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## G = C52⋊10SD16order 400 = 24·52

### 2nd semidirect product of C52 and SD16 acting via SD16/Q8=C2

Aliases: C20.18D10, C5210SD16, Q8⋊(C5⋊D5), (C5×Q8)⋊1D5, C53(Q8⋊D5), C527C85C2, (C5×C10).36D4, (Q8×C52)⋊2C2, C20⋊D5.3C2, C10.24(C5⋊D4), (C5×C20).14C22, C2.6(C527D4), C4.3(C2×C5⋊D5), SmallGroup(400,105)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C20 — C52⋊10SD16
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C20⋊D5 — C52⋊10SD16
 Lower central C52 — C5×C10 — C5×C20 — C52⋊10SD16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C5210SD16
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c3 >

Subgroups: 552 in 80 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, D4, Q8, D5, C10, SD16, C20, C20, D10, C52, C52C8, D20, C5×Q8, C5⋊D5, C5×C10, Q8⋊D5, C5×C20, C5×C20, C2×C5⋊D5, C527C8, C20⋊D5, Q8×C52, C5210SD16
Quotients: C1, C2, C22, D4, D5, SD16, D10, C5⋊D4, C5⋊D5, Q8⋊D5, C2×C5⋊D5, C527D4, C5210SD16

Smallest permutation representation of C5210SD16
On 200 points
Generators in S200
(1 34 62 46 182)(2 183 47 63 35)(3 36 64 48 184)(4 177 41 57 37)(5 38 58 42 178)(6 179 43 59 39)(7 40 60 44 180)(8 181 45 61 33)(9 174 192 22 78)(10 79 23 185 175)(11 176 186 24 80)(12 73 17 187 169)(13 170 188 18 74)(14 75 19 189 171)(15 172 190 20 76)(16 77 21 191 173)(25 143 125 158 197)(26 198 159 126 144)(27 137 127 160 199)(28 200 153 128 138)(29 139 121 154 193)(30 194 155 122 140)(31 141 123 156 195)(32 196 157 124 142)(49 65 106 103 113)(50 114 104 107 66)(51 67 108 97 115)(52 116 98 109 68)(53 69 110 99 117)(54 118 100 111 70)(55 71 112 101 119)(56 120 102 105 72)(81 166 94 148 131)(82 132 149 95 167)(83 168 96 150 133)(84 134 151 89 161)(85 162 90 152 135)(86 136 145 91 163)(87 164 92 146 129)(88 130 147 93 165)
(1 104 87 143 10)(2 11 144 88 97)(3 98 81 137 12)(4 13 138 82 99)(5 100 83 139 14)(6 15 140 84 101)(7 102 85 141 16)(8 9 142 86 103)(17 64 68 94 160)(18 153 95 69 57)(19 58 70 96 154)(20 155 89 71 59)(21 60 72 90 156)(22 157 91 65 61)(23 62 66 92 158)(24 159 93 67 63)(25 175 182 114 129)(26 130 115 183 176)(27 169 184 116 131)(28 132 117 177 170)(29 171 178 118 133)(30 134 119 179 172)(31 173 180 120 135)(32 136 113 181 174)(33 78 124 163 106)(34 107 164 125 79)(35 80 126 165 108)(36 109 166 127 73)(37 74 128 167 110)(38 111 168 121 75)(39 76 122 161 112)(40 105 162 123 77)(41 188 200 149 53)(42 54 150 193 189)(43 190 194 151 55)(44 56 152 195 191)(45 192 196 145 49)(46 50 146 197 185)(47 186 198 147 51)(48 52 148 199 187)
(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)(9 101)(10 104)(11 99)(12 102)(13 97)(14 100)(15 103)(16 98)(17 56)(18 51)(19 54)(20 49)(21 52)(22 55)(23 50)(24 53)(25 164)(26 167)(27 162)(28 165)(29 168)(30 163)(31 166)(32 161)(33 179)(34 182)(35 177)(36 180)(37 183)(38 178)(39 181)(40 184)(41 63)(42 58)(43 61)(44 64)(45 59)(46 62)(47 57)(48 60)(65 190)(66 185)(67 188)(68 191)(69 186)(70 189)(71 192)(72 187)(73 120)(74 115)(75 118)(76 113)(77 116)(78 119)(79 114)(80 117)(81 141)(82 144)(83 139)(84 142)(85 137)(86 140)(87 143)(88 138)(89 196)(90 199)(91 194)(92 197)(93 200)(94 195)(95 198)(96 193)(105 169)(106 172)(107 175)(108 170)(109 173)(110 176)(111 171)(112 174)(121 133)(122 136)(123 131)(124 134)(125 129)(126 132)(127 135)(128 130)(145 155)(146 158)(147 153)(148 156)(149 159)(150 154)(151 157)(152 160)

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

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

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

67 conjugacy classes

 class 1 2A 2B 4A 4B 5A ··· 5L 8A 8B 10A ··· 10L 20A ··· 20AJ order 1 2 2 4 4 5 ··· 5 8 8 10 ··· 10 20 ··· 20 size 1 1 100 2 4 2 ··· 2 50 50 2 ··· 2 4 ··· 4

67 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 D4 D5 SD16 D10 C5⋊D4 Q8⋊D5 kernel C52⋊10SD16 C52⋊7C8 C20⋊D5 Q8×C52 C5×C10 C5×Q8 C52 C20 C10 C5 # reps 1 1 1 1 1 12 2 12 24 12

Matrix representation of C5210SD16 in GL6(𝔽41)

 40 6 0 0 0 0 35 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 6 0 0 0 0 0 0 35 1 0 0 0 0 5 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 18 21 0 0 0 0 6 23 0 0 0 0 0 0 35 34 0 0 0 0 5 6 0 0 0 0 0 0 0 26 0 0 0 0 30 11
,
 1 35 0 0 0 0 0 40 0 0 0 0 0 0 6 7 0 0 0 0 36 35 0 0 0 0 0 0 1 0 0 0 0 0 2 40

G:=sub<GL(6,GF(41))| [40,35,0,0,0,0,6,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,6,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,6,0,0,0,0,21,23,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,0,30,0,0,0,0,26,11],[1,0,0,0,0,0,35,40,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,1,2,0,0,0,0,0,40] >;

C5210SD16 in GAP, Magma, Sage, TeX

C_5^2\rtimes_{10}{\rm SD}_{16}
% in TeX

G:=Group("C5^2:10SD16");
// GroupNames label

G:=SmallGroup(400,105);
// by ID

G=gap.SmallGroup(400,105);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,55,218,116,50,1924,11525]);
// Polycyclic

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

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