direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic26, C26⋊Q8, C4.11D26, C26.1C23, C22.8D26, C52.11C22, Dic13.1C22, C13⋊1(C2×Q8), (C2×C52).4C2, (C2×C4).4D13, (C2×C26).8C22, C2.3(C22×D13), (C2×Dic13).3C2, SmallGroup(208,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic26
G = < a,b,c | a2=b52=1, c2=b26, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic26
►Regular action on 208 points
Generators in S208
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(113 191)(114 192)(115 193)(116 194)(117 195)(118 196)(119 197)(120 198)(121 199)(122 200)(123 201)(124 202)(125 203)(126 204)(127 205)(128 206)(129 207)(130 208)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 169)(144 170)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 177)(152 178)(153 179)(154 180)(155 181)(156 182)
(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)
(1 131 27 105)(2 130 28 156)(3 129 29 155)(4 128 30 154)(5 127 31 153)(6 126 32 152)(7 125 33 151)(8 124 34 150)(9 123 35 149)(10 122 36 148)(11 121 37 147)(12 120 38 146)(13 119 39 145)(14 118 40 144)(15 117 41 143)(16 116 42 142)(17 115 43 141)(18 114 44 140)(19 113 45 139)(20 112 46 138)(21 111 47 137)(22 110 48 136)(23 109 49 135)(24 108 50 134)(25 107 51 133)(26 106 52 132)(53 202 79 176)(54 201 80 175)(55 200 81 174)(56 199 82 173)(57 198 83 172)(58 197 84 171)(59 196 85 170)(60 195 86 169)(61 194 87 168)(62 193 88 167)(63 192 89 166)(64 191 90 165)(65 190 91 164)(66 189 92 163)(67 188 93 162)(68 187 94 161)(69 186 95 160)(70 185 96 159)(71 184 97 158)(72 183 98 157)(73 182 99 208)(74 181 100 207)(75 180 101 206)(76 179 102 205)(77 178 103 204)(78 177 104 203)
G:=sub<Sym(208)| (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,181)(156,182), (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), (1,131,27,105)(2,130,28,156)(3,129,29,155)(4,128,30,154)(5,127,31,153)(6,126,32,152)(7,125,33,151)(8,124,34,150)(9,123,35,149)(10,122,36,148)(11,121,37,147)(12,120,38,146)(13,119,39,145)(14,118,40,144)(15,117,41,143)(16,116,42,142)(17,115,43,141)(18,114,44,140)(19,113,45,139)(20,112,46,138)(21,111,47,137)(22,110,48,136)(23,109,49,135)(24,108,50,134)(25,107,51,133)(26,106,52,132)(53,202,79,176)(54,201,80,175)(55,200,81,174)(56,199,82,173)(57,198,83,172)(58,197,84,171)(59,196,85,170)(60,195,86,169)(61,194,87,168)(62,193,88,167)(63,192,89,166)(64,191,90,165)(65,190,91,164)(66,189,92,163)(67,188,93,162)(68,187,94,161)(69,186,95,160)(70,185,96,159)(71,184,97,158)(72,183,98,157)(73,182,99,208)(74,181,100,207)(75,180,101,206)(76,179,102,205)(77,178,103,204)(78,177,104,203)>;
G:=Group( (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,181)(156,182), (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), (1,131,27,105)(2,130,28,156)(3,129,29,155)(4,128,30,154)(5,127,31,153)(6,126,32,152)(7,125,33,151)(8,124,34,150)(9,123,35,149)(10,122,36,148)(11,121,37,147)(12,120,38,146)(13,119,39,145)(14,118,40,144)(15,117,41,143)(16,116,42,142)(17,115,43,141)(18,114,44,140)(19,113,45,139)(20,112,46,138)(21,111,47,137)(22,110,48,136)(23,109,49,135)(24,108,50,134)(25,107,51,133)(26,106,52,132)(53,202,79,176)(54,201,80,175)(55,200,81,174)(56,199,82,173)(57,198,83,172)(58,197,84,171)(59,196,85,170)(60,195,86,169)(61,194,87,168)(62,193,88,167)(63,192,89,166)(64,191,90,165)(65,190,91,164)(66,189,92,163)(67,188,93,162)(68,187,94,161)(69,186,95,160)(70,185,96,159)(71,184,97,158)(72,183,98,157)(73,182,99,208)(74,181,100,207)(75,180,101,206)(76,179,102,205)(77,178,103,204)(78,177,104,203) );
G=PermutationGroup([[(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(37,82),(38,83),(39,84),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(105,183),(106,184),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(113,191),(114,192),(115,193),(116,194),(117,195),(118,196),(119,197),(120,198),(121,199),(122,200),(123,201),(124,202),(125,203),(126,204),(127,205),(128,206),(129,207),(130,208),(131,157),(132,158),(133,159),(134,160),(135,161),(136,162),(137,163),(138,164),(139,165),(140,166),(141,167),(142,168),(143,169),(144,170),(145,171),(146,172),(147,173),(148,174),(149,175),(150,176),(151,177),(152,178),(153,179),(154,180),(155,181),(156,182)], [(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)], [(1,131,27,105),(2,130,28,156),(3,129,29,155),(4,128,30,154),(5,127,31,153),(6,126,32,152),(7,125,33,151),(8,124,34,150),(9,123,35,149),(10,122,36,148),(11,121,37,147),(12,120,38,146),(13,119,39,145),(14,118,40,144),(15,117,41,143),(16,116,42,142),(17,115,43,141),(18,114,44,140),(19,113,45,139),(20,112,46,138),(21,111,47,137),(22,110,48,136),(23,109,49,135),(24,108,50,134),(25,107,51,133),(26,106,52,132),(53,202,79,176),(54,201,80,175),(55,200,81,174),(56,199,82,173),(57,198,83,172),(58,197,84,171),(59,196,85,170),(60,195,86,169),(61,194,87,168),(62,193,88,167),(63,192,89,166),(64,191,90,165),(65,190,91,164),(66,189,92,163),(67,188,93,162),(68,187,94,161),(69,186,95,160),(70,185,96,159),(71,184,97,158),(72,183,98,157),(73,182,99,208),(74,181,100,207),(75,180,101,206),(76,179,102,205),(77,178,103,204),(78,177,104,203)]])
C2×Dic26 is a maximal subgroup of
C26.Q16 C52.44D4 C4.12D52 Dic13.D4 C52⋊2Q8 C4.D52 C22⋊Dic26 C23.6D26 Dic13⋊3Q8 C52⋊Q8 D26⋊Q8 D26⋊2Q8 C8.D26 C52.48D4 C52.17D4 Dic13⋊Q8 D4.9D26 C2×Q8×D13 D4.10D26
C2×Dic26 is a maximal quotient of
C52⋊2Q8 C52.6Q8 C22⋊Dic26 C52⋊Q8 C4.Dic26 C52.48D4
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 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D13 | D26 | D26 | Dic26 |
| kernel | C2×Dic26 | Dic26 | C2×Dic13 | C2×C52 | C26 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 6 | 12 | 6 | 24 |
Matrix representation of C2×Dic26 ►in GL3(𝔽53) generated by
| 52 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 52 | 0 | 0 |
| 0 | 19 | 28 |
| 0 | 25 | 9 |
| 52 | 0 | 0 |
| 0 | 0 | 30 |
| 0 | 30 | 0 |
G:=sub<GL(3,GF(53))| [52,0,0,0,1,0,0,0,1],[52,0,0,0,19,25,0,28,9],[52,0,0,0,0,30,0,30,0] >;
C2×Dic26 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{26} % in TeX
G:=Group("C2xDic26"); // GroupNames label
G:=SmallGroup(208,35);
// by ID
G=gap.SmallGroup(208,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,40,182,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^52=1,c^2=b^26,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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