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

 Derived series C1 — C4 — SD16×C25
 Chief series C1 — C2 — C10 — C20 — C100 — Q8×C25 — SD16×C25
 Lower central C1 — C2 — C4 — SD16×C25
 Upper central C1 — C50 — C100 — SD16×C25

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

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

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

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

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

175 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E 20F 20G 20H 25A ··· 25T 40A ··· 40H 50A ··· 50T 50U ··· 50AN 100A ··· 100T 100U ··· 100AN 200A ··· 200AN order 1 2 2 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 25 ··· 25 40 ··· 40 50 ··· 50 50 ··· 50 100 ··· 100 100 ··· 100 200 ··· 200 size 1 1 4 2 4 1 1 1 1 2 2 1 1 1 1 4 4 4 4 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

175 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 C25 C50 C50 C50 D4 SD16 C5×D4 C5×SD16 D4×C25 SD16×C25 kernel SD16×C25 C200 D4×C25 Q8×C25 C5×SD16 C40 C5×D4 C5×Q8 SD16 C8 D4 Q8 C50 C25 C10 C5 C2 C1 # reps 1 1 1 1 4 4 4 4 20 20 20 20 1 2 4 8 20 40

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

 25 0 0 25
,
 143 143 129 0
,
 1 0 400 400
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

׿
×
𝔽