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G = C13×D16order 416 = 25·13

Direct product of C13 and D16

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C13×D16, C2085C2, C161C26, D81C26, C26.15D8, C52.36D4, C104.24C22, (C13×D8)⋊5C2, C8.2(C2×C26), C4.1(D4×C13), C2.3(C13×D8), SmallGroup(416,61)

Series: Derived Chief Lower central Upper central

C1C8 — C13×D16
C1C2C4C8C104C13×D8 — C13×D16
C1C2C4C8 — C13×D16
C1C26C52C104 — C13×D16

Generators and relations for C13×D16
 G = < a,b,c | a13=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

8C2
8C2
4C22
4C22
8C26
8C26
2D4
2D4
4C2×C26
4C2×C26
2D4×C13
2D4×C13

Smallest permutation representation of C13×D16
On 208 points
Generators in S208
(1 46 150 55 197 131 95 189 32 106 71 172 124)(2 47 151 56 198 132 96 190 17 107 72 173 125)(3 48 152 57 199 133 81 191 18 108 73 174 126)(4 33 153 58 200 134 82 192 19 109 74 175 127)(5 34 154 59 201 135 83 177 20 110 75 176 128)(6 35 155 60 202 136 84 178 21 111 76 161 113)(7 36 156 61 203 137 85 179 22 112 77 162 114)(8 37 157 62 204 138 86 180 23 97 78 163 115)(9 38 158 63 205 139 87 181 24 98 79 164 116)(10 39 159 64 206 140 88 182 25 99 80 165 117)(11 40 160 49 207 141 89 183 26 100 65 166 118)(12 41 145 50 208 142 90 184 27 101 66 167 119)(13 42 146 51 193 143 91 185 28 102 67 168 120)(14 43 147 52 194 144 92 186 29 103 68 169 121)(15 44 148 53 195 129 93 187 30 104 69 170 122)(16 45 149 54 196 130 94 188 31 105 70 171 123)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(33 43)(34 42)(35 41)(36 40)(37 39)(44 48)(45 47)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(62 64)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(78 80)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 96)(97 99)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(113 119)(114 118)(115 117)(120 128)(121 127)(122 126)(123 125)(129 133)(130 132)(134 144)(135 143)(136 142)(137 141)(138 140)(145 155)(146 154)(147 153)(148 152)(149 151)(156 160)(157 159)(161 167)(162 166)(163 165)(168 176)(169 175)(170 174)(171 173)(177 185)(178 184)(179 183)(180 182)(186 192)(187 191)(188 190)(193 201)(194 200)(195 199)(196 198)(202 208)(203 207)(204 206)

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

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

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

143 conjugacy classes

class 1 2A2B2C 4 8A8B13A···13L16A16B16C16D26A···26L26M···26AJ52A···52L104A···104X208A···208AV
order122248813···131616161626···2626···2652···52104···104208···208
size11882221···122221···18···82···22···22···2

143 irreducible representations

dim111111222222
type++++++
imageC1C2C2C13C26C26D4D8D16D4×C13C13×D8C13×D16
kernelC13×D16C208C13×D8D16C16D8C52C26C13C4C2C1
# reps112121224124122448

Matrix representation of C13×D16 in GL4(𝔽1249) generated by

933000
093300
0010
0001
,
0124800
1000
0035678
00910713
,
1000
0124800
0010
0011248
G:=sub<GL(4,GF(1249))| [933,0,0,0,0,933,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1248,0,0,0,0,0,35,910,0,0,678,713],[1,0,0,0,0,1248,0,0,0,0,1,1,0,0,0,1248] >;

C13×D16 in GAP, Magma, Sage, TeX

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

G:=Group("C13xD16");
// GroupNames label

G:=SmallGroup(416,61);
// by ID

G=gap.SmallGroup(416,61);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,649,3747,1881,165,9364,4690,88]);
// Polycyclic

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

Export

Subgroup lattice of C13×D16 in TeX

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