metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.23D4, (Q8×C26)⋊4C2, (C2×Q8)⋊4D13, (C2×C4).57D26, C26.58(C2×D4), (C4×Dic13)⋊7C2, (C2×D52).10C2, C13⋊4(C4.4D4), D26⋊C4⋊16C2, C26.37(C4○D4), C4.11(C13⋊D4), (C2×C26).59C23, (C2×C52).40C22, C2.9(D52⋊C2), C22.65(C22×D13), (C2×Dic13).44C22, (C22×D13).13C22, C2.22(C2×C13⋊D4), SmallGroup(416,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.23D4
G = < a,b,c | a52=b4=c2=1, bab-1=a25, cac=a-1, cbc=a26b-1 >
Subgroups: 584 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C13, C42, C22⋊C4, C2×D4, C2×Q8, D13, C26, C26, C4.4D4, Dic13, C52, C52, D26, C2×C26, D52, C2×Dic13, C2×C52, C2×C52, Q8×C13, C22×D13, C4×Dic13, D26⋊C4, C2×D52, Q8×C26, C52.23D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4.4D4, D26, C13⋊D4, C22×D13, D52⋊C2, C2×C13⋊D4, C52.23D4
►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 54 187 137)(2 79 188 110)(3 104 189 135)(4 77 190 108)(5 102 191 133)(6 75 192 106)(7 100 193 131)(8 73 194 156)(9 98 195 129)(10 71 196 154)(11 96 197 127)(12 69 198 152)(13 94 199 125)(14 67 200 150)(15 92 201 123)(16 65 202 148)(17 90 203 121)(18 63 204 146)(19 88 205 119)(20 61 206 144)(21 86 207 117)(22 59 208 142)(23 84 157 115)(24 57 158 140)(25 82 159 113)(26 55 160 138)(27 80 161 111)(28 53 162 136)(29 78 163 109)(30 103 164 134)(31 76 165 107)(32 101 166 132)(33 74 167 105)(34 99 168 130)(35 72 169 155)(36 97 170 128)(37 70 171 153)(38 95 172 126)(39 68 173 151)(40 93 174 124)(41 66 175 149)(42 91 176 122)(43 64 177 147)(44 89 178 120)(45 62 179 145)(46 87 180 118)(47 60 181 143)(48 85 182 116)(49 58 183 141)(50 83 184 114)(51 56 185 139)(52 81 186 112)
(1 200)(2 199)(3 198)(4 197)(5 196)(6 195)(7 194)(8 193)(9 192)(10 191)(11 190)(12 189)(13 188)(14 187)(15 186)(16 185)(17 184)(18 183)(19 182)(20 181)(21 180)(22 179)(23 178)(24 177)(25 176)(26 175)(27 174)(28 173)(29 172)(30 171)(31 170)(32 169)(33 168)(34 167)(35 166)(36 165)(37 164)(38 163)(39 162)(40 161)(41 160)(42 159)(43 158)(44 157)(45 208)(46 207)(47 206)(48 205)(49 204)(50 203)(51 202)(52 201)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(95 104)(96 103)(97 102)(98 101)(99 100)(105 156)(106 155)(107 154)(108 153)(109 152)(110 151)(111 150)(112 149)(113 148)(114 147)(115 146)(116 145)(117 144)(118 143)(119 142)(120 141)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)
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), (1,54,187,137)(2,79,188,110)(3,104,189,135)(4,77,190,108)(5,102,191,133)(6,75,192,106)(7,100,193,131)(8,73,194,156)(9,98,195,129)(10,71,196,154)(11,96,197,127)(12,69,198,152)(13,94,199,125)(14,67,200,150)(15,92,201,123)(16,65,202,148)(17,90,203,121)(18,63,204,146)(19,88,205,119)(20,61,206,144)(21,86,207,117)(22,59,208,142)(23,84,157,115)(24,57,158,140)(25,82,159,113)(26,55,160,138)(27,80,161,111)(28,53,162,136)(29,78,163,109)(30,103,164,134)(31,76,165,107)(32,101,166,132)(33,74,167,105)(34,99,168,130)(35,72,169,155)(36,97,170,128)(37,70,171,153)(38,95,172,126)(39,68,173,151)(40,93,174,124)(41,66,175,149)(42,91,176,122)(43,64,177,147)(44,89,178,120)(45,62,179,145)(46,87,180,118)(47,60,181,143)(48,85,182,116)(49,58,183,141)(50,83,184,114)(51,56,185,139)(52,81,186,112), (1,200)(2,199)(3,198)(4,197)(5,196)(6,195)(7,194)(8,193)(9,192)(10,191)(11,190)(12,189)(13,188)(14,187)(15,186)(16,185)(17,184)(18,183)(19,182)(20,181)(21,180)(22,179)(23,178)(24,177)(25,176)(26,175)(27,174)(28,173)(29,172)(30,171)(31,170)(32,169)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,208)(46,207)(47,206)(48,205)(49,204)(50,203)(51,202)(52,201)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(95,104)(96,103)(97,102)(98,101)(99,100)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)>;
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), (1,54,187,137)(2,79,188,110)(3,104,189,135)(4,77,190,108)(5,102,191,133)(6,75,192,106)(7,100,193,131)(8,73,194,156)(9,98,195,129)(10,71,196,154)(11,96,197,127)(12,69,198,152)(13,94,199,125)(14,67,200,150)(15,92,201,123)(16,65,202,148)(17,90,203,121)(18,63,204,146)(19,88,205,119)(20,61,206,144)(21,86,207,117)(22,59,208,142)(23,84,157,115)(24,57,158,140)(25,82,159,113)(26,55,160,138)(27,80,161,111)(28,53,162,136)(29,78,163,109)(30,103,164,134)(31,76,165,107)(32,101,166,132)(33,74,167,105)(34,99,168,130)(35,72,169,155)(36,97,170,128)(37,70,171,153)(38,95,172,126)(39,68,173,151)(40,93,174,124)(41,66,175,149)(42,91,176,122)(43,64,177,147)(44,89,178,120)(45,62,179,145)(46,87,180,118)(47,60,181,143)(48,85,182,116)(49,58,183,141)(50,83,184,114)(51,56,185,139)(52,81,186,112), (1,200)(2,199)(3,198)(4,197)(5,196)(6,195)(7,194)(8,193)(9,192)(10,191)(11,190)(12,189)(13,188)(14,187)(15,186)(16,185)(17,184)(18,183)(19,182)(20,181)(21,180)(22,179)(23,178)(24,177)(25,176)(26,175)(27,174)(28,173)(29,172)(30,171)(31,170)(32,169)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,208)(46,207)(47,206)(48,205)(49,204)(50,203)(51,202)(52,201)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(95,104)(96,103)(97,102)(98,101)(99,100)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131) );
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)], [(1,54,187,137),(2,79,188,110),(3,104,189,135),(4,77,190,108),(5,102,191,133),(6,75,192,106),(7,100,193,131),(8,73,194,156),(9,98,195,129),(10,71,196,154),(11,96,197,127),(12,69,198,152),(13,94,199,125),(14,67,200,150),(15,92,201,123),(16,65,202,148),(17,90,203,121),(18,63,204,146),(19,88,205,119),(20,61,206,144),(21,86,207,117),(22,59,208,142),(23,84,157,115),(24,57,158,140),(25,82,159,113),(26,55,160,138),(27,80,161,111),(28,53,162,136),(29,78,163,109),(30,103,164,134),(31,76,165,107),(32,101,166,132),(33,74,167,105),(34,99,168,130),(35,72,169,155),(36,97,170,128),(37,70,171,153),(38,95,172,126),(39,68,173,151),(40,93,174,124),(41,66,175,149),(42,91,176,122),(43,64,177,147),(44,89,178,120),(45,62,179,145),(46,87,180,118),(47,60,181,143),(48,85,182,116),(49,58,183,141),(50,83,184,114),(51,56,185,139),(52,81,186,112)], [(1,200),(2,199),(3,198),(4,197),(5,196),(6,195),(7,194),(8,193),(9,192),(10,191),(11,190),(12,189),(13,188),(14,187),(15,186),(16,185),(17,184),(18,183),(19,182),(20,181),(21,180),(22,179),(23,178),(24,177),(25,176),(26,175),(27,174),(28,173),(29,172),(30,171),(31,170),(32,169),(33,168),(34,167),(35,166),(36,165),(37,164),(38,163),(39,162),(40,161),(41,160),(42,159),(43,158),(44,157),(45,208),(46,207),(47,206),(48,205),(49,204),(50,203),(51,202),(52,201),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(95,104),(96,103),(97,102),(98,101),(99,100),(105,156),(106,155),(107,154),(108,153),(109,152),(110,151),(111,150),(112,149),(113,148),(114,147),(115,146),(116,145),(117,144),(118,143),(119,142),(120,141),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| 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 | 52 | 52 | 2 | 2 | 4 | 4 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D13 | D26 | C13⋊D4 | D52⋊C2 |
| kernel | C52.23D4 | C4×Dic13 | D26⋊C4 | C2×D52 | Q8×C26 | C52 | C26 | C2×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 6 | 18 | 24 | 12 |
Matrix representation of C52.23D4 ►in GL4(𝔽53) generated by
| 0 | 52 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 50 | 12 |
| 0 | 0 | 44 | 18 |
| 0 | 30 | 0 | 0 |
| 23 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 45 | 52 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 23 |
| 0 | 0 | 9 | 35 |
G:=sub<GL(4,GF(53))| [0,1,0,0,52,0,0,0,0,0,50,44,0,0,12,18],[0,23,0,0,30,0,0,0,0,0,1,45,0,0,40,52],[0,1,0,0,1,0,0,0,0,0,18,9,0,0,23,35] >;
C52.23D4 in GAP, Magma, Sage, TeX
C_{52}._{23}D_4 % in TeX
G:=Group("C52.23D4"); // GroupNames label
G:=SmallGroup(416,168);
// by ID
G=gap.SmallGroup(416,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,218,188,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^25,c*a*c=a^-1,c*b*c=a^26*b^-1>;
// generators/relations