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## G = SD16×C52order 400 = 24·52

### Direct product of C52 and SD16

direct product, metacyclic, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C52
 Chief series C1 — C2 — C4 — C20 — C5×C20 — Q8×C52 — SD16×C52
 Lower central C1 — C2 — C4 — SD16×C52
 Upper central C1 — C5×C10 — C5×C20 — SD16×C52

Generators and relations for SD16×C52
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 120 in 80 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, D4, Q8, C10, C10, SD16, C20, C20, C2×C10, C52, C40, C5×D4, C5×Q8, C5×C10, C5×C10, C5×SD16, C5×C20, C5×C20, C102, C5×C40, D4×C52, Q8×C52, SD16×C52
Quotients: C1, C2, C22, C5, D4, C10, SD16, C2×C10, C52, C5×D4, C5×C10, C5×SD16, C102, D4×C52, SD16×C52

Smallest permutation representation of SD16×C52
On 200 points
Generators in S200
(1 98 94 171 70)(2 99 95 172 71)(3 100 96 173 72)(4 101 89 174 65)(5 102 90 175 66)(6 103 91 176 67)(7 104 92 169 68)(8 97 93 170 69)(9 195 179 78 36)(10 196 180 79 37)(11 197 181 80 38)(12 198 182 73 39)(13 199 183 74 40)(14 200 184 75 33)(15 193 177 76 34)(16 194 178 77 35)(17 192 46 123 118)(18 185 47 124 119)(19 186 48 125 120)(20 187 41 126 113)(21 188 42 127 114)(22 189 43 128 115)(23 190 44 121 116)(24 191 45 122 117)(25 110 155 149 52)(26 111 156 150 53)(27 112 157 151 54)(28 105 158 152 55)(29 106 159 145 56)(30 107 160 146 49)(31 108 153 147 50)(32 109 154 148 51)(57 137 132 81 166)(58 138 133 82 167)(59 139 134 83 168)(60 140 135 84 161)(61 141 136 85 162)(62 142 129 86 163)(63 143 130 87 164)(64 144 131 88 165)
(1 155 9 62 20)(2 156 10 63 21)(3 157 11 64 22)(4 158 12 57 23)(5 159 13 58 24)(6 160 14 59 17)(7 153 15 60 18)(8 154 16 61 19)(25 78 86 126 171)(26 79 87 127 172)(27 80 88 128 173)(28 73 81 121 174)(29 74 82 122 175)(30 75 83 123 176)(31 76 84 124 169)(32 77 85 125 170)(33 168 118 67 107)(34 161 119 68 108)(35 162 120 69 109)(36 163 113 70 110)(37 164 114 71 111)(38 165 115 72 112)(39 166 116 65 105)(40 167 117 66 106)(41 94 52 179 129)(42 95 53 180 130)(43 96 54 181 131)(44 89 55 182 132)(45 90 56 183 133)(46 91 49 184 134)(47 92 50 177 135)(48 93 51 178 136)(97 148 194 141 186)(98 149 195 142 187)(99 150 196 143 188)(100 151 197 144 189)(101 152 198 137 190)(102 145 199 138 191)(103 146 200 139 192)(104 147 193 140 185)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)(49 51)(50 54)(53 55)(57 63)(59 61)(60 64)(65 71)(67 69)(68 72)(73 79)(75 77)(76 80)(81 87)(83 85)(84 88)(89 95)(91 93)(92 96)(97 103)(99 101)(100 104)(105 111)(107 109)(108 112)(114 116)(115 119)(118 120)(121 127)(123 125)(124 128)(130 132)(131 135)(134 136)(137 143)(139 141)(140 144)(146 148)(147 151)(150 152)(153 157)(154 160)(156 158)(161 165)(162 168)(164 166)(169 173)(170 176)(172 174)(177 181)(178 184)(180 182)(185 189)(186 192)(188 190)(193 197)(194 200)(196 198)

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

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

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

175 conjugacy classes

 class 1 2A 2B 4A 4B 5A ··· 5X 8A 8B 10A ··· 10X 10Y ··· 10AV 20A ··· 20X 20Y ··· 20AV 40A ··· 40AV order 1 2 2 4 4 5 ··· 5 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 4 2 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 2 2 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 SD16 C5×D4 C5×SD16 kernel SD16×C52 C5×C40 D4×C52 Q8×C52 C5×SD16 C40 C5×D4 C5×Q8 C5×C10 C52 C10 C5 # reps 1 1 1 1 24 24 24 24 1 2 24 48

Matrix representation of SD16×C52 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 40 39 0 0 1 1 0 0 0 0 15 26 0 0 15 15
,
 1 0 0 0 40 40 0 0 0 0 1 0 0 0 0 40
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[40,1,0,0,39,1,0,0,0,0,15,15,0,0,26,15],[1,40,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

SD16×C52 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_5^2
% in TeX

G:=Group("SD16xC5^2");
// GroupNames label

G:=SmallGroup(400,114);
// by ID

G=gap.SmallGroup(400,114);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,1200,1225,9004,4510,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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