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G = C16×D13order 416 = 25·13

Direct product of C16 and D13

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C16×D13, C2083C2, D26.4C8, C8.19D26, Dic13.4C8, C104.19C22, C133(C2×C16), C2.1(C8×D13), C132C166C2, C132C8.7C4, C26.10(C2×C8), C52.42(C2×C4), (C4×D13).9C4, C4.16(C4×D13), (C8×D13).11C2, SmallGroup(416,4)

Series: Derived Chief Lower central Upper central

C1C13 — C16×D13
C1C13C26C52C104C8×D13 — C16×D13
C13 — C16×D13
C1C16

Generators and relations for C16×D13
 G = < a,b,c | a16=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C2
13C22
13C4
13C2×C4
13C8
13C2×C8
13C16
13C2×C16

Smallest permutation representation of C16×D13
On 208 points
Generators in S208
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 77 123 25 186 196 146 164 40 82 101 135 56)(2 78 124 26 187 197 147 165 41 83 102 136 57)(3 79 125 27 188 198 148 166 42 84 103 137 58)(4 80 126 28 189 199 149 167 43 85 104 138 59)(5 65 127 29 190 200 150 168 44 86 105 139 60)(6 66 128 30 191 201 151 169 45 87 106 140 61)(7 67 113 31 192 202 152 170 46 88 107 141 62)(8 68 114 32 177 203 153 171 47 89 108 142 63)(9 69 115 17 178 204 154 172 48 90 109 143 64)(10 70 116 18 179 205 155 173 33 91 110 144 49)(11 71 117 19 180 206 156 174 34 92 111 129 50)(12 72 118 20 181 207 157 175 35 93 112 130 51)(13 73 119 21 182 208 158 176 36 94 97 131 52)(14 74 120 22 183 193 159 161 37 95 98 132 53)(15 75 121 23 184 194 160 162 38 96 99 133 54)(16 76 122 24 185 195 145 163 39 81 100 134 55)
(1 64)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 81)(33 187)(34 188)(35 189)(36 190)(37 191)(38 192)(39 177)(40 178)(41 179)(42 180)(43 181)(44 182)(45 183)(46 184)(47 185)(48 186)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 129)(80 130)(97 127)(98 128)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(145 153)(146 154)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(161 201)(162 202)(163 203)(164 204)(165 205)(166 206)(167 207)(168 208)(169 193)(170 194)(171 195)(172 196)(173 197)(174 198)(175 199)(176 200)

G:=sub<Sym(208)| (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,77,123,25,186,196,146,164,40,82,101,135,56)(2,78,124,26,187,197,147,165,41,83,102,136,57)(3,79,125,27,188,198,148,166,42,84,103,137,58)(4,80,126,28,189,199,149,167,43,85,104,138,59)(5,65,127,29,190,200,150,168,44,86,105,139,60)(6,66,128,30,191,201,151,169,45,87,106,140,61)(7,67,113,31,192,202,152,170,46,88,107,141,62)(8,68,114,32,177,203,153,171,47,89,108,142,63)(9,69,115,17,178,204,154,172,48,90,109,143,64)(10,70,116,18,179,205,155,173,33,91,110,144,49)(11,71,117,19,180,206,156,174,34,92,111,129,50)(12,72,118,20,181,207,157,175,35,93,112,130,51)(13,73,119,21,182,208,158,176,36,94,97,131,52)(14,74,120,22,183,193,159,161,37,95,98,132,53)(15,75,121,23,184,194,160,162,38,96,99,133,54)(16,76,122,24,185,195,145,163,39,81,100,134,55), (1,64)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,81)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,177)(40,178)(41,179)(42,180)(43,181)(44,182)(45,183)(46,184)(47,185)(48,186)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,129)(80,130)(97,127)(98,128)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(161,201)(162,202)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)(176,200)>;

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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,77,123,25,186,196,146,164,40,82,101,135,56)(2,78,124,26,187,197,147,165,41,83,102,136,57)(3,79,125,27,188,198,148,166,42,84,103,137,58)(4,80,126,28,189,199,149,167,43,85,104,138,59)(5,65,127,29,190,200,150,168,44,86,105,139,60)(6,66,128,30,191,201,151,169,45,87,106,140,61)(7,67,113,31,192,202,152,170,46,88,107,141,62)(8,68,114,32,177,203,153,171,47,89,108,142,63)(9,69,115,17,178,204,154,172,48,90,109,143,64)(10,70,116,18,179,205,155,173,33,91,110,144,49)(11,71,117,19,180,206,156,174,34,92,111,129,50)(12,72,118,20,181,207,157,175,35,93,112,130,51)(13,73,119,21,182,208,158,176,36,94,97,131,52)(14,74,120,22,183,193,159,161,37,95,98,132,53)(15,75,121,23,184,194,160,162,38,96,99,133,54)(16,76,122,24,185,195,145,163,39,81,100,134,55), (1,64)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,81)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,177)(40,178)(41,179)(42,180)(43,181)(44,182)(45,183)(46,184)(47,185)(48,186)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,129)(80,130)(97,127)(98,128)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(161,201)(162,202)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)(176,200) );

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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,77,123,25,186,196,146,164,40,82,101,135,56),(2,78,124,26,187,197,147,165,41,83,102,136,57),(3,79,125,27,188,198,148,166,42,84,103,137,58),(4,80,126,28,189,199,149,167,43,85,104,138,59),(5,65,127,29,190,200,150,168,44,86,105,139,60),(6,66,128,30,191,201,151,169,45,87,106,140,61),(7,67,113,31,192,202,152,170,46,88,107,141,62),(8,68,114,32,177,203,153,171,47,89,108,142,63),(9,69,115,17,178,204,154,172,48,90,109,143,64),(10,70,116,18,179,205,155,173,33,91,110,144,49),(11,71,117,19,180,206,156,174,34,92,111,129,50),(12,72,118,20,181,207,157,175,35,93,112,130,51),(13,73,119,21,182,208,158,176,36,94,97,131,52),(14,74,120,22,183,193,159,161,37,95,98,132,53),(15,75,121,23,184,194,160,162,38,96,99,133,54),(16,76,122,24,185,195,145,163,39,81,100,134,55)], [(1,64),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,81),(33,187),(34,188),(35,189),(36,190),(37,191),(38,192),(39,177),(40,178),(41,179),(42,180),(43,181),(44,182),(45,183),(46,184),(47,185),(48,186),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,129),(80,130),(97,127),(98,128),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,119),(106,120),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(145,153),(146,154),(147,155),(148,156),(149,157),(150,158),(151,159),(152,160),(161,201),(162,202),(163,203),(164,204),(165,205),(166,206),(167,207),(168,208),(169,193),(170,194),(171,195),(172,196),(173,197),(174,198),(175,199),(176,200)])

128 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H13A···13F16A···16H16I···16P26A···26F52A···52L104A···104X208A···208AV
order122244448888888813···1316···1616···1626···2652···52104···104208···208
size1113131113131111131313132···21···113···132···22···22···22···2

128 irreducible representations

dim11111111122222
type++++++
imageC1C2C2C2C4C4C8C8C16D13D26C4×D13C8×D13C16×D13
kernelC16×D13C132C16C208C8×D13C132C8C4×D13Dic13D26D13C16C8C4C2C1
# reps111122441666122448

Matrix representation of C16×D13 in GL3(𝔽1249) generated by

9800
05850
00585
,
100
010251
01008804
,
100
0807580
0152442
G:=sub<GL(3,GF(1249))| [98,0,0,0,585,0,0,0,585],[1,0,0,0,1025,1008,0,1,804],[1,0,0,0,807,152,0,580,442] >;

C16×D13 in GAP, Magma, Sage, TeX

C_{16}\times D_{13}
% in TeX

G:=Group("C16xD13");
// GroupNames label

G:=SmallGroup(416,4);
// by ID

G=gap.SmallGroup(416,4);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,31,50,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^16=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C16×D13 in TeX

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