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## G = D4×C50order 400 = 24·52

### Direct product of C50 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C50
 Chief series C1 — C5 — C10 — C50 — C2×C50 — D4×C25 — D4×C50
 Lower central C1 — C2 — D4×C50
 Upper central C1 — C2×C50 — D4×C50

Generators and relations for D4×C50
G = < a,b,c | a50=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 105 in 81 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C10, C10, C10, C2×D4, C20, C2×C10, C2×C10, C2×C10, C25, C2×C20, C5×D4, C22×C10, C50, C50, C50, D4×C10, C100, C2×C50, C2×C50, C2×C50, C2×C100, D4×C25, C22×C50, D4×C50
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C25, C5×D4, C22×C10, C50, D4×C10, C2×C50, D4×C25, C22×C50, D4×C50

Smallest permutation representation of D4×C50
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)
(1 104 100 198)(2 105 51 199)(3 106 52 200)(4 107 53 151)(5 108 54 152)(6 109 55 153)(7 110 56 154)(8 111 57 155)(9 112 58 156)(10 113 59 157)(11 114 60 158)(12 115 61 159)(13 116 62 160)(14 117 63 161)(15 118 64 162)(16 119 65 163)(17 120 66 164)(18 121 67 165)(19 122 68 166)(20 123 69 167)(21 124 70 168)(22 125 71 169)(23 126 72 170)(24 127 73 171)(25 128 74 172)(26 129 75 173)(27 130 76 174)(28 131 77 175)(29 132 78 176)(30 133 79 177)(31 134 80 178)(32 135 81 179)(33 136 82 180)(34 137 83 181)(35 138 84 182)(36 139 85 183)(37 140 86 184)(38 141 87 185)(39 142 88 186)(40 143 89 187)(41 144 90 188)(42 145 91 189)(43 146 92 190)(44 147 93 191)(45 148 94 192)(46 149 95 193)(47 150 96 194)(48 101 97 195)(49 102 98 196)(50 103 99 197)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(101 126)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 145)(121 146)(122 147)(123 148)(124 149)(125 150)(151 176)(152 177)(153 178)(154 179)(155 180)(156 181)(157 182)(158 183)(159 184)(160 185)(161 186)(162 187)(163 188)(164 189)(165 190)(166 191)(167 192)(168 193)(169 194)(170 195)(171 196)(172 197)(173 198)(174 199)(175 200)

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), (1,104,100,198)(2,105,51,199)(3,106,52,200)(4,107,53,151)(5,108,54,152)(6,109,55,153)(7,110,56,154)(8,111,57,155)(9,112,58,156)(10,113,59,157)(11,114,60,158)(12,115,61,159)(13,116,62,160)(14,117,63,161)(15,118,64,162)(16,119,65,163)(17,120,66,164)(18,121,67,165)(19,122,68,166)(20,123,69,167)(21,124,70,168)(22,125,71,169)(23,126,72,170)(24,127,73,171)(25,128,74,172)(26,129,75,173)(27,130,76,174)(28,131,77,175)(29,132,78,176)(30,133,79,177)(31,134,80,178)(32,135,81,179)(33,136,82,180)(34,137,83,181)(35,138,84,182)(36,139,85,183)(37,140,86,184)(38,141,87,185)(39,142,88,186)(40,143,89,187)(41,144,90,188)(42,145,91,189)(43,146,92,190)(44,147,93,191)(45,148,94,192)(46,149,95,193)(47,150,96,194)(48,101,97,195)(49,102,98,196)(50,103,99,197), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181)(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)(174,199)(175,200)>;

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), (1,104,100,198)(2,105,51,199)(3,106,52,200)(4,107,53,151)(5,108,54,152)(6,109,55,153)(7,110,56,154)(8,111,57,155)(9,112,58,156)(10,113,59,157)(11,114,60,158)(12,115,61,159)(13,116,62,160)(14,117,63,161)(15,118,64,162)(16,119,65,163)(17,120,66,164)(18,121,67,165)(19,122,68,166)(20,123,69,167)(21,124,70,168)(22,125,71,169)(23,126,72,170)(24,127,73,171)(25,128,74,172)(26,129,75,173)(27,130,76,174)(28,131,77,175)(29,132,78,176)(30,133,79,177)(31,134,80,178)(32,135,81,179)(33,136,82,180)(34,137,83,181)(35,138,84,182)(36,139,85,183)(37,140,86,184)(38,141,87,185)(39,142,88,186)(40,143,89,187)(41,144,90,188)(42,145,91,189)(43,146,92,190)(44,147,93,191)(45,148,94,192)(46,149,95,193)(47,150,96,194)(48,101,97,195)(49,102,98,196)(50,103,99,197), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181)(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)(174,199)(175,200) );

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)], [(1,104,100,198),(2,105,51,199),(3,106,52,200),(4,107,53,151),(5,108,54,152),(6,109,55,153),(7,110,56,154),(8,111,57,155),(9,112,58,156),(10,113,59,157),(11,114,60,158),(12,115,61,159),(13,116,62,160),(14,117,63,161),(15,118,64,162),(16,119,65,163),(17,120,66,164),(18,121,67,165),(19,122,68,166),(20,123,69,167),(21,124,70,168),(22,125,71,169),(23,126,72,170),(24,127,73,171),(25,128,74,172),(26,129,75,173),(27,130,76,174),(28,131,77,175),(29,132,78,176),(30,133,79,177),(31,134,80,178),(32,135,81,179),(33,136,82,180),(34,137,83,181),(35,138,84,182),(36,139,85,183),(37,140,86,184),(38,141,87,185),(39,142,88,186),(40,143,89,187),(41,144,90,188),(42,145,91,189),(43,146,92,190),(44,147,93,191),(45,148,94,192),(46,149,95,193),(47,150,96,194),(48,101,97,195),(49,102,98,196),(50,103,99,197)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(101,126),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,145),(121,146),(122,147),(123,148),(124,149),(125,150),(151,176),(152,177),(153,178),(154,179),(155,180),(156,181),(157,182),(158,183),(159,184),(160,185),(161,186),(162,187),(163,188),(164,189),(165,190),(166,191),(167,192),(168,193),(169,194),(170,195),(171,196),(172,197),(173,198),(174,199),(175,200)]])

250 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AB 20A ··· 20H 25A ··· 25T 50A ··· 50BH 50BI ··· 50EJ 100A ··· 100AN order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 50 ··· 50 100 ··· 100 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 C25 C50 C50 C50 D4 C5×D4 D4×C25 kernel D4×C50 C2×C100 D4×C25 C22×C50 D4×C10 C2×C20 C5×D4 C22×C10 C2×D4 C2×C4 D4 C23 C50 C10 C2 # reps 1 1 4 2 4 4 16 8 20 20 80 40 2 8 40

Matrix representation of D4×C50 in GL3(𝔽101) generated by

 100 0 0 0 33 0 0 0 33
,
 1 0 0 0 100 90 0 92 1
,
 1 0 0 0 1 11 0 0 100
G:=sub<GL(3,GF(101))| [100,0,0,0,33,0,0,0,33],[1,0,0,0,100,92,0,90,1],[1,0,0,0,1,0,0,11,100] >;

D4×C50 in GAP, Magma, Sage, TeX

D_4\times C_{50}
% in TeX

G:=Group("D4xC50");
// GroupNames label

G:=SmallGroup(400,46);
// by ID

G=gap.SmallGroup(400,46);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,505,261]);
// Polycyclic

G:=Group<a,b,c|a^50=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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