metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C26.5D4, C26.1Q8, Dic13⋊1C4, C2.1Dic26, C22.4D26, C13⋊2(C4⋊C4), (C2×C52).1C2, C2.4(C4×D13), (C2×C4).1D13, C26.11(C2×C4), C2.1(C13⋊D4), (C2×C26).4C22, (C2×Dic13).1C2, SmallGroup(208,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C26.D4
G = < a,b,c | a26=b4=1, c2=a13, bab-1=cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C26.D4
►Regular action 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)
(1 112 201 30)(2 111 202 29)(3 110 203 28)(4 109 204 27)(5 108 205 52)(6 107 206 51)(7 106 207 50)(8 105 208 49)(9 130 183 48)(10 129 184 47)(11 128 185 46)(12 127 186 45)(13 126 187 44)(14 125 188 43)(15 124 189 42)(16 123 190 41)(17 122 191 40)(18 121 192 39)(19 120 193 38)(20 119 194 37)(21 118 195 36)(22 117 196 35)(23 116 197 34)(24 115 198 33)(25 114 199 32)(26 113 200 31)(53 98 166 152)(54 97 167 151)(55 96 168 150)(56 95 169 149)(57 94 170 148)(58 93 171 147)(59 92 172 146)(60 91 173 145)(61 90 174 144)(62 89 175 143)(63 88 176 142)(64 87 177 141)(65 86 178 140)(66 85 179 139)(67 84 180 138)(68 83 181 137)(69 82 182 136)(70 81 157 135)(71 80 158 134)(72 79 159 133)(73 104 160 132)(74 103 161 131)(75 102 162 156)(76 101 163 155)(77 100 164 154)(78 99 165 153)
(1 95 14 82)(2 94 15 81)(3 93 16 80)(4 92 17 79)(5 91 18 104)(6 90 19 103)(7 89 20 102)(8 88 21 101)(9 87 22 100)(10 86 23 99)(11 85 24 98)(12 84 25 97)(13 83 26 96)(27 172 40 159)(28 171 41 158)(29 170 42 157)(30 169 43 182)(31 168 44 181)(32 167 45 180)(33 166 46 179)(34 165 47 178)(35 164 48 177)(36 163 49 176)(37 162 50 175)(38 161 51 174)(39 160 52 173)(53 128 66 115)(54 127 67 114)(55 126 68 113)(56 125 69 112)(57 124 70 111)(58 123 71 110)(59 122 72 109)(60 121 73 108)(61 120 74 107)(62 119 75 106)(63 118 76 105)(64 117 77 130)(65 116 78 129)(131 206 144 193)(132 205 145 192)(133 204 146 191)(134 203 147 190)(135 202 148 189)(136 201 149 188)(137 200 150 187)(138 199 151 186)(139 198 152 185)(140 197 153 184)(141 196 154 183)(142 195 155 208)(143 194 156 207)
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), (1,112,201,30)(2,111,202,29)(3,110,203,28)(4,109,204,27)(5,108,205,52)(6,107,206,51)(7,106,207,50)(8,105,208,49)(9,130,183,48)(10,129,184,47)(11,128,185,46)(12,127,186,45)(13,126,187,44)(14,125,188,43)(15,124,189,42)(16,123,190,41)(17,122,191,40)(18,121,192,39)(19,120,193,38)(20,119,194,37)(21,118,195,36)(22,117,196,35)(23,116,197,34)(24,115,198,33)(25,114,199,32)(26,113,200,31)(53,98,166,152)(54,97,167,151)(55,96,168,150)(56,95,169,149)(57,94,170,148)(58,93,171,147)(59,92,172,146)(60,91,173,145)(61,90,174,144)(62,89,175,143)(63,88,176,142)(64,87,177,141)(65,86,178,140)(66,85,179,139)(67,84,180,138)(68,83,181,137)(69,82,182,136)(70,81,157,135)(71,80,158,134)(72,79,159,133)(73,104,160,132)(74,103,161,131)(75,102,162,156)(76,101,163,155)(77,100,164,154)(78,99,165,153), (1,95,14,82)(2,94,15,81)(3,93,16,80)(4,92,17,79)(5,91,18,104)(6,90,19,103)(7,89,20,102)(8,88,21,101)(9,87,22,100)(10,86,23,99)(11,85,24,98)(12,84,25,97)(13,83,26,96)(27,172,40,159)(28,171,41,158)(29,170,42,157)(30,169,43,182)(31,168,44,181)(32,167,45,180)(33,166,46,179)(34,165,47,178)(35,164,48,177)(36,163,49,176)(37,162,50,175)(38,161,51,174)(39,160,52,173)(53,128,66,115)(54,127,67,114)(55,126,68,113)(56,125,69,112)(57,124,70,111)(58,123,71,110)(59,122,72,109)(60,121,73,108)(61,120,74,107)(62,119,75,106)(63,118,76,105)(64,117,77,130)(65,116,78,129)(131,206,144,193)(132,205,145,192)(133,204,146,191)(134,203,147,190)(135,202,148,189)(136,201,149,188)(137,200,150,187)(138,199,151,186)(139,198,152,185)(140,197,153,184)(141,196,154,183)(142,195,155,208)(143,194,156,207)>;
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), (1,112,201,30)(2,111,202,29)(3,110,203,28)(4,109,204,27)(5,108,205,52)(6,107,206,51)(7,106,207,50)(8,105,208,49)(9,130,183,48)(10,129,184,47)(11,128,185,46)(12,127,186,45)(13,126,187,44)(14,125,188,43)(15,124,189,42)(16,123,190,41)(17,122,191,40)(18,121,192,39)(19,120,193,38)(20,119,194,37)(21,118,195,36)(22,117,196,35)(23,116,197,34)(24,115,198,33)(25,114,199,32)(26,113,200,31)(53,98,166,152)(54,97,167,151)(55,96,168,150)(56,95,169,149)(57,94,170,148)(58,93,171,147)(59,92,172,146)(60,91,173,145)(61,90,174,144)(62,89,175,143)(63,88,176,142)(64,87,177,141)(65,86,178,140)(66,85,179,139)(67,84,180,138)(68,83,181,137)(69,82,182,136)(70,81,157,135)(71,80,158,134)(72,79,159,133)(73,104,160,132)(74,103,161,131)(75,102,162,156)(76,101,163,155)(77,100,164,154)(78,99,165,153), (1,95,14,82)(2,94,15,81)(3,93,16,80)(4,92,17,79)(5,91,18,104)(6,90,19,103)(7,89,20,102)(8,88,21,101)(9,87,22,100)(10,86,23,99)(11,85,24,98)(12,84,25,97)(13,83,26,96)(27,172,40,159)(28,171,41,158)(29,170,42,157)(30,169,43,182)(31,168,44,181)(32,167,45,180)(33,166,46,179)(34,165,47,178)(35,164,48,177)(36,163,49,176)(37,162,50,175)(38,161,51,174)(39,160,52,173)(53,128,66,115)(54,127,67,114)(55,126,68,113)(56,125,69,112)(57,124,70,111)(58,123,71,110)(59,122,72,109)(60,121,73,108)(61,120,74,107)(62,119,75,106)(63,118,76,105)(64,117,77,130)(65,116,78,129)(131,206,144,193)(132,205,145,192)(133,204,146,191)(134,203,147,190)(135,202,148,189)(136,201,149,188)(137,200,150,187)(138,199,151,186)(139,198,152,185)(140,197,153,184)(141,196,154,183)(142,195,155,208)(143,194,156,207) );
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)], [(1,112,201,30),(2,111,202,29),(3,110,203,28),(4,109,204,27),(5,108,205,52),(6,107,206,51),(7,106,207,50),(8,105,208,49),(9,130,183,48),(10,129,184,47),(11,128,185,46),(12,127,186,45),(13,126,187,44),(14,125,188,43),(15,124,189,42),(16,123,190,41),(17,122,191,40),(18,121,192,39),(19,120,193,38),(20,119,194,37),(21,118,195,36),(22,117,196,35),(23,116,197,34),(24,115,198,33),(25,114,199,32),(26,113,200,31),(53,98,166,152),(54,97,167,151),(55,96,168,150),(56,95,169,149),(57,94,170,148),(58,93,171,147),(59,92,172,146),(60,91,173,145),(61,90,174,144),(62,89,175,143),(63,88,176,142),(64,87,177,141),(65,86,178,140),(66,85,179,139),(67,84,180,138),(68,83,181,137),(69,82,182,136),(70,81,157,135),(71,80,158,134),(72,79,159,133),(73,104,160,132),(74,103,161,131),(75,102,162,156),(76,101,163,155),(77,100,164,154),(78,99,165,153)], [(1,95,14,82),(2,94,15,81),(3,93,16,80),(4,92,17,79),(5,91,18,104),(6,90,19,103),(7,89,20,102),(8,88,21,101),(9,87,22,100),(10,86,23,99),(11,85,24,98),(12,84,25,97),(13,83,26,96),(27,172,40,159),(28,171,41,158),(29,170,42,157),(30,169,43,182),(31,168,44,181),(32,167,45,180),(33,166,46,179),(34,165,47,178),(35,164,48,177),(36,163,49,176),(37,162,50,175),(38,161,51,174),(39,160,52,173),(53,128,66,115),(54,127,67,114),(55,126,68,113),(56,125,69,112),(57,124,70,111),(58,123,71,110),(59,122,72,109),(60,121,73,108),(61,120,74,107),(62,119,75,106),(63,118,76,105),(64,117,77,130),(65,116,78,129),(131,206,144,193),(132,205,145,192),(133,204,146,191),(134,203,147,190),(135,202,148,189),(136,201,149,188),(137,200,150,187),(138,199,151,186),(139,198,152,185),(140,197,153,184),(141,196,154,183),(142,195,155,208),(143,194,156,207)]])
C26.D4 is a maximal subgroup of
C4×Dic26 C52.6Q8 C42⋊D13 C42⋊2D13 C23.11D26 C22⋊Dic26 C23.D26 Dic13⋊4D4 D26.12D4 D26⋊D4 Dic13⋊3Q8 C52⋊Q8 Dic13.Q8 C4.Dic26 C4⋊C4×D13 D26.13D4 D26⋊Q8 C4⋊C4⋊D13 C52.48D4 C4×C13⋊D4 C23.23D26 C23.18D26 Dic13⋊D4 Dic13⋊Q8 D26⋊3Q8
C26.D4 is a maximal quotient of
C26.D8 C52.Q8 C52.8Q8 C52.53D4 C26.10C42
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D13 | D26 | Dic26 | C4×D13 | C13⋊D4 |
| kernel | C26.D4 | C2×Dic13 | C2×C52 | Dic13 | C26 | C26 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of C26.D4 ►in GL3(𝔽53) generated by
| 52 | 0 | 0 |
| 0 | 47 | 47 |
| 0 | 6 | 50 |
| 1 | 0 | 0 |
| 0 | 7 | 11 |
| 0 | 34 | 46 |
| 30 | 0 | 0 |
| 0 | 13 | 37 |
| 0 | 4 | 40 |
G:=sub<GL(3,GF(53))| [52,0,0,0,47,6,0,47,50],[1,0,0,0,7,34,0,11,46],[30,0,0,0,13,4,0,37,40] >;
C26.D4 in GAP, Magma, Sage, TeX
C_{26}.D_4 % in TeX
G:=Group("C26.D4"); // GroupNames label
G:=SmallGroup(208,12);
// by ID
G=gap.SmallGroup(208,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,40,101,26,4804]);
// Polycyclic
G:=Group<a,b,c|a^26=b^4=1,c^2=a^13,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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