43 56 174 151)(18 103 173 130 44 77 199 156)(19 72 198 135 45 98 172 109)(20 93 171 140 46 67 197 114)(21 62 196 145 47 88 170 119)(22 83 169 150 48 57 195 124)(23 104 194 155 49 78 168 129)(24 73 167 108 50 99 193 134)(25 94 192 113 51 68 166 139)(26 63 165 118 52 89 191 144)
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), (1,177,27,203)(2,176,28,202)(3,175,29,201)(4,174,30,200)(5,173,31,199)(6,172,32,198)(7,171,33,197)(8,170,34,196)(9,169,35,195)(10,168,36,194)(11,167,37,193)(12,166,38,192)(13,165,39,191)(14,164,40,190)(15,163,41,189)(16,162,42,188)(17,161,43,187)(18,160,44,186)(19,159,45,185)(20,158,46,184)(21,157,47,183)(22,208,48,182)(23,207,49,181)(24,206,50,180)(25,205,51,179)(26,204,52,178)(53,115,79,141)(54,114,80,140)(55,113,81,139)(56,112,82,138)(57,111,83,137)(58,110,84,136)(59,109,85,135)(60,108,86,134)(61,107,87,133)(62,106,88,132)(63,105,89,131)(64,156,90,130)(65,155,91,129)(66,154,92,128)(67,153,93,127)(68,152,94,126)(69,151,95,125)(70,150,96,124)(71,149,97,123)(72,148,98,122)(73,147,99,121)(74,146,100,120)(75,145,101,119)(76,144,102,118)(77,143,103,117)(78,142,104,116), (1,58,164,149,27,84,190,123)(2,79,189,154,28,53,163,128)(3,100,162,107,29,74,188,133)(4,69,187,112,30,95,161,138)(5,90,160,117,31,64,186,143)(6,59,185,122,32,85,159,148)(7,80,158,127,33,54,184,153)(8,101,183,132,34,75,157,106)(9,70,208,137,35,96,182,111)(10,91,181,142,36,65,207,116)(11,60,206,147,37,86,180,121)(12,81,179,152,38,55,205,126)(13,102,204,105,39,76,178,131)(14,71,177,110,40,97,203,136)(15,92,202,115,41,66,176,141)(16,61,175,120,42,87,201,146)(17,82,200,125,43,56,174,151)(18,103,173,130,44,77,199,156)(19,72,198,135,45,98,172,109)(20,93,171,140,46,67,197,114)(21,62,196,145,47,88,170,119)(22,83,169,150,48,57,195,124)(23,104,194,155,49,78,168,129)(24,73,167,108,50,99,193,134)(25,94,192,113,51,68,166,139)(26,63,165,118,52,89,191,144)>;
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), (1,177,27,203)(2,176,28,202)(3,175,29,201)(4,174,30,200)(5,173,31,199)(6,172,32,198)(7,171,33,197)(8,170,34,196)(9,169,35,195)(10,168,36,194)(11,167,37,193)(12,166,38,192)(13,165,39,191)(14,164,40,190)(15,163,41,189)(16,162,42,188)(17,161,43,187)(18,160,44,186)(19,159,45,185)(20,158,46,184)(21,157,47,183)(22,208,48,182)(23,207,49,181)(24,206,50,180)(25,205,51,179)(26,204,52,178)(53,115,79,141)(54,114,80,140)(55,113,81,139)(56,112,82,138)(57,111,83,137)(58,110,84,136)(59,109,85,135)(60,108,86,134)(61,107,87,133)(62,106,88,132)(63,105,89,131)(64,156,90,130)(65,155,91,129)(66,154,92,128)(67,153,93,127)(68,152,94,126)(69,151,95,125)(70,150,96,124)(71,149,97,123)(72,148,98,122)(73,147,99,121)(74,146,100,120)(75,145,101,119)(76,144,102,118)(77,143,103,117)(78,142,104,116), (1,58,164,149,27,84,190,123)(2,79,189,154,28,53,163,128)(3,100,162,107,29,74,188,133)(4,69,187,112,30,95,161,138)(5,90,160,117,31,64,186,143)(6,59,185,122,32,85,159,148)(7,80,158,127,33,54,184,153)(8,101,183,132,34,75,157,106)(9,70,208,137,35,96,182,111)(10,91,181,142,36,65,207,116)(11,60,206,147,37,86,180,121)(12,81,179,152,38,55,205,126)(13,102,204,105,39,76,178,131)(14,71,177,110,40,97,203,136)(15,92,202,115,41,66,176,141)(16,61,175,120,42,87,201,146)(17,82,200,125,43,56,174,151)(18,103,173,130,44,77,199,156)(19,72,198,135,45,98,172,109)(20,93,171,140,46,67,197,114)(21,62,196,145,47,88,170,119)(22,83,169,150,48,57,195,124)(23,104,194,155,49,78,168,129)(24,73,167,108,50,99,193,134)(25,94,192,113,51,68,166,139)(26,63,165,118,52,89,191,144) );
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)], [(1,177,27,203),(2,176,28,202),(3,175,29,201),(4,174,30,200),(5,173,31,199),(6,172,32,198),(7,171,33,197),(8,170,34,196),(9,169,35,195),(10,168,36,194),(11,167,37,193),(12,166,38,192),(13,165,39,191),(14,164,40,190),(15,163,41,189),(16,162,42,188),(17,161,43,187),(18,160,44,186),(19,159,45,185),(20,158,46,184),(21,157,47,183),(22,208,48,182),(23,207,49,181),(24,206,50,180),(25,205,51,179),(26,204,52,178),(53,115,79,141),(54,114,80,140),(55,113,81,139),(56,112,82,138),(57,111,83,137),(58,110,84,136),(59,109,85,135),(60,108,86,134),(61,107,87,133),(62,106,88,132),(63,105,89,131),(64,156,90,130),(65,155,91,129),(66,154,92,128),(67,153,93,127),(68,152,94,126),(69,151,95,125),(70,150,96,124),(71,149,97,123),(72,148,98,122),(73,147,99,121),(74,146,100,120),(75,145,101,119),(76,144,102,118),(77,143,103,117),(78,142,104,116)], [(1,58,164,149,27,84,190,123),(2,79,189,154,28,53,163,128),(3,100,162,107,29,74,188,133),(4,69,187,112,30,95,161,138),(5,90,160,117,31,64,186,143),(6,59,185,122,32,85,159,148),(7,80,158,127,33,54,184,153),(8,101,183,132,34,75,157,106),(9,70,208,137,35,96,182,111),(10,91,181,142,36,65,207,116),(11,60,206,147,37,86,180,121),(12,81,179,152,38,55,205,126),(13,102,204,105,39,76,178,131),(14,71,177,110,40,97,203,136),(15,92,202,115,41,66,176,141),(16,61,175,120,42,87,201,146),(17,82,200,125,43,56,174,151),(18,103,173,130,44,77,199,156),(19,72,198,135,45,98,172,109),(20,93,171,140,46,67,197,114),(21,62,196,145,47,88,170,119),(22,83,169,150,48,57,195,124),(23,104,194,155,49,78,168,129),(24,73,167,108,50,99,193,134),(25,94,192,113,51,68,166,139),(26,63,165,118,52,89,191,144)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 13A | 13B | 13C | 26A | 26B | 26C | 26D | ··· | 26I | 52A | 52B | 52C |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | ··· | 26 | 52 | 52 | 52 |
| size | 1 | 1 | 2 | 2 | 26 | 2 | 13 | 13 | 26 | 26 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 | C13⋊C4 | C2×C13⋊C4 | C2×C13⋊C4 | Dic26.C4 |
| kernel | Dic26.C4 | D13⋊C8 | C52.C4 | C2×C13⋊C8 | C13⋊M4(2) | D4⋊2D13 | Dic26 | C13⋊D4 | D4×C13 | C13 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 3 | 3 | 6 | 3 |
Matrix representation of Dic26.C4 ►in GL6(𝔽313)
| 0 | 312 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 | 312 | 0 |
| 0 | 0 | 1 | 0 | 0 | 312 |
| 0 | 0 | 103 | 240 | 73 | 211 |
| 0 | 288 | 0 | 0 | 0 | 0 |
| 288 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 33 | 304 | 50 |
| 0 | 0 | 53 | 92 | 289 | 201 |
| 0 | 0 | 211 | 291 | 171 | 41 |
| 0 | 0 | 230 | 40 | 244 | 275 |
| 125 | 0 | 0 | 0 | 0 | 0 |
| 0 | 125 | 0 | 0 | 0 | 0 |
| 0 | 0 | 188 | 45 | 53 | 56 |
| 0 | 0 | 60 | 299 | 163 | 184 |
| 0 | 0 | 143 | 273 | 269 | 254 |
| 0 | 0 | 296 | 26 | 303 | 183 |
G:=sub<GL(6,GF(313))| [0,1,0,0,0,0,312,0,0,0,0,0,0,0,1,1,1,103,0,0,312,0,0,240,0,0,0,312,0,73,0,0,0,0,312,211],[0,288,0,0,0,0,288,0,0,0,0,0,0,0,88,53,211,230,0,0,33,92,291,40,0,0,304,289,171,244,0,0,50,201,41,275],[125,0,0,0,0,0,0,125,0,0,0,0,0,0,188,60,143,296,0,0,45,299,273,26,0,0,53,163,269,303,0,0,56,184,254,183] >;
Dic26.C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{26}.C_4 % in TeX
G:=Group("Dic26.C4"); // GroupNames label
G:=SmallGroup(416,205);
// by ID
G=gap.SmallGroup(416,205);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,69,9221,1751]);
// Polycyclic
G:=Group<a,b,c|a^52=1,b^2=c^4=a^26,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations