metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊2C8, C26.3M4(2), Dic13.20D4, C26.3(C2×C8), (C2×C52).3C4, C13⋊1(C22⋊C8), C2.4(D13⋊C8), C26.2(C22⋊C4), (C22×D13).6C4, C2.1(D13.D4), C2.3(C52.C4), (C2×Dic13).51C22, (C2×C13⋊C8)⋊1C2, (C2×C4×D13).8C2, (C2×C4).3(C13⋊C4), (C2×C26).6(C2×C4), C22.11(C2×C13⋊C4), SmallGroup(416,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊C8
G = < a,b,c | a26=b2=c8=1, bab=a-1, cac-1=a5, cbc-1=a17b >
26C2
26C2
2C4
13C4
13C22
13C22
13C4
26C22
26C22
2D13
2D13
13C23
13C2×C4
26C8
26C2×C4
26C8
26C2×C4
2D26
2C52
2D26
13C2×C8
13C22×C4
13C2×C8
13C22⋊C8
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)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(75 78)(76 77)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(99 104)(100 103)(101 102)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(120 130)(121 129)(122 128)(123 127)(124 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(176 182)(177 181)(178 180)(183 195)(184 194)(185 193)(186 192)(187 191)(188 190)(196 208)(197 207)(198 206)(199 205)(200 204)(201 203)
(1 196 89 47 136 119 77 173)(2 191 88 52 137 114 76 178)(3 186 87 31 138 109 75 157)(4 207 86 36 139 130 74 162)(5 202 85 41 140 125 73 167)(6 197 84 46 141 120 72 172)(7 192 83 51 142 115 71 177)(8 187 82 30 143 110 70 182)(9 208 81 35 144 105 69 161)(10 203 80 40 145 126 68 166)(11 198 79 45 146 121 67 171)(12 193 104 50 147 116 66 176)(13 188 103 29 148 111 65 181)(14 183 102 34 149 106 64 160)(15 204 101 39 150 127 63 165)(16 199 100 44 151 122 62 170)(17 194 99 49 152 117 61 175)(18 189 98 28 153 112 60 180)(19 184 97 33 154 107 59 159)(20 205 96 38 155 128 58 164)(21 200 95 43 156 123 57 169)(22 195 94 48 131 118 56 174)(23 190 93 27 132 113 55 179)(24 185 92 32 133 108 54 158)(25 206 91 37 134 129 53 163)(26 201 90 42 135 124 78 168)
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(75,78)(76,77)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(99,104)(100,103)(101,102)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(120,130)(121,129)(122,128)(123,127)(124,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(176,182)(177,181)(178,180)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(196,208)(197,207)(198,206)(199,205)(200,204)(201,203), (1,196,89,47,136,119,77,173)(2,191,88,52,137,114,76,178)(3,186,87,31,138,109,75,157)(4,207,86,36,139,130,74,162)(5,202,85,41,140,125,73,167)(6,197,84,46,141,120,72,172)(7,192,83,51,142,115,71,177)(8,187,82,30,143,110,70,182)(9,208,81,35,144,105,69,161)(10,203,80,40,145,126,68,166)(11,198,79,45,146,121,67,171)(12,193,104,50,147,116,66,176)(13,188,103,29,148,111,65,181)(14,183,102,34,149,106,64,160)(15,204,101,39,150,127,63,165)(16,199,100,44,151,122,62,170)(17,194,99,49,152,117,61,175)(18,189,98,28,153,112,60,180)(19,184,97,33,154,107,59,159)(20,205,96,38,155,128,58,164)(21,200,95,43,156,123,57,169)(22,195,94,48,131,118,56,174)(23,190,93,27,132,113,55,179)(24,185,92,32,133,108,54,158)(25,206,91,37,134,129,53,163)(26,201,90,42,135,124,78,168)>;
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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(75,78)(76,77)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(99,104)(100,103)(101,102)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(120,130)(121,129)(122,128)(123,127)(124,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(176,182)(177,181)(178,180)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(196,208)(197,207)(198,206)(199,205)(200,204)(201,203), (1,196,89,47,136,119,77,173)(2,191,88,52,137,114,76,178)(3,186,87,31,138,109,75,157)(4,207,86,36,139,130,74,162)(5,202,85,41,140,125,73,167)(6,197,84,46,141,120,72,172)(7,192,83,51,142,115,71,177)(8,187,82,30,143,110,70,182)(9,208,81,35,144,105,69,161)(10,203,80,40,145,126,68,166)(11,198,79,45,146,121,67,171)(12,193,104,50,147,116,66,176)(13,188,103,29,148,111,65,181)(14,183,102,34,149,106,64,160)(15,204,101,39,150,127,63,165)(16,199,100,44,151,122,62,170)(17,194,99,49,152,117,61,175)(18,189,98,28,153,112,60,180)(19,184,97,33,154,107,59,159)(20,205,96,38,155,128,58,164)(21,200,95,43,156,123,57,169)(22,195,94,48,131,118,56,174)(23,190,93,27,132,113,55,179)(24,185,92,32,133,108,54,158)(25,206,91,37,134,129,53,163)(26,201,90,42,135,124,78,168) );
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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(13,14),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(53,74),(54,73),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(75,78),(76,77),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(99,104),(100,103),(101,102),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(120,130),(121,129),(122,128),(123,127),(124,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,175),(158,174),(159,173),(160,172),(161,171),(162,170),(163,169),(164,168),(165,167),(176,182),(177,181),(178,180),(183,195),(184,194),(185,193),(186,192),(187,191),(188,190),(196,208),(197,207),(198,206),(199,205),(200,204),(201,203)], [(1,196,89,47,136,119,77,173),(2,191,88,52,137,114,76,178),(3,186,87,31,138,109,75,157),(4,207,86,36,139,130,74,162),(5,202,85,41,140,125,73,167),(6,197,84,46,141,120,72,172),(7,192,83,51,142,115,71,177),(8,187,82,30,143,110,70,182),(9,208,81,35,144,105,69,161),(10,203,80,40,145,126,68,166),(11,198,79,45,146,121,67,171),(12,193,104,50,147,116,66,176),(13,188,103,29,148,111,65,181),(14,183,102,34,149,106,64,160),(15,204,101,39,150,127,63,165),(16,199,100,44,151,122,62,170),(17,194,99,49,152,117,61,175),(18,189,98,28,153,112,60,180),(19,184,97,33,154,107,59,159),(20,205,96,38,155,128,58,164),(21,200,95,43,156,123,57,169),(22,195,94,48,131,118,56,174),(23,190,93,27,132,113,55,179),(24,185,92,32,133,108,54,158),(25,206,91,37,134,129,53,163),(26,201,90,42,135,124,78,168)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | C13⋊C4 | C2×C13⋊C4 | D13⋊C8 | C52.C4 | D13.D4 |
| kernel | D26⋊C8 | C2×C13⋊C8 | C2×C4×D13 | C2×C52 | C22×D13 | D26 | Dic13 | C26 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 3 | 3 | 6 | 6 | 6 |
Matrix representation of D26⋊C8 ►in GL6(𝔽313)
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 200 | 199 | 100 |
| 0 | 0 | 213 | 84 | 84 | 213 |
| 0 | 0 | 131 | 230 | 231 | 101 |
| 0 | 0 | 243 | 213 | 243 | 0 |
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 243 | 0 | 29 | 29 |
| 0 | 0 | 100 | 312 | 212 | 212 |
| 0 | 0 | 182 | 212 | 312 | 311 |
| 0 | 0 | 70 | 101 | 71 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 268 | 185 | 193 | 311 |
| 0 | 0 | 133 | 304 | 242 | 310 |
| 0 | 0 | 296 | 152 | 20 | 138 |
| 0 | 0 | 98 | 196 | 186 | 34 |
G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,70,213,131,243,0,0,200,84,230,213,0,0,199,84,231,243,0,0,100,213,101,0],[312,0,0,0,0,0,0,1,0,0,0,0,0,0,243,100,182,70,0,0,0,312,212,101,0,0,29,212,312,71,0,0,29,212,311,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,268,133,296,98,0,0,185,304,152,196,0,0,193,242,20,186,0,0,311,310,138,34] >;
D26⋊C8 in GAP, Magma, Sage, TeX
D_{26}\rtimes C_8 % in TeX
G:=Group("D26:C8"); // GroupNames label
G:=SmallGroup(416,78);
// by ID
G=gap.SmallGroup(416,78);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,86,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^26=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^17*b>;
// generators/relations
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