metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊1C8, C52.52D4, C4.19D52, C26.8M4(2), (C2×C8)⋊1D13, (C2×C104)⋊1C2, C2.5(C8×D13), C13⋊3(C22⋊C8), C26.14(C2×C8), (C2×C4).93D26, C2.3(C8⋊D13), C4.27(C13⋊D4), (C2×Dic13).5C4, (C22×D13).3C4, C22.11(C4×D13), C2.1(D26⋊C4), C26.17(C22⋊C4), (C2×C52).107C22, (C2×C4×D13).7C2, (C2×C13⋊2C8)⋊9C2, (C2×C26).32(C2×C4), SmallGroup(416,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊1C8
G = < a,b,c | a26=b2=c8=1, bab=a-1, ac=ca, cbc-1=a13b >
26C2
26C2
13C22
13C22
26C22
26C22
26C4
2D13
2D13
2C8
13C2×C4
13C23
26C8
26C2×C4
26C2×C4
2D26
2D26
13C2×C8
13C22×C4
2C104
13C22⋊C8
Smallest permutation representation of D26⋊1C8
►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 31)(28 30)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(53 56)(54 55)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(79 81)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(105 106)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(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)(158 182)(159 181)(160 180)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)(183 185)(186 208)(187 207)(188 206)(189 205)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)
(1 100 68 191 106 36 136 164)(2 101 69 192 107 37 137 165)(3 102 70 193 108 38 138 166)(4 103 71 194 109 39 139 167)(5 104 72 195 110 40 140 168)(6 79 73 196 111 41 141 169)(7 80 74 197 112 42 142 170)(8 81 75 198 113 43 143 171)(9 82 76 199 114 44 144 172)(10 83 77 200 115 45 145 173)(11 84 78 201 116 46 146 174)(12 85 53 202 117 47 147 175)(13 86 54 203 118 48 148 176)(14 87 55 204 119 49 149 177)(15 88 56 205 120 50 150 178)(16 89 57 206 121 51 151 179)(17 90 58 207 122 52 152 180)(18 91 59 208 123 27 153 181)(19 92 60 183 124 28 154 182)(20 93 61 184 125 29 155 157)(21 94 62 185 126 30 156 158)(22 95 63 186 127 31 131 159)(23 96 64 187 128 32 132 160)(24 97 65 188 129 33 133 161)(25 98 66 189 130 34 134 162)(26 99 67 190 105 35 135 163)
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)(27,31)(28,30)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(53,56)(54,55)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(79,81)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(105,106)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(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)(158,182)(159,181)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171)(183,185)(186,208)(187,207)(188,206)(189,205)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198), (1,100,68,191,106,36,136,164)(2,101,69,192,107,37,137,165)(3,102,70,193,108,38,138,166)(4,103,71,194,109,39,139,167)(5,104,72,195,110,40,140,168)(6,79,73,196,111,41,141,169)(7,80,74,197,112,42,142,170)(8,81,75,198,113,43,143,171)(9,82,76,199,114,44,144,172)(10,83,77,200,115,45,145,173)(11,84,78,201,116,46,146,174)(12,85,53,202,117,47,147,175)(13,86,54,203,118,48,148,176)(14,87,55,204,119,49,149,177)(15,88,56,205,120,50,150,178)(16,89,57,206,121,51,151,179)(17,90,58,207,122,52,152,180)(18,91,59,208,123,27,153,181)(19,92,60,183,124,28,154,182)(20,93,61,184,125,29,155,157)(21,94,62,185,126,30,156,158)(22,95,63,186,127,31,131,159)(23,96,64,187,128,32,132,160)(24,97,65,188,129,33,133,161)(25,98,66,189,130,34,134,162)(26,99,67,190,105,35,135,163)>;
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)(27,31)(28,30)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(53,56)(54,55)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(79,81)(82,104)(83,103)(84,102)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(105,106)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(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)(158,182)(159,181)(160,180)(161,179)(162,178)(163,177)(164,176)(165,175)(166,174)(167,173)(168,172)(169,171)(183,185)(186,208)(187,207)(188,206)(189,205)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198), (1,100,68,191,106,36,136,164)(2,101,69,192,107,37,137,165)(3,102,70,193,108,38,138,166)(4,103,71,194,109,39,139,167)(5,104,72,195,110,40,140,168)(6,79,73,196,111,41,141,169)(7,80,74,197,112,42,142,170)(8,81,75,198,113,43,143,171)(9,82,76,199,114,44,144,172)(10,83,77,200,115,45,145,173)(11,84,78,201,116,46,146,174)(12,85,53,202,117,47,147,175)(13,86,54,203,118,48,148,176)(14,87,55,204,119,49,149,177)(15,88,56,205,120,50,150,178)(16,89,57,206,121,51,151,179)(17,90,58,207,122,52,152,180)(18,91,59,208,123,27,153,181)(19,92,60,183,124,28,154,182)(20,93,61,184,125,29,155,157)(21,94,62,185,126,30,156,158)(22,95,63,186,127,31,131,159)(23,96,64,187,128,32,132,160)(24,97,65,188,129,33,133,161)(25,98,66,189,130,34,134,162)(26,99,67,190,105,35,135,163) );
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),(27,31),(28,30),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(53,56),(54,55),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(79,81),(82,104),(83,103),(84,102),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(105,106),(107,130),(108,129),(109,128),(110,127),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(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),(158,182),(159,181),(160,180),(161,179),(162,178),(163,177),(164,176),(165,175),(166,174),(167,173),(168,172),(169,171),(183,185),(186,208),(187,207),(188,206),(189,205),(190,204),(191,203),(192,202),(193,201),(194,200),(195,199),(196,198)], [(1,100,68,191,106,36,136,164),(2,101,69,192,107,37,137,165),(3,102,70,193,108,38,138,166),(4,103,71,194,109,39,139,167),(5,104,72,195,110,40,140,168),(6,79,73,196,111,41,141,169),(7,80,74,197,112,42,142,170),(8,81,75,198,113,43,143,171),(9,82,76,199,114,44,144,172),(10,83,77,200,115,45,145,173),(11,84,78,201,116,46,146,174),(12,85,53,202,117,47,147,175),(13,86,54,203,118,48,148,176),(14,87,55,204,119,49,149,177),(15,88,56,205,120,50,150,178),(16,89,57,206,121,51,151,179),(17,90,58,207,122,52,152,180),(18,91,59,208,123,27,153,181),(19,92,60,183,124,28,154,182),(20,93,61,184,125,29,155,157),(21,94,62,185,126,30,156,158),(22,95,63,186,127,31,131,159),(23,96,64,187,128,32,132,160),(24,97,65,188,129,33,133,161),(25,98,66,189,130,34,134,162),(26,99,67,190,105,35,135,163)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 1 | 1 | 1 | 1 | 26 | 26 | 2 | 2 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | D13 | D26 | D52 | C13⋊D4 | C4×D13 | C8×D13 | C8⋊D13 |
| kernel | D26⋊1C8 | C2×C13⋊2C8 | C2×C104 | C2×C4×D13 | C2×Dic13 | C22×D13 | D26 | C52 | C26 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 24 | 24 |
Matrix representation of D26⋊1C8 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 232 | 126 |
| 0 | 298 | 4 |
| 312 | 0 | 0 |
| 0 | 1 | 203 |
| 0 | 0 | 312 |
| 188 | 0 | 0 |
| 0 | 44 | 131 |
| 0 | 126 | 269 |
G:=sub<GL(3,GF(313))| [1,0,0,0,232,298,0,126,4],[312,0,0,0,1,0,0,203,312],[188,0,0,0,44,126,0,131,269] >;
D26⋊1C8 in GAP, Magma, Sage, TeX
D_{26}\rtimes_1C_8 % in TeX
G:=Group("D26:1C8"); // GroupNames label
G:=SmallGroup(416,27);
// by ID
G=gap.SmallGroup(416,27);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^26=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^13*b>;
// generators/relations
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