direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C26, C52.41D4, C104⋊12C22, C52.44C23, (C2×C8)⋊3C26, C8⋊2(C2×C26), (C2×D4)⋊4C26, D4⋊1(C2×C26), C4.6(D4×C13), (C2×C104)⋊11C2, (D4×C26)⋊13C2, C26.74(C2×D4), C2.11(D4×C26), (C2×C26).52D4, C4.1(C22×C26), (D4×C13)⋊10C22, C22.14(D4×C13), (C2×C52).129C22, (C2×C4).25(C2×C26), SmallGroup(416,193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C26
G = < a,b,c | a26=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, D4, D4, C23, C13, C2×C8, D8, C2×D4, C26, C26, C26, C2×D8, C52, C2×C26, C2×C26, C104, C2×C52, D4×C13, D4×C13, C22×C26, C2×C104, C13×D8, D4×C26, D8×C26
Quotients: C1, C2, C22, D4, C23, C13, D8, C2×D4, C26, C2×D8, C2×C26, D4×C13, C22×C26, C13×D8, D4×C26, D8×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 53 86 208 117 134 34 162)(2 54 87 183 118 135 35 163)(3 55 88 184 119 136 36 164)(4 56 89 185 120 137 37 165)(5 57 90 186 121 138 38 166)(6 58 91 187 122 139 39 167)(7 59 92 188 123 140 40 168)(8 60 93 189 124 141 41 169)(9 61 94 190 125 142 42 170)(10 62 95 191 126 143 43 171)(11 63 96 192 127 144 44 172)(12 64 97 193 128 145 45 173)(13 65 98 194 129 146 46 174)(14 66 99 195 130 147 47 175)(15 67 100 196 105 148 48 176)(16 68 101 197 106 149 49 177)(17 69 102 198 107 150 50 178)(18 70 103 199 108 151 51 179)(19 71 104 200 109 152 52 180)(20 72 79 201 110 153 27 181)(21 73 80 202 111 154 28 182)(22 74 81 203 112 155 29 157)(23 75 82 204 113 156 30 158)(24 76 83 205 114 131 31 159)(25 77 84 206 115 132 32 160)(26 78 85 207 116 133 33 161)
(1 130)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(53 195)(54 196)(55 197)(56 198)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 189)(74 190)(75 191)(76 192)(77 193)(78 194)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(131 172)(132 173)(133 174)(134 175)(135 176)(136 177)(137 178)(138 179)(139 180)(140 181)(141 182)(142 157)(143 158)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 168)(154 169)(155 170)(156 171)
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), (1,53,86,208,117,134,34,162)(2,54,87,183,118,135,35,163)(3,55,88,184,119,136,36,164)(4,56,89,185,120,137,37,165)(5,57,90,186,121,138,38,166)(6,58,91,187,122,139,39,167)(7,59,92,188,123,140,40,168)(8,60,93,189,124,141,41,169)(9,61,94,190,125,142,42,170)(10,62,95,191,126,143,43,171)(11,63,96,192,127,144,44,172)(12,64,97,193,128,145,45,173)(13,65,98,194,129,146,46,174)(14,66,99,195,130,147,47,175)(15,67,100,196,105,148,48,176)(16,68,101,197,106,149,49,177)(17,69,102,198,107,150,50,178)(18,70,103,199,108,151,51,179)(19,71,104,200,109,152,52,180)(20,72,79,201,110,153,27,181)(21,73,80,202,111,154,28,182)(22,74,81,203,112,155,29,157)(23,75,82,204,113,156,30,158)(24,76,83,205,114,131,31,159)(25,77,84,206,115,132,32,160)(26,78,85,207,116,133,33,161), (1,130)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(53,195)(54,196)(55,197)(56,198)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(131,172)(132,173)(133,174)(134,175)(135,176)(136,177)(137,178)(138,179)(139,180)(140,181)(141,182)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)>;
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), (1,53,86,208,117,134,34,162)(2,54,87,183,118,135,35,163)(3,55,88,184,119,136,36,164)(4,56,89,185,120,137,37,165)(5,57,90,186,121,138,38,166)(6,58,91,187,122,139,39,167)(7,59,92,188,123,140,40,168)(8,60,93,189,124,141,41,169)(9,61,94,190,125,142,42,170)(10,62,95,191,126,143,43,171)(11,63,96,192,127,144,44,172)(12,64,97,193,128,145,45,173)(13,65,98,194,129,146,46,174)(14,66,99,195,130,147,47,175)(15,67,100,196,105,148,48,176)(16,68,101,197,106,149,49,177)(17,69,102,198,107,150,50,178)(18,70,103,199,108,151,51,179)(19,71,104,200,109,152,52,180)(20,72,79,201,110,153,27,181)(21,73,80,202,111,154,28,182)(22,74,81,203,112,155,29,157)(23,75,82,204,113,156,30,158)(24,76,83,205,114,131,31,159)(25,77,84,206,115,132,32,160)(26,78,85,207,116,133,33,161), (1,130)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(53,195)(54,196)(55,197)(56,198)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,183)(68,184)(69,185)(70,186)(71,187)(72,188)(73,189)(74,190)(75,191)(76,192)(77,193)(78,194)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(131,172)(132,173)(133,174)(134,175)(135,176)(136,177)(137,178)(138,179)(139,180)(140,181)(141,182)(142,157)(143,158)(144,159)(145,160)(146,161)(147,162)(148,163)(149,164)(150,165)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171) );
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)], [(1,53,86,208,117,134,34,162),(2,54,87,183,118,135,35,163),(3,55,88,184,119,136,36,164),(4,56,89,185,120,137,37,165),(5,57,90,186,121,138,38,166),(6,58,91,187,122,139,39,167),(7,59,92,188,123,140,40,168),(8,60,93,189,124,141,41,169),(9,61,94,190,125,142,42,170),(10,62,95,191,126,143,43,171),(11,63,96,192,127,144,44,172),(12,64,97,193,128,145,45,173),(13,65,98,194,129,146,46,174),(14,66,99,195,130,147,47,175),(15,67,100,196,105,148,48,176),(16,68,101,197,106,149,49,177),(17,69,102,198,107,150,50,178),(18,70,103,199,108,151,51,179),(19,71,104,200,109,152,52,180),(20,72,79,201,110,153,27,181),(21,73,80,202,111,154,28,182),(22,74,81,203,112,155,29,157),(23,75,82,204,113,156,30,158),(24,76,83,205,114,131,31,159),(25,77,84,206,115,132,32,160),(26,78,85,207,116,133,33,161)], [(1,130),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,121),(19,122),(20,123),(21,124),(22,125),(23,126),(24,127),(25,128),(26,129),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(53,195),(54,196),(55,197),(56,198),(57,199),(58,200),(59,201),(60,202),(61,203),(62,204),(63,205),(64,206),(65,207),(66,208),(67,183),(68,184),(69,185),(70,186),(71,187),(72,188),(73,189),(74,190),(75,191),(76,192),(77,193),(78,194),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104),(131,172),(132,173),(133,174),(134,175),(135,176),(136,177),(137,178),(138,179),(139,180),(140,181),(141,182),(142,157),(143,158),(144,159),(145,160),(146,161),(147,162),(148,163),(149,164),(150,165),(151,166),(152,167),(153,168),(154,169),(155,170),(156,171)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26CF | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | D4 | D4 | D8 | D4×C13 | D4×C13 | C13×D8 |
| kernel | D8×C26 | C2×C104 | C13×D8 | D4×C26 | C2×D8 | C2×C8 | D8 | C2×D4 | C52 | C2×C26 | C26 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 12 | 12 | 48 | 24 | 1 | 1 | 4 | 12 | 12 | 48 |
Matrix representation of D8×C26 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 0 | 120 |
| 0 | 253 | 120 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 312 |
G:=sub<GL(3,GF(313))| [312,0,0,0,64,0,0,0,64],[1,0,0,0,0,253,0,120,120],[1,0,0,0,1,1,0,0,312] >;
D8×C26 in GAP, Magma, Sage, TeX
D_8\times C_{26} % in TeX
G:=Group("D8xC26"); // GroupNames label
G:=SmallGroup(416,193);
// by ID
G=gap.SmallGroup(416,193);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,9364,4690,88]);
// 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^-1>;
// generators/relations