metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104.5C4, D26.3C8, C13⋊1M5(2), Dic13.3C8, C13⋊C16⋊1C2, C8.3(C13⋊C4), C26.2(C2×C8), C52.14(C2×C4), (C4×D13).6C4, (C8×D13).8C2, C2.3(D13⋊C8), C13⋊2C8.15C22, C4.15(C2×C13⋊C4), SmallGroup(416,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26.C8
G = < a,b,c | a26=b2=1, c8=a13, bab=a-1, cac-1=a5, cbc-1=a17b >
Smallest permutation representation of D26.C8
►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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(47 52)(48 51)(49 50)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(77 78)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(95 104)(96 103)(97 102)(98 101)(99 100)(105 107)(108 130)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(157 163)(158 162)(159 161)(164 182)(165 181)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(183 207)(184 206)(185 205)(186 204)(187 203)(188 202)(189 201)(190 200)(191 199)(192 198)(193 197)(194 196)
(1 202 87 144 50 167 65 113 14 189 100 131 37 180 78 126)(2 197 86 149 51 162 64 118 15 184 99 136 38 175 77 105)(3 192 85 154 52 157 63 123 16 205 98 141 39 170 76 110)(4 187 84 133 27 178 62 128 17 200 97 146 40 165 75 115)(5 208 83 138 28 173 61 107 18 195 96 151 41 160 74 120)(6 203 82 143 29 168 60 112 19 190 95 156 42 181 73 125)(7 198 81 148 30 163 59 117 20 185 94 135 43 176 72 130)(8 193 80 153 31 158 58 122 21 206 93 140 44 171 71 109)(9 188 79 132 32 179 57 127 22 201 92 145 45 166 70 114)(10 183 104 137 33 174 56 106 23 196 91 150 46 161 69 119)(11 204 103 142 34 169 55 111 24 191 90 155 47 182 68 124)(12 199 102 147 35 164 54 116 25 186 89 134 48 177 67 129)(13 194 101 152 36 159 53 121 26 207 88 139 49 172 66 108)
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(47,52)(48,51)(49,50)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(77,78)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(95,104)(96,103)(97,102)(98,101)(99,100)(105,107)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(157,163)(158,162)(159,161)(164,182)(165,181)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(183,207)(184,206)(185,205)(186,204)(187,203)(188,202)(189,201)(190,200)(191,199)(192,198)(193,197)(194,196), (1,202,87,144,50,167,65,113,14,189,100,131,37,180,78,126)(2,197,86,149,51,162,64,118,15,184,99,136,38,175,77,105)(3,192,85,154,52,157,63,123,16,205,98,141,39,170,76,110)(4,187,84,133,27,178,62,128,17,200,97,146,40,165,75,115)(5,208,83,138,28,173,61,107,18,195,96,151,41,160,74,120)(6,203,82,143,29,168,60,112,19,190,95,156,42,181,73,125)(7,198,81,148,30,163,59,117,20,185,94,135,43,176,72,130)(8,193,80,153,31,158,58,122,21,206,93,140,44,171,71,109)(9,188,79,132,32,179,57,127,22,201,92,145,45,166,70,114)(10,183,104,137,33,174,56,106,23,196,91,150,46,161,69,119)(11,204,103,142,34,169,55,111,24,191,90,155,47,182,68,124)(12,199,102,147,35,164,54,116,25,186,89,134,48,177,67,129)(13,194,101,152,36,159,53,121,26,207,88,139,49,172,66,108)>;
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(47,52)(48,51)(49,50)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(77,78)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(95,104)(96,103)(97,102)(98,101)(99,100)(105,107)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(157,163)(158,162)(159,161)(164,182)(165,181)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(183,207)(184,206)(185,205)(186,204)(187,203)(188,202)(189,201)(190,200)(191,199)(192,198)(193,197)(194,196), (1,202,87,144,50,167,65,113,14,189,100,131,37,180,78,126)(2,197,86,149,51,162,64,118,15,184,99,136,38,175,77,105)(3,192,85,154,52,157,63,123,16,205,98,141,39,170,76,110)(4,187,84,133,27,178,62,128,17,200,97,146,40,165,75,115)(5,208,83,138,28,173,61,107,18,195,96,151,41,160,74,120)(6,203,82,143,29,168,60,112,19,190,95,156,42,181,73,125)(7,198,81,148,30,163,59,117,20,185,94,135,43,176,72,130)(8,193,80,153,31,158,58,122,21,206,93,140,44,171,71,109)(9,188,79,132,32,179,57,127,22,201,92,145,45,166,70,114)(10,183,104,137,33,174,56,106,23,196,91,150,46,161,69,119)(11,204,103,142,34,169,55,111,24,191,90,155,47,182,68,124)(12,199,102,147,35,164,54,116,25,186,89,134,48,177,67,129)(13,194,101,152,36,159,53,121,26,207,88,139,49,172,66,108) );
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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(13,14),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(47,52),(48,51),(49,50),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(77,78),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(95,104),(96,103),(97,102),(98,101),(99,100),(105,107),(108,130),(109,129),(110,128),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151),(157,163),(158,162),(159,161),(164,182),(165,181),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(183,207),(184,206),(185,205),(186,204),(187,203),(188,202),(189,201),(190,200),(191,199),(192,198),(193,197),(194,196)], [(1,202,87,144,50,167,65,113,14,189,100,131,37,180,78,126),(2,197,86,149,51,162,64,118,15,184,99,136,38,175,77,105),(3,192,85,154,52,157,63,123,16,205,98,141,39,170,76,110),(4,187,84,133,27,178,62,128,17,200,97,146,40,165,75,115),(5,208,83,138,28,173,61,107,18,195,96,151,41,160,74,120),(6,203,82,143,29,168,60,112,19,190,95,156,42,181,73,125),(7,198,81,148,30,163,59,117,20,185,94,135,43,176,72,130),(8,193,80,153,31,158,58,122,21,206,93,140,44,171,71,109),(9,188,79,132,32,179,57,127,22,201,92,145,45,166,70,114),(10,183,104,137,33,174,56,106,23,196,91,150,46,161,69,119),(11,204,103,142,34,169,55,111,24,191,90,155,47,182,68,124),(12,199,102,147,35,164,54,116,25,186,89,134,48,177,67,129),(13,194,101,152,36,159,53,121,26,207,88,139,49,172,66,108)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 13A | 13B | 13C | 16A | ··· | 16H | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 16 | ··· | 16 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 26 | 1 | 1 | 26 | 2 | 2 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) | C13⋊C4 | C2×C13⋊C4 | D13⋊C8 | D26.C8 |
| kernel | D26.C8 | C13⋊C16 | C8×D13 | C104 | C4×D13 | Dic13 | D26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 3 | 3 | 6 | 12 |
Matrix representation of D26.C8 ►in GL4(𝔽5) generated by
| 3 | 4 | 4 | 1 |
| 3 | 4 | 1 | 4 |
| 3 | 3 | 4 | 2 |
| 2 | 4 | 1 | 2 |
| 3 | 2 | 4 | 0 |
| 3 | 3 | 1 | 1 |
| 4 | 2 | 4 | 2 |
| 3 | 4 | 1 | 0 |
| 1 | 2 | 1 | 2 |
| 3 | 0 | 1 | 0 |
| 2 | 0 | 2 | 3 |
| 2 | 0 | 2 | 2 |
G:=sub<GL(4,GF(5))| [3,3,3,2,4,4,3,4,4,1,4,1,1,4,2,2],[3,3,4,3,2,3,2,4,4,1,4,1,0,1,2,0],[1,3,2,2,2,0,0,0,1,1,2,2,2,0,3,2] >;
D26.C8 in GAP, Magma, Sage, TeX
D_{26}.C_8 % in TeX
G:=Group("D26.C8"); // GroupNames label
G:=SmallGroup(416,65);
// by ID
G=gap.SmallGroup(416,65);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,50,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^26=b^2=1,c^8=a^13,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^17*b>;
// generators/relations
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