metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.48D4, C22⋊2Dic26, C23.20D26, (C2×C26)⋊3Q8, C52⋊3C4⋊8C2, C26.8(C2×Q8), C26.39(C2×D4), (C2×C4).83D26, C13⋊4(C22⋊Q8), (C2×Dic26)⋊6C2, C26.D4⋊2C2, (C22×C52).6C2, C2.9(C2×Dic26), (C22×C4).5D13, C26.15(C4○D4), C4.23(C13⋊D4), (C2×C52).91C22, (C2×C26).42C23, C23.D13.4C2, C2.17(D52⋊5C2), (C22×C26).34C22, C22.54(C22×D13), (C2×Dic13).14C22, C2.5(C2×C13⋊D4), SmallGroup(416,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.48D4
G = < a,b,c | a52=b4=1, c2=a26, bab-1=cac-1=a-1, cbc-1=a26b-1 >
Subgroups: 376 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C26, C26, C22⋊Q8, Dic13, C52, C52, C2×C26, C2×C26, C2×C26, Dic26, C2×Dic13, C2×C52, C2×C52, C22×C26, C26.D4, C52⋊3C4, C23.D13, C2×Dic26, C22×C52, C52.48D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26, Dic26, C13⋊D4, C22×D13, C2×Dic26, D52⋊5C2, C2×C13⋊D4, C52.48D4
►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 86 142 166)(2 85 143 165)(3 84 144 164)(4 83 145 163)(5 82 146 162)(6 81 147 161)(7 80 148 160)(8 79 149 159)(9 78 150 158)(10 77 151 157)(11 76 152 208)(12 75 153 207)(13 74 154 206)(14 73 155 205)(15 72 156 204)(16 71 105 203)(17 70 106 202)(18 69 107 201)(19 68 108 200)(20 67 109 199)(21 66 110 198)(22 65 111 197)(23 64 112 196)(24 63 113 195)(25 62 114 194)(26 61 115 193)(27 60 116 192)(28 59 117 191)(29 58 118 190)(30 57 119 189)(31 56 120 188)(32 55 121 187)(33 54 122 186)(34 53 123 185)(35 104 124 184)(36 103 125 183)(37 102 126 182)(38 101 127 181)(39 100 128 180)(40 99 129 179)(41 98 130 178)(42 97 131 177)(43 96 132 176)(44 95 133 175)(45 94 134 174)(46 93 135 173)(47 92 136 172)(48 91 137 171)(49 90 138 170)(50 89 139 169)(51 88 140 168)(52 87 141 167)
(1 192 27 166)(2 191 28 165)(3 190 29 164)(4 189 30 163)(5 188 31 162)(6 187 32 161)(7 186 33 160)(8 185 34 159)(9 184 35 158)(10 183 36 157)(11 182 37 208)(12 181 38 207)(13 180 39 206)(14 179 40 205)(15 178 41 204)(16 177 42 203)(17 176 43 202)(18 175 44 201)(19 174 45 200)(20 173 46 199)(21 172 47 198)(22 171 48 197)(23 170 49 196)(24 169 50 195)(25 168 51 194)(26 167 52 193)(53 123 79 149)(54 122 80 148)(55 121 81 147)(56 120 82 146)(57 119 83 145)(58 118 84 144)(59 117 85 143)(60 116 86 142)(61 115 87 141)(62 114 88 140)(63 113 89 139)(64 112 90 138)(65 111 91 137)(66 110 92 136)(67 109 93 135)(68 108 94 134)(69 107 95 133)(70 106 96 132)(71 105 97 131)(72 156 98 130)(73 155 99 129)(74 154 100 128)(75 153 101 127)(76 152 102 126)(77 151 103 125)(78 150 104 124)
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), (1,86,142,166)(2,85,143,165)(3,84,144,164)(4,83,145,163)(5,82,146,162)(6,81,147,161)(7,80,148,160)(8,79,149,159)(9,78,150,158)(10,77,151,157)(11,76,152,208)(12,75,153,207)(13,74,154,206)(14,73,155,205)(15,72,156,204)(16,71,105,203)(17,70,106,202)(18,69,107,201)(19,68,108,200)(20,67,109,199)(21,66,110,198)(22,65,111,197)(23,64,112,196)(24,63,113,195)(25,62,114,194)(26,61,115,193)(27,60,116,192)(28,59,117,191)(29,58,118,190)(30,57,119,189)(31,56,120,188)(32,55,121,187)(33,54,122,186)(34,53,123,185)(35,104,124,184)(36,103,125,183)(37,102,126,182)(38,101,127,181)(39,100,128,180)(40,99,129,179)(41,98,130,178)(42,97,131,177)(43,96,132,176)(44,95,133,175)(45,94,134,174)(46,93,135,173)(47,92,136,172)(48,91,137,171)(49,90,138,170)(50,89,139,169)(51,88,140,168)(52,87,141,167), (1,192,27,166)(2,191,28,165)(3,190,29,164)(4,189,30,163)(5,188,31,162)(6,187,32,161)(7,186,33,160)(8,185,34,159)(9,184,35,158)(10,183,36,157)(11,182,37,208)(12,181,38,207)(13,180,39,206)(14,179,40,205)(15,178,41,204)(16,177,42,203)(17,176,43,202)(18,175,44,201)(19,174,45,200)(20,173,46,199)(21,172,47,198)(22,171,48,197)(23,170,49,196)(24,169,50,195)(25,168,51,194)(26,167,52,193)(53,123,79,149)(54,122,80,148)(55,121,81,147)(56,120,82,146)(57,119,83,145)(58,118,84,144)(59,117,85,143)(60,116,86,142)(61,115,87,141)(62,114,88,140)(63,113,89,139)(64,112,90,138)(65,111,91,137)(66,110,92,136)(67,109,93,135)(68,108,94,134)(69,107,95,133)(70,106,96,132)(71,105,97,131)(72,156,98,130)(73,155,99,129)(74,154,100,128)(75,153,101,127)(76,152,102,126)(77,151,103,125)(78,150,104,124)>;
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), (1,86,142,166)(2,85,143,165)(3,84,144,164)(4,83,145,163)(5,82,146,162)(6,81,147,161)(7,80,148,160)(8,79,149,159)(9,78,150,158)(10,77,151,157)(11,76,152,208)(12,75,153,207)(13,74,154,206)(14,73,155,205)(15,72,156,204)(16,71,105,203)(17,70,106,202)(18,69,107,201)(19,68,108,200)(20,67,109,199)(21,66,110,198)(22,65,111,197)(23,64,112,196)(24,63,113,195)(25,62,114,194)(26,61,115,193)(27,60,116,192)(28,59,117,191)(29,58,118,190)(30,57,119,189)(31,56,120,188)(32,55,121,187)(33,54,122,186)(34,53,123,185)(35,104,124,184)(36,103,125,183)(37,102,126,182)(38,101,127,181)(39,100,128,180)(40,99,129,179)(41,98,130,178)(42,97,131,177)(43,96,132,176)(44,95,133,175)(45,94,134,174)(46,93,135,173)(47,92,136,172)(48,91,137,171)(49,90,138,170)(50,89,139,169)(51,88,140,168)(52,87,141,167), (1,192,27,166)(2,191,28,165)(3,190,29,164)(4,189,30,163)(5,188,31,162)(6,187,32,161)(7,186,33,160)(8,185,34,159)(9,184,35,158)(10,183,36,157)(11,182,37,208)(12,181,38,207)(13,180,39,206)(14,179,40,205)(15,178,41,204)(16,177,42,203)(17,176,43,202)(18,175,44,201)(19,174,45,200)(20,173,46,199)(21,172,47,198)(22,171,48,197)(23,170,49,196)(24,169,50,195)(25,168,51,194)(26,167,52,193)(53,123,79,149)(54,122,80,148)(55,121,81,147)(56,120,82,146)(57,119,83,145)(58,118,84,144)(59,117,85,143)(60,116,86,142)(61,115,87,141)(62,114,88,140)(63,113,89,139)(64,112,90,138)(65,111,91,137)(66,110,92,136)(67,109,93,135)(68,108,94,134)(69,107,95,133)(70,106,96,132)(71,105,97,131)(72,156,98,130)(73,155,99,129)(74,154,100,128)(75,153,101,127)(76,152,102,126)(77,151,103,125)(78,150,104,124) );
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)], [(1,86,142,166),(2,85,143,165),(3,84,144,164),(4,83,145,163),(5,82,146,162),(6,81,147,161),(7,80,148,160),(8,79,149,159),(9,78,150,158),(10,77,151,157),(11,76,152,208),(12,75,153,207),(13,74,154,206),(14,73,155,205),(15,72,156,204),(16,71,105,203),(17,70,106,202),(18,69,107,201),(19,68,108,200),(20,67,109,199),(21,66,110,198),(22,65,111,197),(23,64,112,196),(24,63,113,195),(25,62,114,194),(26,61,115,193),(27,60,116,192),(28,59,117,191),(29,58,118,190),(30,57,119,189),(31,56,120,188),(32,55,121,187),(33,54,122,186),(34,53,123,185),(35,104,124,184),(36,103,125,183),(37,102,126,182),(38,101,127,181),(39,100,128,180),(40,99,129,179),(41,98,130,178),(42,97,131,177),(43,96,132,176),(44,95,133,175),(45,94,134,174),(46,93,135,173),(47,92,136,172),(48,91,137,171),(49,90,138,170),(50,89,139,169),(51,88,140,168),(52,87,141,167)], [(1,192,27,166),(2,191,28,165),(3,190,29,164),(4,189,30,163),(5,188,31,162),(6,187,32,161),(7,186,33,160),(8,185,34,159),(9,184,35,158),(10,183,36,157),(11,182,37,208),(12,181,38,207),(13,180,39,206),(14,179,40,205),(15,178,41,204),(16,177,42,203),(17,176,43,202),(18,175,44,201),(19,174,45,200),(20,173,46,199),(21,172,47,198),(22,171,48,197),(23,170,49,196),(24,169,50,195),(25,168,51,194),(26,167,52,193),(53,123,79,149),(54,122,80,148),(55,121,81,147),(56,120,82,146),(57,119,83,145),(58,118,84,144),(59,117,85,143),(60,116,86,142),(61,115,87,141),(62,114,88,140),(63,113,89,139),(64,112,90,138),(65,111,91,137),(66,110,92,136),(67,109,93,135),(68,108,94,134),(69,107,95,133),(70,106,96,132),(71,105,97,131),(72,156,98,130),(73,155,99,129),(74,154,100,128),(75,153,101,127),(76,152,102,126),(77,151,103,125),(78,150,104,124)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26AP | 52A | ··· | 52AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 52 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | D13 | D26 | D26 | C13⋊D4 | Dic26 | D52⋊5C2 |
| kernel | C52.48D4 | C26.D4 | C52⋊3C4 | C23.D13 | C2×Dic26 | C22×C52 | C52 | C2×C26 | C26 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 12 | 6 | 24 | 24 | 24 |
Matrix representation of C52.48D4 ►in GL4(𝔽53) generated by
| 33 | 0 | 0 | 0 |
| 33 | 45 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 52 | 9 |
| 30 | 35 | 0 | 0 |
| 47 | 23 | 0 | 0 |
| 0 | 0 | 49 | 12 |
| 0 | 0 | 3 | 4 |
| 30 | 35 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 49 | 12 |
| 0 | 0 | 12 | 4 |
G:=sub<GL(4,GF(53))| [33,33,0,0,0,45,0,0,0,0,6,52,0,0,0,9],[30,47,0,0,35,23,0,0,0,0,49,3,0,0,12,4],[30,0,0,0,35,23,0,0,0,0,49,12,0,0,12,4] >;
C52.48D4 in GAP, Magma, Sage, TeX
C_{52}._{48}D_4 % in TeX
G:=Group("C52.48D4"); // GroupNames label
G:=SmallGroup(416,145);
// by ID
G=gap.SmallGroup(416,145);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,217,103,218,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=a^26,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^26*b^-1>;
// generators/relations