direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic13, C52⋊4C4, C13⋊2C42, C22.3D26, (C2×C52).7C2, C2.2(C4×D13), (C2×C4).6D13, C26.10(C2×C4), (C2×C26).3C22, C2.2(C2×Dic13), (C2×Dic13).6C2, SmallGroup(208,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C4×Dic13 |
Generators and relations for C4×Dic13
G = < a,b,c | a4=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic13
►Regular action on 208 points
Generators in S208
(1 66 33 181)(2 67 34 182)(3 68 35 157)(4 69 36 158)(5 70 37 159)(6 71 38 160)(7 72 39 161)(8 73 40 162)(9 74 41 163)(10 75 42 164)(11 76 43 165)(12 77 44 166)(13 78 45 167)(14 53 46 168)(15 54 47 169)(16 55 48 170)(17 56 49 171)(18 57 50 172)(19 58 51 173)(20 59 52 174)(21 60 27 175)(22 61 28 176)(23 62 29 177)(24 63 30 178)(25 64 31 179)(26 65 32 180)(79 197 152 110)(80 198 153 111)(81 199 154 112)(82 200 155 113)(83 201 156 114)(84 202 131 115)(85 203 132 116)(86 204 133 117)(87 205 134 118)(88 206 135 119)(89 207 136 120)(90 208 137 121)(91 183 138 122)(92 184 139 123)(93 185 140 124)(94 186 141 125)(95 187 142 126)(96 188 143 127)(97 189 144 128)(98 190 145 129)(99 191 146 130)(100 192 147 105)(101 193 148 106)(102 194 149 107)(103 195 150 108)(104 196 151 109)
(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 97 14 84)(2 96 15 83)(3 95 16 82)(4 94 17 81)(5 93 18 80)(6 92 19 79)(7 91 20 104)(8 90 21 103)(9 89 22 102)(10 88 23 101)(11 87 24 100)(12 86 25 99)(13 85 26 98)(27 150 40 137)(28 149 41 136)(29 148 42 135)(30 147 43 134)(31 146 44 133)(32 145 45 132)(33 144 46 131)(34 143 47 156)(35 142 48 155)(36 141 49 154)(37 140 50 153)(38 139 51 152)(39 138 52 151)(53 202 66 189)(54 201 67 188)(55 200 68 187)(56 199 69 186)(57 198 70 185)(58 197 71 184)(59 196 72 183)(60 195 73 208)(61 194 74 207)(62 193 75 206)(63 192 76 205)(64 191 77 204)(65 190 78 203)(105 165 118 178)(106 164 119 177)(107 163 120 176)(108 162 121 175)(109 161 122 174)(110 160 123 173)(111 159 124 172)(112 158 125 171)(113 157 126 170)(114 182 127 169)(115 181 128 168)(116 180 129 167)(117 179 130 166)
G:=sub<Sym(208)| (1,66,33,181)(2,67,34,182)(3,68,35,157)(4,69,36,158)(5,70,37,159)(6,71,38,160)(7,72,39,161)(8,73,40,162)(9,74,41,163)(10,75,42,164)(11,76,43,165)(12,77,44,166)(13,78,45,167)(14,53,46,168)(15,54,47,169)(16,55,48,170)(17,56,49,171)(18,57,50,172)(19,58,51,173)(20,59,52,174)(21,60,27,175)(22,61,28,176)(23,62,29,177)(24,63,30,178)(25,64,31,179)(26,65,32,180)(79,197,152,110)(80,198,153,111)(81,199,154,112)(82,200,155,113)(83,201,156,114)(84,202,131,115)(85,203,132,116)(86,204,133,117)(87,205,134,118)(88,206,135,119)(89,207,136,120)(90,208,137,121)(91,183,138,122)(92,184,139,123)(93,185,140,124)(94,186,141,125)(95,187,142,126)(96,188,143,127)(97,189,144,128)(98,190,145,129)(99,191,146,130)(100,192,147,105)(101,193,148,106)(102,194,149,107)(103,195,150,108)(104,196,151,109), (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), (1,97,14,84)(2,96,15,83)(3,95,16,82)(4,94,17,81)(5,93,18,80)(6,92,19,79)(7,91,20,104)(8,90,21,103)(9,89,22,102)(10,88,23,101)(11,87,24,100)(12,86,25,99)(13,85,26,98)(27,150,40,137)(28,149,41,136)(29,148,42,135)(30,147,43,134)(31,146,44,133)(32,145,45,132)(33,144,46,131)(34,143,47,156)(35,142,48,155)(36,141,49,154)(37,140,50,153)(38,139,51,152)(39,138,52,151)(53,202,66,189)(54,201,67,188)(55,200,68,187)(56,199,69,186)(57,198,70,185)(58,197,71,184)(59,196,72,183)(60,195,73,208)(61,194,74,207)(62,193,75,206)(63,192,76,205)(64,191,77,204)(65,190,78,203)(105,165,118,178)(106,164,119,177)(107,163,120,176)(108,162,121,175)(109,161,122,174)(110,160,123,173)(111,159,124,172)(112,158,125,171)(113,157,126,170)(114,182,127,169)(115,181,128,168)(116,180,129,167)(117,179,130,166)>;
G:=Group( (1,66,33,181)(2,67,34,182)(3,68,35,157)(4,69,36,158)(5,70,37,159)(6,71,38,160)(7,72,39,161)(8,73,40,162)(9,74,41,163)(10,75,42,164)(11,76,43,165)(12,77,44,166)(13,78,45,167)(14,53,46,168)(15,54,47,169)(16,55,48,170)(17,56,49,171)(18,57,50,172)(19,58,51,173)(20,59,52,174)(21,60,27,175)(22,61,28,176)(23,62,29,177)(24,63,30,178)(25,64,31,179)(26,65,32,180)(79,197,152,110)(80,198,153,111)(81,199,154,112)(82,200,155,113)(83,201,156,114)(84,202,131,115)(85,203,132,116)(86,204,133,117)(87,205,134,118)(88,206,135,119)(89,207,136,120)(90,208,137,121)(91,183,138,122)(92,184,139,123)(93,185,140,124)(94,186,141,125)(95,187,142,126)(96,188,143,127)(97,189,144,128)(98,190,145,129)(99,191,146,130)(100,192,147,105)(101,193,148,106)(102,194,149,107)(103,195,150,108)(104,196,151,109), (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), (1,97,14,84)(2,96,15,83)(3,95,16,82)(4,94,17,81)(5,93,18,80)(6,92,19,79)(7,91,20,104)(8,90,21,103)(9,89,22,102)(10,88,23,101)(11,87,24,100)(12,86,25,99)(13,85,26,98)(27,150,40,137)(28,149,41,136)(29,148,42,135)(30,147,43,134)(31,146,44,133)(32,145,45,132)(33,144,46,131)(34,143,47,156)(35,142,48,155)(36,141,49,154)(37,140,50,153)(38,139,51,152)(39,138,52,151)(53,202,66,189)(54,201,67,188)(55,200,68,187)(56,199,69,186)(57,198,70,185)(58,197,71,184)(59,196,72,183)(60,195,73,208)(61,194,74,207)(62,193,75,206)(63,192,76,205)(64,191,77,204)(65,190,78,203)(105,165,118,178)(106,164,119,177)(107,163,120,176)(108,162,121,175)(109,161,122,174)(110,160,123,173)(111,159,124,172)(112,158,125,171)(113,157,126,170)(114,182,127,169)(115,181,128,168)(116,180,129,167)(117,179,130,166) );
G=PermutationGroup([[(1,66,33,181),(2,67,34,182),(3,68,35,157),(4,69,36,158),(5,70,37,159),(6,71,38,160),(7,72,39,161),(8,73,40,162),(9,74,41,163),(10,75,42,164),(11,76,43,165),(12,77,44,166),(13,78,45,167),(14,53,46,168),(15,54,47,169),(16,55,48,170),(17,56,49,171),(18,57,50,172),(19,58,51,173),(20,59,52,174),(21,60,27,175),(22,61,28,176),(23,62,29,177),(24,63,30,178),(25,64,31,179),(26,65,32,180),(79,197,152,110),(80,198,153,111),(81,199,154,112),(82,200,155,113),(83,201,156,114),(84,202,131,115),(85,203,132,116),(86,204,133,117),(87,205,134,118),(88,206,135,119),(89,207,136,120),(90,208,137,121),(91,183,138,122),(92,184,139,123),(93,185,140,124),(94,186,141,125),(95,187,142,126),(96,188,143,127),(97,189,144,128),(98,190,145,129),(99,191,146,130),(100,192,147,105),(101,193,148,106),(102,194,149,107),(103,195,150,108),(104,196,151,109)], [(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)], [(1,97,14,84),(2,96,15,83),(3,95,16,82),(4,94,17,81),(5,93,18,80),(6,92,19,79),(7,91,20,104),(8,90,21,103),(9,89,22,102),(10,88,23,101),(11,87,24,100),(12,86,25,99),(13,85,26,98),(27,150,40,137),(28,149,41,136),(29,148,42,135),(30,147,43,134),(31,146,44,133),(32,145,45,132),(33,144,46,131),(34,143,47,156),(35,142,48,155),(36,141,49,154),(37,140,50,153),(38,139,51,152),(39,138,52,151),(53,202,66,189),(54,201,67,188),(55,200,68,187),(56,199,69,186),(57,198,70,185),(58,197,71,184),(59,196,72,183),(60,195,73,208),(61,194,74,207),(62,193,75,206),(63,192,76,205),(64,191,77,204),(65,190,78,203),(105,165,118,178),(106,164,119,177),(107,163,120,176),(108,162,121,175),(109,161,122,174),(110,160,123,173),(111,159,124,172),(112,158,125,171),(113,157,126,170),(114,182,127,169),(115,181,128,168),(116,180,129,167),(117,179,130,166)]])
C4×Dic13 is a maximal subgroup of
C52.8Q8 C104⋊8C4 D52⋊7C4 C52.56D4 C52⋊C8 C26.C42 Dic13⋊C8 C42×D13 C42⋊D13 C23.11D26 C23.D26 Dic13⋊4D4 C23.6D26 Dic13⋊3Q8 C52⋊Q8 Dic13.Q8 C4.Dic26 C4⋊C4⋊7D13 D52⋊8C4 C4⋊C4⋊D13 C23.21D26 C52.17D4 C52⋊D4 Dic13⋊Q8 C52.23D4
C4×Dic13 is a maximal quotient of
C26.7C42 C104⋊8C4 C26.10C42
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D13 | Dic13 | D26 | C4×D13 |
| kernel | C4×Dic13 | C2×Dic13 | C2×C52 | Dic13 | C52 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 6 | 12 | 6 | 24 |
Matrix representation of C4×Dic13 ►in GL3(𝔽53) generated by
| 30 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 1 | 0 | 0 |
| 0 | 0 | 52 |
| 0 | 1 | 27 |
| 52 | 0 | 0 |
| 0 | 15 | 30 |
| 0 | 49 | 38 |
G:=sub<GL(3,GF(53))| [30,0,0,0,23,0,0,0,23],[1,0,0,0,0,1,0,52,27],[52,0,0,0,15,49,0,30,38] >;
C4×Dic13 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{13} % in TeX
G:=Group("C4xDic13"); // GroupNames label
G:=SmallGroup(208,11);
// by ID
G=gap.SmallGroup(208,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,46,4804]);
// Polycyclic
G:=Group<a,b,c|a^4=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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