Copied to
clipboard

G = SD16×C25order 400 = 24·52

Direct product of C25 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: SD16×C25, Q8⋊C50, C82C50, D4.C50, C2006C2, C40.6C10, C50.15D4, C100.18C22, C5.(C5×SD16), C4.2(C2×C50), (Q8×C25)⋊4C2, C2.4(D4×C25), (C5×SD16).C5, (D4×C25).2C2, (C5×D4).3C10, C10.15(C5×D4), (C5×Q8).2C10, C20.18(C2×C10), SmallGroup(400,26)

Series: Derived Chief Lower central Upper central

C1C4 — SD16×C25
C1C2C10C20C100Q8×C25 — SD16×C25
C1C2C4 — SD16×C25
C1C50C100 — SD16×C25

Generators and relations for SD16×C25
 G = < a,b,c | a25=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C22
2C4
4C10
2C20
2C2×C10
4C50
2C100
2C2×C50

Smallest permutation representation of SD16×C25
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 97 101 26 184 149 164 70)(2 98 102 27 185 150 165 71)(3 99 103 28 186 126 166 72)(4 100 104 29 187 127 167 73)(5 76 105 30 188 128 168 74)(6 77 106 31 189 129 169 75)(7 78 107 32 190 130 170 51)(8 79 108 33 191 131 171 52)(9 80 109 34 192 132 172 53)(10 81 110 35 193 133 173 54)(11 82 111 36 194 134 174 55)(12 83 112 37 195 135 175 56)(13 84 113 38 196 136 151 57)(14 85 114 39 197 137 152 58)(15 86 115 40 198 138 153 59)(16 87 116 41 199 139 154 60)(17 88 117 42 200 140 155 61)(18 89 118 43 176 141 156 62)(19 90 119 44 177 142 157 63)(20 91 120 45 178 143 158 64)(21 92 121 46 179 144 159 65)(22 93 122 47 180 145 160 66)(23 94 123 48 181 146 161 67)(24 95 124 49 182 147 162 68)(25 96 125 50 183 148 163 69)
(26 97)(27 98)(28 99)(29 100)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 126)(73 127)(74 128)(75 129)(101 164)(102 165)(103 166)(104 167)(105 168)(106 169)(107 170)(108 171)(109 172)(110 173)(111 174)(112 175)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)(121 159)(122 160)(123 161)(124 162)(125 163)

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

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

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

175 conjugacy classes

class 1 2A2B4A4B5A5B5C5D8A8B10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H25A···25T40A···40H50A···50T50U···50AN100A···100T100U···100AN200A···200AN
order122445555881010101010101010202020202020202025···2540···4050···5050···50100···100100···100200···200
size1142411112211114444222244441···12···21···14···42···24···42···2

175 irreducible representations

dim111111111111222222
type+++++
imageC1C2C2C2C5C10C10C10C25C50C50C50D4SD16C5×D4C5×SD16D4×C25SD16×C25
kernelSD16×C25C200D4×C25Q8×C25C5×SD16C40C5×D4C5×Q8SD16C8D4Q8C50C25C10C5C2C1
# reps111144442020202012482040

Matrix representation of SD16×C25 in GL2(𝔽401) generated by

250
025
,
143143
1290
,
10
400400
G:=sub<GL(2,GF(401))| [25,0,0,25],[143,129,143,0],[1,400,0,400] >;

SD16×C25 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{25}
% in TeX

G:=Group("SD16xC25");
// GroupNames label

G:=SmallGroup(400,26);
// by ID

G=gap.SmallGroup(400,26);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-5,-2,1200,265,194,5283,2649,261]);
// Polycyclic

G:=Group<a,b,c|a^25=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

Export

Subgroup lattice of SD16×C25 in TeX

׿
×
𝔽