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G = C16⋊D13order 416 = 25·13

2nd semidirect product of C16 and D13 acting via D13/C13=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C2082C2, C162D13, C26.2D8, C4.2D52, C131SD32, C2.4D104, C52.25D4, C8.14D26, Dic521C2, D104.1C2, C104.15C22, SmallGroup(416,7)

Series: Derived Chief Lower central Upper central

C1C104 — C16⋊D13
C1C13C26C52C104D104 — C16⋊D13
C13C26C52C104 — C16⋊D13
C1C2C4C8C16

Generators and relations for C16⋊D13
 G = < a,b,c | a16=b13=c2=1, ab=ba, cac=a7, cbc=b-1 >

104C2
52C4
52C22
8D13
26Q8
26D4
4D26
4Dic13
13Q16
13D8
2Dic26
2D52
13SD32

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

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

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

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

107 conjugacy classes

class 1 2A2B4A4B8A8B13A···13F16A16B16C16D26A···26F52A···52L104A···104X208A···208AV
order122448813···131616161626···2652···52104···104208···208
size111042104222···222222···22···22···22···2

107 irreducible representations

dim111122222222
type++++++++++
imageC1C2C2C2D4D8D13SD32D26D52D104C16⋊D13
kernelC16⋊D13C208D104Dic52C52C26C16C13C8C4C2C1
# reps111112646122448

Matrix representation of C16⋊D13 in GL2(𝔽1249) generated by

69358
251361
,
3931
68824
,
1235972
89814
G:=sub<GL(2,GF(1249))| [69,251,358,361],[393,688,1,24],[1235,898,972,14] >;

C16⋊D13 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,7);
// by ID

G=gap.SmallGroup(416,7);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,79,506,50,579,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^16=b^13=c^2=1,a*b=b*a,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C16⋊D13 in TeX

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