direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C42⋊2C2, C42⋊2C26, C4⋊C4⋊5C26, (C4×C52)⋊3C2, C22⋊C4.2C26, C23.3(C2×C26), C26.46(C4○D4), (C2×C52).68C22, (C2×C26).81C23, (C22×C26).3C22, C22.16(C22×C26), (C13×C4⋊C4)⋊14C2, (C2×C4).8(C2×C26), C2.9(C13×C4○D4), (C13×C22⋊C4).5C2, SmallGroup(416,187)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C42⋊2C2
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
Subgroups: 84 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C23, C13, C42, C22⋊C4, C4⋊C4, C26, C26, C42⋊2C2, C52, C2×C26, C2×C26, C2×C52, C22×C26, C4×C52, C13×C22⋊C4, C13×C4⋊C4, C13×C42⋊2C2
Quotients: C1, C2, C22, C23, C13, C4○D4, C26, C42⋊2C2, C2×C26, C22×C26, C13×C4○D4, C13×C42⋊2C2
►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)
(1 142 41 82)(2 143 42 83)(3 131 43 84)(4 132 44 85)(5 133 45 86)(6 134 46 87)(7 135 47 88)(8 136 48 89)(9 137 49 90)(10 138 50 91)(11 139 51 79)(12 140 52 80)(13 141 40 81)(14 72 28 152)(15 73 29 153)(16 74 30 154)(17 75 31 155)(18 76 32 156)(19 77 33 144)(20 78 34 145)(21 66 35 146)(22 67 36 147)(23 68 37 148)(24 69 38 149)(25 70 39 150)(26 71 27 151)(53 117 159 203)(54 105 160 204)(55 106 161 205)(56 107 162 206)(57 108 163 207)(58 109 164 208)(59 110 165 196)(60 111 166 197)(61 112 167 198)(62 113 168 199)(63 114 169 200)(64 115 157 201)(65 116 158 202)(92 128 194 173)(93 129 195 174)(94 130 183 175)(95 118 184 176)(96 119 185 177)(97 120 186 178)(98 121 187 179)(99 122 188 180)(100 123 189 181)(101 124 190 182)(102 125 191 170)(103 126 192 171)(104 127 193 172)
(1 203 68 127)(2 204 69 128)(3 205 70 129)(4 206 71 130)(5 207 72 118)(6 208 73 119)(7 196 74 120)(8 197 75 121)(9 198 76 122)(10 199 77 123)(11 200 78 124)(12 201 66 125)(13 202 67 126)(14 95 86 163)(15 96 87 164)(16 97 88 165)(17 98 89 166)(18 99 90 167)(19 100 91 168)(20 101 79 169)(21 102 80 157)(22 103 81 158)(23 104 82 159)(24 92 83 160)(25 93 84 161)(26 94 85 162)(27 183 132 56)(28 184 133 57)(29 185 134 58)(30 186 135 59)(31 187 136 60)(32 188 137 61)(33 189 138 62)(34 190 139 63)(35 191 140 64)(36 192 141 65)(37 193 142 53)(38 194 143 54)(39 195 131 55)(40 116 147 171)(41 117 148 172)(42 105 149 173)(43 106 150 174)(44 107 151 175)(45 108 152 176)(46 109 153 177)(47 110 154 178)(48 111 155 179)(49 112 156 180)(50 113 144 181)(51 114 145 182)(52 115 146 170)
(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 85)(27 132)(28 133)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(39 131)(53 159)(54 160)(55 161)(56 162)(57 163)(58 164)(59 165)(60 166)(61 167)(62 168)(63 169)(64 157)(65 158)(92 194)(93 195)(94 183)(95 184)(96 185)(97 186)(98 187)(99 188)(100 189)(101 190)(102 191)(103 192)(104 193)(105 128)(106 129)(107 130)(108 118)(109 119)(110 120)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(170 201)(171 202)(172 203)(173 204)(174 205)(175 206)(176 207)(177 208)(178 196)(179 197)(180 198)(181 199)(182 200)
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), (1,142,41,82)(2,143,42,83)(3,131,43,84)(4,132,44,85)(5,133,45,86)(6,134,46,87)(7,135,47,88)(8,136,48,89)(9,137,49,90)(10,138,50,91)(11,139,51,79)(12,140,52,80)(13,141,40,81)(14,72,28,152)(15,73,29,153)(16,74,30,154)(17,75,31,155)(18,76,32,156)(19,77,33,144)(20,78,34,145)(21,66,35,146)(22,67,36,147)(23,68,37,148)(24,69,38,149)(25,70,39,150)(26,71,27,151)(53,117,159,203)(54,105,160,204)(55,106,161,205)(56,107,162,206)(57,108,163,207)(58,109,164,208)(59,110,165,196)(60,111,166,197)(61,112,167,198)(62,113,168,199)(63,114,169,200)(64,115,157,201)(65,116,158,202)(92,128,194,173)(93,129,195,174)(94,130,183,175)(95,118,184,176)(96,119,185,177)(97,120,186,178)(98,121,187,179)(99,122,188,180)(100,123,189,181)(101,124,190,182)(102,125,191,170)(103,126,192,171)(104,127,193,172), (1,203,68,127)(2,204,69,128)(3,205,70,129)(4,206,71,130)(5,207,72,118)(6,208,73,119)(7,196,74,120)(8,197,75,121)(9,198,76,122)(10,199,77,123)(11,200,78,124)(12,201,66,125)(13,202,67,126)(14,95,86,163)(15,96,87,164)(16,97,88,165)(17,98,89,166)(18,99,90,167)(19,100,91,168)(20,101,79,169)(21,102,80,157)(22,103,81,158)(23,104,82,159)(24,92,83,160)(25,93,84,161)(26,94,85,162)(27,183,132,56)(28,184,133,57)(29,185,134,58)(30,186,135,59)(31,187,136,60)(32,188,137,61)(33,189,138,62)(34,190,139,63)(35,191,140,64)(36,192,141,65)(37,193,142,53)(38,194,143,54)(39,195,131,55)(40,116,147,171)(41,117,148,172)(42,105,149,173)(43,106,150,174)(44,107,151,175)(45,108,152,176)(46,109,153,177)(47,110,154,178)(48,111,155,179)(49,112,156,180)(50,113,144,181)(51,114,145,182)(52,115,146,170), (14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,131)(53,159)(54,160)(55,161)(56,162)(57,163)(58,164)(59,165)(60,166)(61,167)(62,168)(63,169)(64,157)(65,158)(92,194)(93,195)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,128)(106,129)(107,130)(108,118)(109,119)(110,120)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,208)(178,196)(179,197)(180,198)(181,199)(182,200)>;
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), (1,142,41,82)(2,143,42,83)(3,131,43,84)(4,132,44,85)(5,133,45,86)(6,134,46,87)(7,135,47,88)(8,136,48,89)(9,137,49,90)(10,138,50,91)(11,139,51,79)(12,140,52,80)(13,141,40,81)(14,72,28,152)(15,73,29,153)(16,74,30,154)(17,75,31,155)(18,76,32,156)(19,77,33,144)(20,78,34,145)(21,66,35,146)(22,67,36,147)(23,68,37,148)(24,69,38,149)(25,70,39,150)(26,71,27,151)(53,117,159,203)(54,105,160,204)(55,106,161,205)(56,107,162,206)(57,108,163,207)(58,109,164,208)(59,110,165,196)(60,111,166,197)(61,112,167,198)(62,113,168,199)(63,114,169,200)(64,115,157,201)(65,116,158,202)(92,128,194,173)(93,129,195,174)(94,130,183,175)(95,118,184,176)(96,119,185,177)(97,120,186,178)(98,121,187,179)(99,122,188,180)(100,123,189,181)(101,124,190,182)(102,125,191,170)(103,126,192,171)(104,127,193,172), (1,203,68,127)(2,204,69,128)(3,205,70,129)(4,206,71,130)(5,207,72,118)(6,208,73,119)(7,196,74,120)(8,197,75,121)(9,198,76,122)(10,199,77,123)(11,200,78,124)(12,201,66,125)(13,202,67,126)(14,95,86,163)(15,96,87,164)(16,97,88,165)(17,98,89,166)(18,99,90,167)(19,100,91,168)(20,101,79,169)(21,102,80,157)(22,103,81,158)(23,104,82,159)(24,92,83,160)(25,93,84,161)(26,94,85,162)(27,183,132,56)(28,184,133,57)(29,185,134,58)(30,186,135,59)(31,187,136,60)(32,188,137,61)(33,189,138,62)(34,190,139,63)(35,191,140,64)(36,192,141,65)(37,193,142,53)(38,194,143,54)(39,195,131,55)(40,116,147,171)(41,117,148,172)(42,105,149,173)(43,106,150,174)(44,107,151,175)(45,108,152,176)(46,109,153,177)(47,110,154,178)(48,111,155,179)(49,112,156,180)(50,113,144,181)(51,114,145,182)(52,115,146,170), (14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,131)(53,159)(54,160)(55,161)(56,162)(57,163)(58,164)(59,165)(60,166)(61,167)(62,168)(63,169)(64,157)(65,158)(92,194)(93,195)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,128)(106,129)(107,130)(108,118)(109,119)(110,120)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,208)(178,196)(179,197)(180,198)(181,199)(182,200) );
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)], [(1,142,41,82),(2,143,42,83),(3,131,43,84),(4,132,44,85),(5,133,45,86),(6,134,46,87),(7,135,47,88),(8,136,48,89),(9,137,49,90),(10,138,50,91),(11,139,51,79),(12,140,52,80),(13,141,40,81),(14,72,28,152),(15,73,29,153),(16,74,30,154),(17,75,31,155),(18,76,32,156),(19,77,33,144),(20,78,34,145),(21,66,35,146),(22,67,36,147),(23,68,37,148),(24,69,38,149),(25,70,39,150),(26,71,27,151),(53,117,159,203),(54,105,160,204),(55,106,161,205),(56,107,162,206),(57,108,163,207),(58,109,164,208),(59,110,165,196),(60,111,166,197),(61,112,167,198),(62,113,168,199),(63,114,169,200),(64,115,157,201),(65,116,158,202),(92,128,194,173),(93,129,195,174),(94,130,183,175),(95,118,184,176),(96,119,185,177),(97,120,186,178),(98,121,187,179),(99,122,188,180),(100,123,189,181),(101,124,190,182),(102,125,191,170),(103,126,192,171),(104,127,193,172)], [(1,203,68,127),(2,204,69,128),(3,205,70,129),(4,206,71,130),(5,207,72,118),(6,208,73,119),(7,196,74,120),(8,197,75,121),(9,198,76,122),(10,199,77,123),(11,200,78,124),(12,201,66,125),(13,202,67,126),(14,95,86,163),(15,96,87,164),(16,97,88,165),(17,98,89,166),(18,99,90,167),(19,100,91,168),(20,101,79,169),(21,102,80,157),(22,103,81,158),(23,104,82,159),(24,92,83,160),(25,93,84,161),(26,94,85,162),(27,183,132,56),(28,184,133,57),(29,185,134,58),(30,186,135,59),(31,187,136,60),(32,188,137,61),(33,189,138,62),(34,190,139,63),(35,191,140,64),(36,192,141,65),(37,193,142,53),(38,194,143,54),(39,195,131,55),(40,116,147,171),(41,117,148,172),(42,105,149,173),(43,106,150,174),(44,107,151,175),(45,108,152,176),(46,109,153,177),(47,110,154,178),(48,111,155,179),(49,112,156,180),(50,113,144,181),(51,114,145,182),(52,115,146,170)], [(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,85),(27,132),(28,133),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,142),(38,143),(39,131),(53,159),(54,160),(55,161),(56,162),(57,163),(58,164),(59,165),(60,166),(61,167),(62,168),(63,169),(64,157),(65,158),(92,194),(93,195),(94,183),(95,184),(96,185),(97,186),(98,187),(99,188),(100,189),(101,190),(102,191),(103,192),(104,193),(105,128),(106,129),(107,130),(108,118),(109,119),(110,120),(111,121),(112,122),(113,123),(114,124),(115,125),(116,126),(117,127),(170,201),(171,202),(172,203),(173,204),(174,205),(175,206),(176,207),(177,208),(178,196),(179,197),(180,198),(181,199),(182,200)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26AV | 52A | ··· | 52BT | 52BU | ··· | 52DD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C4○D4 | C13×C4○D4 |
| kernel | C13×C42⋊2C2 | C4×C52 | C13×C22⋊C4 | C13×C4⋊C4 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C26 | C2 |
| # reps | 1 | 1 | 3 | 3 | 12 | 12 | 36 | 36 | 6 | 72 |
Matrix representation of C13×C42⋊2C2 ►in GL4(𝔽53) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 46 | 0 | 0 |
| 16 | 22 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 24 | 51 | 0 | 0 |
| 50 | 29 | 0 | 0 |
| 0 | 0 | 3 | 51 |
| 0 | 0 | 4 | 50 |
| 1 | 0 | 0 | 0 |
| 24 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 52 |
G:=sub<GL(4,GF(53))| [46,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[31,16,0,0,46,22,0,0,0,0,23,0,0,0,0,23],[24,50,0,0,51,29,0,0,0,0,3,4,0,0,51,50],[1,24,0,0,0,52,0,0,0,0,1,3,0,0,0,52] >;
C13×C42⋊2C2 in GAP, Magma, Sage, TeX
C_{13}\times C_4^2\rtimes_2C_2 % in TeX
G:=Group("C13xC4^2:2C2"); // GroupNames label
G:=SmallGroup(416,187);
// by ID
G=gap.SmallGroup(416,187);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,1879,3818,482]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;
// generators/relations