metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D25, C4.5D50, C20.6D10, Dic50⋊3C2, C50.6C23, C22.1D50, C100.5C22, D50.2C22, Dic25.3C22, (D4×C25)⋊3C2, (C4×D25)⋊2C2, C25⋊2(C4○D4), C25⋊D4⋊2C2, (C5×D4).4D5, (C2×C50).C22, C5.(D4⋊2D5), (C2×C10).2D10, (C2×Dic25)⋊3C2, C2.7(C22×D25), C10.24(C22×D5), SmallGroup(400,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D25
G = < a,b,c,d | a4=b2=c25=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 433 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, D10, C2×C10, C25, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, D25, C50, C50, D4⋊2D5, Dic25, Dic25, C100, D50, C2×C50, Dic50, C4×D25, C2×Dic25, C25⋊D4, D4×C25, D4⋊2D25
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, D25, D4⋊2D5, D50, C22×D25, D4⋊2D25
►On 200 points
Generators in S200
(1 59 36 97)(2 60 37 98)(3 61 38 99)(4 62 39 100)(5 63 40 76)(6 64 41 77)(7 65 42 78)(8 66 43 79)(9 67 44 80)(10 68 45 81)(11 69 46 82)(12 70 47 83)(13 71 48 84)(14 72 49 85)(15 73 50 86)(16 74 26 87)(17 75 27 88)(18 51 28 89)(19 52 29 90)(20 53 30 91)(21 54 31 92)(22 55 32 93)(23 56 33 94)(24 57 34 95)(25 58 35 96)(101 151 136 178)(102 152 137 179)(103 153 138 180)(104 154 139 181)(105 155 140 182)(106 156 141 183)(107 157 142 184)(108 158 143 185)(109 159 144 186)(110 160 145 187)(111 161 146 188)(112 162 147 189)(113 163 148 190)(114 164 149 191)(115 165 150 192)(116 166 126 193)(117 167 127 194)(118 168 128 195)(119 169 129 196)(120 170 130 197)(121 171 131 198)(122 172 132 199)(123 173 133 200)(124 174 134 176)(125 175 135 177)
(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(101 136)(102 137)(103 138)(104 139)(105 140)(106 141)(107 142)(108 143)(109 144)(110 145)(111 146)(112 147)(113 148)(114 149)(115 150)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 125)(18 124)(19 123)(20 122)(21 121)(22 120)(23 119)(24 118)(25 117)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 150)(38 149)(39 148)(40 147)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 174)(52 173)(53 172)(54 171)(55 170)(56 169)(57 168)(58 167)(59 166)(60 165)(61 164)(62 163)(63 162)(64 161)(65 160)(66 159)(67 158)(68 157)(69 156)(70 155)(71 154)(72 153)(73 152)(74 151)(75 175)(76 189)(77 188)(78 187)(79 186)(80 185)(81 184)(82 183)(83 182)(84 181)(85 180)(86 179)(87 178)(88 177)(89 176)(90 200)(91 199)(92 198)(93 197)(94 196)(95 195)(96 194)(97 193)(98 192)(99 191)(100 190)
G:=sub<Sym(200)| (1,59,36,97)(2,60,37,98)(3,61,38,99)(4,62,39,100)(5,63,40,76)(6,64,41,77)(7,65,42,78)(8,66,43,79)(9,67,44,80)(10,68,45,81)(11,69,46,82)(12,70,47,83)(13,71,48,84)(14,72,49,85)(15,73,50,86)(16,74,26,87)(17,75,27,88)(18,51,28,89)(19,52,29,90)(20,53,30,91)(21,54,31,92)(22,55,32,93)(23,56,33,94)(24,57,34,95)(25,58,35,96)(101,151,136,178)(102,152,137,179)(103,153,138,180)(104,154,139,181)(105,155,140,182)(106,156,141,183)(107,157,142,184)(108,158,143,185)(109,159,144,186)(110,160,145,187)(111,161,146,188)(112,162,147,189)(113,163,148,190)(114,164,149,191)(115,165,150,192)(116,166,126,193)(117,167,127,194)(118,168,128,195)(119,169,129,196)(120,170,130,197)(121,171,131,198)(122,172,132,199)(123,173,133,200)(124,174,134,176)(125,175,135,177), (51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(101,136)(102,137)(103,138)(104,139)(105,140)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,125)(18,124)(19,123)(20,122)(21,121)(22,120)(23,119)(24,118)(25,117)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,174)(52,173)(53,172)(54,171)(55,170)(56,169)(57,168)(58,167)(59,166)(60,165)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,175)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,180)(86,179)(87,178)(88,177)(89,176)(90,200)(91,199)(92,198)(93,197)(94,196)(95,195)(96,194)(97,193)(98,192)(99,191)(100,190)>;
G:=Group( (1,59,36,97)(2,60,37,98)(3,61,38,99)(4,62,39,100)(5,63,40,76)(6,64,41,77)(7,65,42,78)(8,66,43,79)(9,67,44,80)(10,68,45,81)(11,69,46,82)(12,70,47,83)(13,71,48,84)(14,72,49,85)(15,73,50,86)(16,74,26,87)(17,75,27,88)(18,51,28,89)(19,52,29,90)(20,53,30,91)(21,54,31,92)(22,55,32,93)(23,56,33,94)(24,57,34,95)(25,58,35,96)(101,151,136,178)(102,152,137,179)(103,153,138,180)(104,154,139,181)(105,155,140,182)(106,156,141,183)(107,157,142,184)(108,158,143,185)(109,159,144,186)(110,160,145,187)(111,161,146,188)(112,162,147,189)(113,163,148,190)(114,164,149,191)(115,165,150,192)(116,166,126,193)(117,167,127,194)(118,168,128,195)(119,169,129,196)(120,170,130,197)(121,171,131,198)(122,172,132,199)(123,173,133,200)(124,174,134,176)(125,175,135,177), (51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(101,136)(102,137)(103,138)(104,139)(105,140)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135), (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), (1,116)(2,115)(3,114)(4,113)(5,112)(6,111)(7,110)(8,109)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,125)(18,124)(19,123)(20,122)(21,121)(22,120)(23,119)(24,118)(25,117)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,174)(52,173)(53,172)(54,171)(55,170)(56,169)(57,168)(58,167)(59,166)(60,165)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,175)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,180)(86,179)(87,178)(88,177)(89,176)(90,200)(91,199)(92,198)(93,197)(94,196)(95,195)(96,194)(97,193)(98,192)(99,191)(100,190) );
G=PermutationGroup([[(1,59,36,97),(2,60,37,98),(3,61,38,99),(4,62,39,100),(5,63,40,76),(6,64,41,77),(7,65,42,78),(8,66,43,79),(9,67,44,80),(10,68,45,81),(11,69,46,82),(12,70,47,83),(13,71,48,84),(14,72,49,85),(15,73,50,86),(16,74,26,87),(17,75,27,88),(18,51,28,89),(19,52,29,90),(20,53,30,91),(21,54,31,92),(22,55,32,93),(23,56,33,94),(24,57,34,95),(25,58,35,96),(101,151,136,178),(102,152,137,179),(103,153,138,180),(104,154,139,181),(105,155,140,182),(106,156,141,183),(107,157,142,184),(108,158,143,185),(109,159,144,186),(110,160,145,187),(111,161,146,188),(112,162,147,189),(113,163,148,190),(114,164,149,191),(115,165,150,192),(116,166,126,193),(117,167,127,194),(118,168,128,195),(119,169,129,196),(120,170,130,197),(121,171,131,198),(122,172,132,199),(123,173,133,200),(124,174,134,176),(125,175,135,177)], [(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98),(61,99),(62,100),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(101,136),(102,137),(103,138),(104,139),(105,140),(106,141),(107,142),(108,143),(109,144),(110,145),(111,146),(112,147),(113,148),(114,149),(115,150),(116,126),(117,127),(118,128),(119,129),(120,130),(121,131),(122,132),(123,133),(124,134),(125,135)], [(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)], [(1,116),(2,115),(3,114),(4,113),(5,112),(6,111),(7,110),(8,109),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,125),(18,124),(19,123),(20,122),(21,121),(22,120),(23,119),(24,118),(25,117),(26,136),(27,135),(28,134),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,150),(38,149),(39,148),(40,147),(41,146),(42,145),(43,144),(44,143),(45,142),(46,141),(47,140),(48,139),(49,138),(50,137),(51,174),(52,173),(53,172),(54,171),(55,170),(56,169),(57,168),(58,167),(59,166),(60,165),(61,164),(62,163),(63,162),(64,161),(65,160),(66,159),(67,158),(68,157),(69,156),(70,155),(71,154),(72,153),(73,152),(74,151),(75,175),(76,189),(77,188),(78,187),(79,186),(80,185),(81,184),(82,183),(83,182),(84,181),(85,180),(86,179),(87,178),(88,177),(89,176),(90,200),(91,199),(92,198),(93,197),(94,196),(95,195),(96,194),(97,193),(98,192),(99,191),(100,190)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 25A | ··· | 25J | 50A | ··· | 50J | 50K | ··· | 50AD | 100A | ··· | 100J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 2 | 2 | 50 | 2 | 25 | 25 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D25 | D50 | D50 | D4⋊2D5 | D4⋊2D25 |
| kernel | D4⋊2D25 | Dic50 | C4×D25 | C2×Dic25 | C25⋊D4 | D4×C25 | C5×D4 | C25 | C20 | C2×C10 | D4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 10 | 10 | 20 | 2 | 10 |
Matrix representation of D4⋊2D25 ►in GL4(𝔽101) generated by
| 100 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 |
| 0 | 0 | 0 | 100 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 100 |
| 88 | 62 | 0 | 0 |
| 90 | 99 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 56 | 20 | 0 | 0 |
| 20 | 45 | 0 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(101))| [100,0,0,0,0,100,0,0,0,0,0,1,0,0,100,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,100],[88,90,0,0,62,99,0,0,0,0,1,0,0,0,0,1],[56,20,0,0,20,45,0,0,0,0,0,10,0,0,91,0] >;
D4⋊2D25 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{25} % in TeX
G:=Group("D4:2D25"); // GroupNames label
G:=SmallGroup(400,40);
// by ID
G=gap.SmallGroup(400,40);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^25=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations