metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D25, C4.1D50, C50.7D4, C25⋊2SD16, C20.1D10, Dic50⋊2C2, C100.1C22, C25⋊2C8⋊1C2, (C5×D4).1D5, C5.(D4.D5), (D4×C25).1C2, C2.4(C25⋊D4), C10.14(C5⋊D4), SmallGroup(400,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D25
G = < a,b,c,d | a4=b2=c25=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D25
►On 200 points
Generators in S200
(1 90 32 63)(2 91 33 64)(3 92 34 65)(4 93 35 66)(5 94 36 67)(6 95 37 68)(7 96 38 69)(8 97 39 70)(9 98 40 71)(10 99 41 72)(11 100 42 73)(12 76 43 74)(13 77 44 75)(14 78 45 51)(15 79 46 52)(16 80 47 53)(17 81 48 54)(18 82 49 55)(19 83 50 56)(20 84 26 57)(21 85 27 58)(22 86 28 59)(23 87 29 60)(24 88 30 61)(25 89 31 62)(101 167 138 178)(102 168 139 179)(103 169 140 180)(104 170 141 181)(105 171 142 182)(106 172 143 183)(107 173 144 184)(108 174 145 185)(109 175 146 186)(110 151 147 187)(111 152 148 188)(112 153 149 189)(113 154 150 190)(114 155 126 191)(115 156 127 192)(116 157 128 193)(117 158 129 194)(118 159 130 195)(119 160 131 196)(120 161 132 197)(121 162 133 198)(122 163 134 199)(123 164 135 200)(124 165 136 176)(125 166 137 177)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(101 138)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)(121 133)(122 134)(123 135)(124 136)(125 137)
(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)
(1 130 32 118)(2 129 33 117)(3 128 34 116)(4 127 35 115)(5 126 36 114)(6 150 37 113)(7 149 38 112)(8 148 39 111)(9 147 40 110)(10 146 41 109)(11 145 42 108)(12 144 43 107)(13 143 44 106)(14 142 45 105)(15 141 46 104)(16 140 47 103)(17 139 48 102)(18 138 49 101)(19 137 50 125)(20 136 26 124)(21 135 27 123)(22 134 28 122)(23 133 29 121)(24 132 30 120)(25 131 31 119)(51 182 78 171)(52 181 79 170)(53 180 80 169)(54 179 81 168)(55 178 82 167)(56 177 83 166)(57 176 84 165)(58 200 85 164)(59 199 86 163)(60 198 87 162)(61 197 88 161)(62 196 89 160)(63 195 90 159)(64 194 91 158)(65 193 92 157)(66 192 93 156)(67 191 94 155)(68 190 95 154)(69 189 96 153)(70 188 97 152)(71 187 98 151)(72 186 99 175)(73 185 100 174)(74 184 76 173)(75 183 77 172)
G:=sub<Sym(200)| (1,90,32,63)(2,91,33,64)(3,92,34,65)(4,93,35,66)(5,94,36,67)(6,95,37,68)(7,96,38,69)(8,97,39,70)(9,98,40,71)(10,99,41,72)(11,100,42,73)(12,76,43,74)(13,77,44,75)(14,78,45,51)(15,79,46,52)(16,80,47,53)(17,81,48,54)(18,82,49,55)(19,83,50,56)(20,84,26,57)(21,85,27,58)(22,86,28,59)(23,87,29,60)(24,88,30,61)(25,89,31,62)(101,167,138,178)(102,168,139,179)(103,169,140,180)(104,170,141,181)(105,171,142,182)(106,172,143,183)(107,173,144,184)(108,174,145,185)(109,175,146,186)(110,151,147,187)(111,152,148,188)(112,153,149,189)(113,154,150,190)(114,155,126,191)(115,156,127,192)(116,157,128,193)(117,158,129,194)(118,159,130,195)(119,160,131,196)(120,161,132,197)(121,162,133,198)(122,163,134,199)(123,164,135,200)(124,165,136,176)(125,166,137,177), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(101,138)(102,139)(103,140)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(113,150)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132)(121,133)(122,134)(123,135)(124,136)(125,137), (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), (1,130,32,118)(2,129,33,117)(3,128,34,116)(4,127,35,115)(5,126,36,114)(6,150,37,113)(7,149,38,112)(8,148,39,111)(9,147,40,110)(10,146,41,109)(11,145,42,108)(12,144,43,107)(13,143,44,106)(14,142,45,105)(15,141,46,104)(16,140,47,103)(17,139,48,102)(18,138,49,101)(19,137,50,125)(20,136,26,124)(21,135,27,123)(22,134,28,122)(23,133,29,121)(24,132,30,120)(25,131,31,119)(51,182,78,171)(52,181,79,170)(53,180,80,169)(54,179,81,168)(55,178,82,167)(56,177,83,166)(57,176,84,165)(58,200,85,164)(59,199,86,163)(60,198,87,162)(61,197,88,161)(62,196,89,160)(63,195,90,159)(64,194,91,158)(65,193,92,157)(66,192,93,156)(67,191,94,155)(68,190,95,154)(69,189,96,153)(70,188,97,152)(71,187,98,151)(72,186,99,175)(73,185,100,174)(74,184,76,173)(75,183,77,172)>;
G:=Group( (1,90,32,63)(2,91,33,64)(3,92,34,65)(4,93,35,66)(5,94,36,67)(6,95,37,68)(7,96,38,69)(8,97,39,70)(9,98,40,71)(10,99,41,72)(11,100,42,73)(12,76,43,74)(13,77,44,75)(14,78,45,51)(15,79,46,52)(16,80,47,53)(17,81,48,54)(18,82,49,55)(19,83,50,56)(20,84,26,57)(21,85,27,58)(22,86,28,59)(23,87,29,60)(24,88,30,61)(25,89,31,62)(101,167,138,178)(102,168,139,179)(103,169,140,180)(104,170,141,181)(105,171,142,182)(106,172,143,183)(107,173,144,184)(108,174,145,185)(109,175,146,186)(110,151,147,187)(111,152,148,188)(112,153,149,189)(113,154,150,190)(114,155,126,191)(115,156,127,192)(116,157,128,193)(117,158,129,194)(118,159,130,195)(119,160,131,196)(120,161,132,197)(121,162,133,198)(122,163,134,199)(123,164,135,200)(124,165,136,176)(125,166,137,177), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(101,138)(102,139)(103,140)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(113,150)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132)(121,133)(122,134)(123,135)(124,136)(125,137), (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), (1,130,32,118)(2,129,33,117)(3,128,34,116)(4,127,35,115)(5,126,36,114)(6,150,37,113)(7,149,38,112)(8,148,39,111)(9,147,40,110)(10,146,41,109)(11,145,42,108)(12,144,43,107)(13,143,44,106)(14,142,45,105)(15,141,46,104)(16,140,47,103)(17,139,48,102)(18,138,49,101)(19,137,50,125)(20,136,26,124)(21,135,27,123)(22,134,28,122)(23,133,29,121)(24,132,30,120)(25,131,31,119)(51,182,78,171)(52,181,79,170)(53,180,80,169)(54,179,81,168)(55,178,82,167)(56,177,83,166)(57,176,84,165)(58,200,85,164)(59,199,86,163)(60,198,87,162)(61,197,88,161)(62,196,89,160)(63,195,90,159)(64,194,91,158)(65,193,92,157)(66,192,93,156)(67,191,94,155)(68,190,95,154)(69,189,96,153)(70,188,97,152)(71,187,98,151)(72,186,99,175)(73,185,100,174)(74,184,76,173)(75,183,77,172) );
G=PermutationGroup([[(1,90,32,63),(2,91,33,64),(3,92,34,65),(4,93,35,66),(5,94,36,67),(6,95,37,68),(7,96,38,69),(8,97,39,70),(9,98,40,71),(10,99,41,72),(11,100,42,73),(12,76,43,74),(13,77,44,75),(14,78,45,51),(15,79,46,52),(16,80,47,53),(17,81,48,54),(18,82,49,55),(19,83,50,56),(20,84,26,57),(21,85,27,58),(22,86,28,59),(23,87,29,60),(24,88,30,61),(25,89,31,62),(101,167,138,178),(102,168,139,179),(103,169,140,180),(104,170,141,181),(105,171,142,182),(106,172,143,183),(107,173,144,184),(108,174,145,185),(109,175,146,186),(110,151,147,187),(111,152,148,188),(112,153,149,189),(113,154,150,190),(114,155,126,191),(115,156,127,192),(116,157,128,193),(117,158,129,194),(118,159,130,195),(119,160,131,196),(120,161,132,197),(121,162,133,198),(122,163,134,199),(123,164,135,200),(124,165,136,176),(125,166,137,177)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,99),(42,100),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(101,138),(102,139),(103,140),(104,141),(105,142),(106,143),(107,144),(108,145),(109,146),(110,147),(111,148),(112,149),(113,150),(114,126),(115,127),(116,128),(117,129),(118,130),(119,131),(120,132),(121,133),(122,134),(123,135),(124,136),(125,137)], [(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)], [(1,130,32,118),(2,129,33,117),(3,128,34,116),(4,127,35,115),(5,126,36,114),(6,150,37,113),(7,149,38,112),(8,148,39,111),(9,147,40,110),(10,146,41,109),(11,145,42,108),(12,144,43,107),(13,143,44,106),(14,142,45,105),(15,141,46,104),(16,140,47,103),(17,139,48,102),(18,138,49,101),(19,137,50,125),(20,136,26,124),(21,135,27,123),(22,134,28,122),(23,133,29,121),(24,132,30,120),(25,131,31,119),(51,182,78,171),(52,181,79,170),(53,180,80,169),(54,179,81,168),(55,178,82,167),(56,177,83,166),(57,176,84,165),(58,200,85,164),(59,199,86,163),(60,198,87,162),(61,197,88,161),(62,196,89,160),(63,195,90,159),(64,194,91,158),(65,193,92,157),(66,192,93,156),(67,191,94,155),(68,190,95,154),(69,189,96,153),(70,188,97,152),(71,187,98,151),(72,186,99,175),(73,185,100,174),(74,184,76,173),(75,183,77,172)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 25A | ··· | 25J | 50A | ··· | 50J | 50K | ··· | 50AD | 100A | ··· | 100J |
| order | 1 | 2 | 2 | 4 | 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 | 2 | 100 | 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 | SD16 | D10 | C5⋊D4 | D25 | D50 | C25⋊D4 | D4.D5 | D4.D25 |
| kernel | D4.D25 | C25⋊2C8 | Dic50 | 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
| 400 | 396 | 0 | 0 |
| 241 | 1 | 0 | 0 |
| 0 | 0 | 400 | 0 |
| 0 | 0 | 0 | 400 |
| 1 | 0 | 0 | 0 |
| 160 | 400 | 0 | 0 |
| 0 | 0 | 400 | 0 |
| 0 | 0 | 209 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 125 | 0 |
| 0 | 0 | 197 | 77 |
| 0 | 244 | 0 | 0 |
| 212 | 0 | 0 | 0 |
| 0 | 0 | 77 | 254 |
| 0 | 0 | 204 | 324 |
G:=sub<GL(4,GF(401))| [400,241,0,0,396,1,0,0,0,0,400,0,0,0,0,400],[1,160,0,0,0,400,0,0,0,0,400,209,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,125,197,0,0,0,77],[0,212,0,0,244,0,0,0,0,0,77,204,0,0,254,324] >;
D4.D25 in GAP, Magma, Sage, TeX
D_4.D_{25} % in TeX
G:=Group("D4.D25"); // GroupNames label
G:=SmallGroup(400,15);
// by ID
G=gap.SmallGroup(400,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,218,116,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^25=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export