metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D50⋊1C4, C50.6D4, C2.2D100, C10.5D20, C22.6D50, (C2×C4)⋊1D25, (C2×C100)⋊1C2, C2.5(C4×D25), (C2×C20).2D5, C25⋊2(C22⋊C4), C50.12(C2×C4), C10.16(C4×D5), C5.(D10⋊C4), (C2×Dic25)⋊1C2, (C2×C10).21D10, C2.2(C25⋊D4), (C2×C50).6C22, C10.13(C5⋊D4), (C22×D25).1C2, SmallGroup(400,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D50⋊C4
G = < a,b,c | a50=b2=c4=1, bab=a-1, ac=ca, cbc-1=a25b >
50C2
50C2
2C4
25C22
25C22
50C4
50C22
50C22
10D5
10D5
25C23
25C2×C4
2C20
5D10
5D10
10Dic5
10D10
10D10
2D25
2D25
25C22⋊C4
2C100
2D50
2D50
Smallest permutation representation of D50⋊C4
►On 200 points
Generators in S200
(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 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 84)(12 83)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(101 152)(102 151)(103 200)(104 199)(105 198)(106 197)(107 196)(108 195)(109 194)(110 193)(111 192)(112 191)(113 190)(114 189)(115 188)(116 187)(117 186)(118 185)(119 184)(120 183)(121 182)(122 181)(123 180)(124 179)(125 178)(126 177)(127 176)(128 175)(129 174)(130 173)(131 172)(132 171)(133 170)(134 169)(135 168)(136 167)(137 166)(138 165)(139 164)(140 163)(141 162)(142 161)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)
(1 159 95 120)(2 160 96 121)(3 161 97 122)(4 162 98 123)(5 163 99 124)(6 164 100 125)(7 165 51 126)(8 166 52 127)(9 167 53 128)(10 168 54 129)(11 169 55 130)(12 170 56 131)(13 171 57 132)(14 172 58 133)(15 173 59 134)(16 174 60 135)(17 175 61 136)(18 176 62 137)(19 177 63 138)(20 178 64 139)(21 179 65 140)(22 180 66 141)(23 181 67 142)(24 182 68 143)(25 183 69 144)(26 184 70 145)(27 185 71 146)(28 186 72 147)(29 187 73 148)(30 188 74 149)(31 189 75 150)(32 190 76 101)(33 191 77 102)(34 192 78 103)(35 193 79 104)(36 194 80 105)(37 195 81 106)(38 196 82 107)(39 197 83 108)(40 198 84 109)(41 199 85 110)(42 200 86 111)(43 151 87 112)(44 152 88 113)(45 153 89 114)(46 154 90 115)(47 155 91 116)(48 156 92 117)(49 157 93 118)(50 158 94 119)
G:=sub<Sym(200)| (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), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,84)(12,83)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(101,152)(102,151)(103,200)(104,199)(105,198)(106,197)(107,196)(108,195)(109,194)(110,193)(111,192)(112,191)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153), (1,159,95,120)(2,160,96,121)(3,161,97,122)(4,162,98,123)(5,163,99,124)(6,164,100,125)(7,165,51,126)(8,166,52,127)(9,167,53,128)(10,168,54,129)(11,169,55,130)(12,170,56,131)(13,171,57,132)(14,172,58,133)(15,173,59,134)(16,174,60,135)(17,175,61,136)(18,176,62,137)(19,177,63,138)(20,178,64,139)(21,179,65,140)(22,180,66,141)(23,181,67,142)(24,182,68,143)(25,183,69,144)(26,184,70,145)(27,185,71,146)(28,186,72,147)(29,187,73,148)(30,188,74,149)(31,189,75,150)(32,190,76,101)(33,191,77,102)(34,192,78,103)(35,193,79,104)(36,194,80,105)(37,195,81,106)(38,196,82,107)(39,197,83,108)(40,198,84,109)(41,199,85,110)(42,200,86,111)(43,151,87,112)(44,152,88,113)(45,153,89,114)(46,154,90,115)(47,155,91,116)(48,156,92,117)(49,157,93,118)(50,158,94,119)>;
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), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,84)(12,83)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(101,152)(102,151)(103,200)(104,199)(105,198)(106,197)(107,196)(108,195)(109,194)(110,193)(111,192)(112,191)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153), (1,159,95,120)(2,160,96,121)(3,161,97,122)(4,162,98,123)(5,163,99,124)(6,164,100,125)(7,165,51,126)(8,166,52,127)(9,167,53,128)(10,168,54,129)(11,169,55,130)(12,170,56,131)(13,171,57,132)(14,172,58,133)(15,173,59,134)(16,174,60,135)(17,175,61,136)(18,176,62,137)(19,177,63,138)(20,178,64,139)(21,179,65,140)(22,180,66,141)(23,181,67,142)(24,182,68,143)(25,183,69,144)(26,184,70,145)(27,185,71,146)(28,186,72,147)(29,187,73,148)(30,188,74,149)(31,189,75,150)(32,190,76,101)(33,191,77,102)(34,192,78,103)(35,193,79,104)(36,194,80,105)(37,195,81,106)(38,196,82,107)(39,197,83,108)(40,198,84,109)(41,199,85,110)(42,200,86,111)(43,151,87,112)(44,152,88,113)(45,153,89,114)(46,154,90,115)(47,155,91,116)(48,156,92,117)(49,157,93,118)(50,158,94,119) );
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)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,84),(12,83),(13,82),(14,81),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,74),(22,73),(23,72),(24,71),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(101,152),(102,151),(103,200),(104,199),(105,198),(106,197),(107,196),(108,195),(109,194),(110,193),(111,192),(112,191),(113,190),(114,189),(115,188),(116,187),(117,186),(118,185),(119,184),(120,183),(121,182),(122,181),(123,180),(124,179),(125,178),(126,177),(127,176),(128,175),(129,174),(130,173),(131,172),(132,171),(133,170),(134,169),(135,168),(136,167),(137,166),(138,165),(139,164),(140,163),(141,162),(142,161),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153)], [(1,159,95,120),(2,160,96,121),(3,161,97,122),(4,162,98,123),(5,163,99,124),(6,164,100,125),(7,165,51,126),(8,166,52,127),(9,167,53,128),(10,168,54,129),(11,169,55,130),(12,170,56,131),(13,171,57,132),(14,172,58,133),(15,173,59,134),(16,174,60,135),(17,175,61,136),(18,176,62,137),(19,177,63,138),(20,178,64,139),(21,179,65,140),(22,180,66,141),(23,181,67,142),(24,182,68,143),(25,183,69,144),(26,184,70,145),(27,185,71,146),(28,186,72,147),(29,187,73,148),(30,188,74,149),(31,189,75,150),(32,190,76,101),(33,191,77,102),(34,192,78,103),(35,193,79,104),(36,194,80,105),(37,195,81,106),(38,196,82,107),(39,197,83,108),(40,198,84,109),(41,199,85,110),(42,200,86,111),(43,151,87,112),(44,152,88,113),(45,153,89,114),(46,154,90,115),(47,155,91,116),(48,156,92,117),(49,157,93,118),(50,158,94,119)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 25A | ··· | 25J | 50A | ··· | 50AD | 100A | ··· | 100AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 50 | 50 | 2 | 2 | 50 | 50 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D5 | D10 | C4×D5 | D20 | C5⋊D4 | D25 | D50 | C4×D25 | D100 | C25⋊D4 |
| kernel | D50⋊C4 | C2×Dic25 | C2×C100 | C22×D25 | D50 | C50 | C2×C20 | C2×C10 | C10 | C10 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 |
Matrix representation of D50⋊C4 ►in GL3(𝔽101) generated by
| 1 | 0 | 0 |
| 0 | 83 | 40 |
| 0 | 61 | 72 |
| 1 | 0 | 0 |
| 0 | 18 | 61 |
| 0 | 51 | 83 |
| 91 | 0 | 0 |
| 0 | 76 | 11 |
| 0 | 90 | 25 |
G:=sub<GL(3,GF(101))| [1,0,0,0,83,61,0,40,72],[1,0,0,0,18,51,0,61,83],[91,0,0,0,76,90,0,11,25] >;
D50⋊C4 in GAP, Magma, Sage, TeX
D_{50}\rtimes C_4 % in TeX
G:=Group("D50:C4"); // GroupNames label
G:=SmallGroup(400,14);
// by ID
G=gap.SmallGroup(400,14);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^50=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^25*b>;
// generators/relations
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