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G = D13⋊C16order 416 = 25·13

The semidirect product of D13 and C16 acting via C16/C8=C2

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

Aliases: D13⋊C16, C104.4C4, D26.2C8, Dic13.2C8, C13⋊C163C2, C131(C2×C16), C8.5(C13⋊C4), C26.1(C2×C8), C52.13(C2×C4), (C8×D13).7C2, (C4×D13).5C4, C2.1(D13⋊C8), C132C8.14C22, C4.14(C2×C13⋊C4), SmallGroup(416,64)

Series: Derived Chief Lower central Upper central

C1C13 — D13⋊C16
C1C13C26C52C132C8C13⋊C16 — D13⋊C16
C13 — D13⋊C16
C1C8

Generators and relations for D13⋊C16
 G = < a,b,c | a13=b2=c16=1, bab=a-1, cac-1=a5, cbc-1=a4b >

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

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

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H13A13B13C16A···16P26A26B26C52A···52F104A···104L
order122244448888888813131316···1626262652···52104···104
size11131311131311111313131344413···134444···44···4

56 irreducible representations

dim111111114444
type+++++
imageC1C2C2C4C4C8C8C16C13⋊C4C2×C13⋊C4D13⋊C8D13⋊C16
kernelD13⋊C16C13⋊C16C8×D13C104C4×D13Dic13D26D13C8C4C2C1
# reps12122441633612

Matrix representation of D13⋊C16 in GL4(𝔽1249) generated by

1207120812071207
42424342
641641641642
1248124812481248
,
1248416501247
0120711641206
060841608
016412
,
9783879151247
50586211391139
9351099871004
1210894571
G:=sub<GL(4,GF(1249))| [1207,42,641,1248,1208,42,641,1248,1207,43,641,1248,1207,42,642,1248],[1248,0,0,0,41,1207,608,1,650,1164,41,641,1247,1206,608,2],[978,505,935,121,387,862,1099,0,915,1139,87,894,1247,1139,1004,571] >;

D13⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("D13:C16");
// GroupNames label

G:=SmallGroup(416,64);
// by ID

G=gap.SmallGroup(416,64);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,50,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^13=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations

Export

Subgroup lattice of D13⋊C16 in TeX

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