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G = C13⋊D16order 416 = 25·13

The semidirect product of C13 and D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C132D16, D81D13, C8.4D26, C26.8D8, C52.3D4, D1043C2, C104.2C22, (C13×D8)⋊1C2, C132C161C2, C2.4(D4⋊D13), C4.1(C13⋊D4), SmallGroup(416,33)

Series: Derived Chief Lower central Upper central

C1C104 — C13⋊D16
C1C13C26C52C104D104 — C13⋊D16
C13C26C52C104 — C13⋊D16
C1C2C4C8D8

Generators and relations for C13⋊D16
 G = < a,b,c | a13=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

8C2
104C2
4C22
52C22
8D13
8C26
2D4
26D4
4D26
4C2×C26
13C16
13D8
2D52
2D4×C13
13D16

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

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

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

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

53 conjugacy classes

class 1 2A2B2C 4 8A8B13A···13F16A16B16C16D26A···26F26G···26R52A···52F104A···104L
order122248813···131616161626···2626···2652···52104···104
size1181042222···2262626262···28···84···44···4

53 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2D4D8D13D16D26C13⋊D4D4⋊D13C13⋊D16
kernelC13⋊D16C132C16D104C13×D8C52C26D8C13C8C4C2C1
# reps11111264612612

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

1064100
89546100
0010
0001
,
41924100
2383000
00536571
003391214
,
68020900
117656900
0010
0012481248
G:=sub<GL(4,GF(1249))| [1064,895,0,0,1,461,0,0,0,0,1,0,0,0,0,1],[419,23,0,0,241,830,0,0,0,0,536,339,0,0,571,1214],[680,1176,0,0,209,569,0,0,0,0,1,1248,0,0,0,1248] >;

C13⋊D16 in GAP, Magma, Sage, TeX

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

G:=Group("C13:D16");
// GroupNames label

G:=SmallGroup(416,33);
// by ID

G=gap.SmallGroup(416,33);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,218,116,122,579,297,69,13829]);
// Polycyclic

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

Export

Subgroup lattice of C13⋊D16 in TeX

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