metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D25, C25⋊2D8, C50.8D4, C4.2D50, D100⋊2C2, C20.2D10, C100.2C22, C5.(D4⋊D5), C25⋊2C8⋊2C2, (D4×C25)⋊1C2, (C5×D4).2D5, C2.5(C25⋊D4), C10.15(C5⋊D4), SmallGroup(400,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D25
G = < a,b,c,d | a4=b2=c25=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D25
►On 200 points
Generators in S200
(1 88 46 64)(2 89 47 65)(3 90 48 66)(4 91 49 67)(5 92 50 68)(6 93 26 69)(7 94 27 70)(8 95 28 71)(9 96 29 72)(10 97 30 73)(11 98 31 74)(12 99 32 75)(13 100 33 51)(14 76 34 52)(15 77 35 53)(16 78 36 54)(17 79 37 55)(18 80 38 56)(19 81 39 57)(20 82 40 58)(21 83 41 59)(22 84 42 60)(23 85 43 61)(24 86 44 62)(25 87 45 63)(101 164 129 185)(102 165 130 186)(103 166 131 187)(104 167 132 188)(105 168 133 189)(106 169 134 190)(107 170 135 191)(108 171 136 192)(109 172 137 193)(110 173 138 194)(111 174 139 195)(112 175 140 196)(113 151 141 197)(114 152 142 198)(115 153 143 199)(116 154 144 200)(117 155 145 176)(118 156 146 177)(119 157 147 178)(120 158 148 179)(121 159 149 180)(122 160 150 181)(123 161 126 182)(124 162 127 183)(125 163 128 184)
(1 197)(2 198)(3 199)(4 200)(5 176)(6 177)(7 178)(8 179)(9 180)(10 181)(11 182)(12 183)(13 184)(14 185)(15 186)(16 187)(17 188)(18 189)(19 190)(20 191)(21 192)(22 193)(23 194)(24 195)(25 196)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 175)(46 151)(47 152)(48 153)(49 154)(50 155)(51 125)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 113)(65 114)(66 115)(67 116)(68 117)(69 118)(70 119)(71 120)(72 121)(73 122)(74 123)(75 124)(76 129)(77 130)(78 131)(79 132)(80 133)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 126)(99 127)(100 128)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 50)(42 49)(43 48)(44 47)(45 46)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(101 162)(102 161)(103 160)(104 159)(105 158)(106 157)(107 156)(108 155)(109 154)(110 153)(111 152)(112 151)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 168)(121 167)(122 166)(123 165)(124 164)(125 163)(126 186)(127 185)(128 184)(129 183)(130 182)(131 181)(132 180)(133 179)(134 178)(135 177)(136 176)(137 200)(138 199)(139 198)(140 197)(141 196)(142 195)(143 194)(144 193)(145 192)(146 191)(147 190)(148 189)(149 188)(150 187)
G:=sub<Sym(200)| (1,88,46,64)(2,89,47,65)(3,90,48,66)(4,91,49,67)(5,92,50,68)(6,93,26,69)(7,94,27,70)(8,95,28,71)(9,96,29,72)(10,97,30,73)(11,98,31,74)(12,99,32,75)(13,100,33,51)(14,76,34,52)(15,77,35,53)(16,78,36,54)(17,79,37,55)(18,80,38,56)(19,81,39,57)(20,82,40,58)(21,83,41,59)(22,84,42,60)(23,85,43,61)(24,86,44,62)(25,87,45,63)(101,164,129,185)(102,165,130,186)(103,166,131,187)(104,167,132,188)(105,168,133,189)(106,169,134,190)(107,170,135,191)(108,171,136,192)(109,172,137,193)(110,173,138,194)(111,174,139,195)(112,175,140,196)(113,151,141,197)(114,152,142,198)(115,153,143,199)(116,154,144,200)(117,155,145,176)(118,156,146,177)(119,157,147,178)(120,158,148,179)(121,159,149,180)(122,160,150,181)(123,161,126,182)(124,162,127,183)(125,163,128,184), (1,197)(2,198)(3,199)(4,200)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,151)(47,152)(48,153)(49,154)(50,155)(51,125)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,113)(65,114)(66,115)(67,116)(68,117)(69,118)(70,119)(71,120)(72,121)(73,122)(74,123)(75,124)(76,129)(77,130)(78,131)(79,132)(80,133)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,126)(99,127)(100,128), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(101,162)(102,161)(103,160)(104,159)(105,158)(106,157)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,186)(127,185)(128,184)(129,183)(130,182)(131,181)(132,180)(133,179)(134,178)(135,177)(136,176)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187)>;
G:=Group( (1,88,46,64)(2,89,47,65)(3,90,48,66)(4,91,49,67)(5,92,50,68)(6,93,26,69)(7,94,27,70)(8,95,28,71)(9,96,29,72)(10,97,30,73)(11,98,31,74)(12,99,32,75)(13,100,33,51)(14,76,34,52)(15,77,35,53)(16,78,36,54)(17,79,37,55)(18,80,38,56)(19,81,39,57)(20,82,40,58)(21,83,41,59)(22,84,42,60)(23,85,43,61)(24,86,44,62)(25,87,45,63)(101,164,129,185)(102,165,130,186)(103,166,131,187)(104,167,132,188)(105,168,133,189)(106,169,134,190)(107,170,135,191)(108,171,136,192)(109,172,137,193)(110,173,138,194)(111,174,139,195)(112,175,140,196)(113,151,141,197)(114,152,142,198)(115,153,143,199)(116,154,144,200)(117,155,145,176)(118,156,146,177)(119,157,147,178)(120,158,148,179)(121,159,149,180)(122,160,150,181)(123,161,126,182)(124,162,127,183)(125,163,128,184), (1,197)(2,198)(3,199)(4,200)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,151)(47,152)(48,153)(49,154)(50,155)(51,125)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,113)(65,114)(66,115)(67,116)(68,117)(69,118)(70,119)(71,120)(72,121)(73,122)(74,123)(75,124)(76,129)(77,130)(78,131)(79,132)(80,133)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,126)(99,127)(100,128), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(101,162)(102,161)(103,160)(104,159)(105,158)(106,157)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,186)(127,185)(128,184)(129,183)(130,182)(131,181)(132,180)(133,179)(134,178)(135,177)(136,176)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187) );
G=PermutationGroup([[(1,88,46,64),(2,89,47,65),(3,90,48,66),(4,91,49,67),(5,92,50,68),(6,93,26,69),(7,94,27,70),(8,95,28,71),(9,96,29,72),(10,97,30,73),(11,98,31,74),(12,99,32,75),(13,100,33,51),(14,76,34,52),(15,77,35,53),(16,78,36,54),(17,79,37,55),(18,80,38,56),(19,81,39,57),(20,82,40,58),(21,83,41,59),(22,84,42,60),(23,85,43,61),(24,86,44,62),(25,87,45,63),(101,164,129,185),(102,165,130,186),(103,166,131,187),(104,167,132,188),(105,168,133,189),(106,169,134,190),(107,170,135,191),(108,171,136,192),(109,172,137,193),(110,173,138,194),(111,174,139,195),(112,175,140,196),(113,151,141,197),(114,152,142,198),(115,153,143,199),(116,154,144,200),(117,155,145,176),(118,156,146,177),(119,157,147,178),(120,158,148,179),(121,159,149,180),(122,160,150,181),(123,161,126,182),(124,162,127,183),(125,163,128,184)], [(1,197),(2,198),(3,199),(4,200),(5,176),(6,177),(7,178),(8,179),(9,180),(10,181),(11,182),(12,183),(13,184),(14,185),(15,186),(16,187),(17,188),(18,189),(19,190),(20,191),(21,192),(22,193),(23,194),(24,195),(25,196),(26,156),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,175),(46,151),(47,152),(48,153),(49,154),(50,155),(51,125),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,113),(65,114),(66,115),(67,116),(68,117),(69,118),(70,119),(71,120),(72,121),(73,122),(74,123),(75,124),(76,129),(77,130),(78,131),(79,132),(80,133),(81,134),(82,135),(83,136),(84,137),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,126),(99,127),(100,128)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,50),(42,49),(43,48),(44,47),(45,46),(51,100),(52,99),(53,98),(54,97),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(101,162),(102,161),(103,160),(104,159),(105,158),(106,157),(107,156),(108,155),(109,154),(110,153),(111,152),(112,151),(113,175),(114,174),(115,173),(116,172),(117,171),(118,170),(119,169),(120,168),(121,167),(122,166),(123,165),(124,164),(125,163),(126,186),(127,185),(128,184),(129,183),(130,182),(131,181),(132,180),(133,179),(134,178),(135,177),(136,176),(137,200),(138,199),(139,198),(140,197),(141,196),(142,195),(143,194),(144,193),(145,192),(146,191),(147,190),(148,189),(149,188),(150,187)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 25A | ··· | 25J | 50A | ··· | 50J | 50K | ··· | 50AD | 100A | ··· | 100J |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 4 | 100 | 2 | 2 | 2 | 50 | 50 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D8 | D10 | C5⋊D4 | D25 | D50 | C25⋊D4 | D4⋊D5 | D4⋊D25 |
| kernel | D4⋊D25 | C25⋊2C8 | D100 | D4×C25 | C50 | C5×D4 | C25 | C20 | C10 | D4 | C4 | C2 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 10 | 10 | 20 | 2 | 10 |
Matrix representation of D4⋊D25 ►in GL4(𝔽401) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 287 |
| 0 | 0 | 197 | 400 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 348 | 214 |
| 0 | 0 | 193 | 53 |
| 266 | 9 | 0 | 0 |
| 392 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 274 | 311 | 0 | 0 |
| 99 | 127 | 0 | 0 |
| 0 | 0 | 400 | 114 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(401))| [1,0,0,0,0,1,0,0,0,0,1,197,0,0,287,400],[1,0,0,0,0,1,0,0,0,0,348,193,0,0,214,53],[266,392,0,0,9,60,0,0,0,0,1,0,0,0,0,1],[274,99,0,0,311,127,0,0,0,0,400,0,0,0,114,1] >;
D4⋊D25 in GAP, Magma, Sage, TeX
D_4\rtimes D_{25} % in TeX
G:=Group("D4:D25"); // GroupNames label
G:=SmallGroup(400,16);
// by ID
G=gap.SmallGroup(400,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,218,116,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^25=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export