Copied to
clipboard

G = C13⋊C16order 208 = 24·13

The semidirect product of C13 and C16 acting via C16/C4=C4

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

Aliases: C13⋊C16, C26.C8, C52.2C4, C2.(C13⋊C8), C4.2(C13⋊C4), C132C8.2C2, SmallGroup(208,3)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C16
C1C13C26C52C132C8 — C13⋊C16
C13 — C13⋊C16
C1C4

Generators and relations for C13⋊C16
 G = < a,b | a13=b16=1, bab-1=a5 >

13C8
13C16

Character table of C13⋊C16

 class 124A4B8A8B8C8D13A13B13C16A16B16C16D16E16F16G16H26A26B26C52A52B52C52D52E52F
 size 1111131313134441313131313131313444444444
ρ11111111111111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ31111-1-1-1-1111i-i-i-iiii-i111111111    linear of order 4
ρ41111-1-1-1-1111-iiii-i-i-ii111111111    linear of order 4
ρ511-1-1i-i-ii111ζ8ζ87ζ83ζ83ζ85ζ85ζ8ζ87111-1-1-1-1-1-1    linear of order 8
ρ611-1-1-iii-i111ζ87ζ8ζ85ζ85ζ83ζ83ζ87ζ8111-1-1-1-1-1-1    linear of order 8
ρ711-1-1i-i-ii111ζ85ζ83ζ87ζ87ζ8ζ8ζ85ζ83111-1-1-1-1-1-1    linear of order 8
ρ811-1-1-iii-i111ζ83ζ85ζ8ζ8ζ87ζ87ζ83ζ85111-1-1-1-1-1-1    linear of order 8
ρ91-1-iiζ1610ζ166ζ1614ζ162111ζ165ζ163ζ167ζ1615ζ16ζ169ζ1613ζ1611-1-1-1-iii-ii-i    linear of order 16
ρ101-1-iiζ162ζ1614ζ166ζ1610111ζ169ζ1615ζ163ζ1611ζ165ζ1613ζ16ζ167-1-1-1-iii-ii-i    linear of order 16
ρ111-1i-iζ1614ζ162ζ1610ζ166111ζ167ζ16ζ1613ζ165ζ1611ζ163ζ1615ζ169-1-1-1i-i-ii-ii    linear of order 16
ρ121-1i-iζ166ζ1610ζ162ζ1614111ζ1611ζ1613ζ169ζ16ζ1615ζ167ζ163ζ165-1-1-1i-i-ii-ii    linear of order 16
ρ131-1-iiζ162ζ1614ζ166ζ1610111ζ16ζ167ζ1611ζ163ζ1613ζ165ζ169ζ1615-1-1-1-iii-ii-i    linear of order 16
ρ141-1-iiζ1610ζ166ζ1614ζ162111ζ1613ζ1611ζ1615ζ167ζ169ζ16ζ165ζ163-1-1-1-iii-ii-i    linear of order 16
ρ151-1i-iζ1614ζ162ζ1610ζ166111ζ1615ζ169ζ165ζ1613ζ163ζ1611ζ167ζ16-1-1-1i-i-ii-ii    linear of order 16
ρ161-1i-iζ166ζ1610ζ162ζ1614111ζ163ζ165ζ16ζ169ζ167ζ1615ζ1611ζ1613-1-1-1i-i-ii-ii    linear of order 16
ρ1744440000ζ13111310133132ζ131213813513ζ13913713613400000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ139137136134    orthogonal lifted from C13⋊C4
ρ1844440000ζ131213813513ζ139137136134ζ1311131013313200000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ13111310133132    orthogonal lifted from C13⋊C4
ρ1944440000ζ139137136134ζ13111310133132ζ13121381351300000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ131213813513    orthogonal lifted from C13⋊C4
ρ2044-4-40000ζ131213813513ζ139137136134ζ1311131013313200000000ζ13111310133132ζ139137136134ζ1312138135131391371361341391371361341312138135131312138135131311131013313213111310133132    symplectic lifted from C13⋊C8, Schur index 2
ρ2144-4-40000ζ13111310133132ζ131213813513ζ13913713613400000000ζ139137136134ζ131213813513ζ131113101331321312138135131312138135131311131013313213111310133132139137136134139137136134    symplectic lifted from C13⋊C8, Schur index 2
ρ2244-4-40000ζ139137136134ζ13111310133132ζ13121381351300000000ζ131213813513ζ13111310133132ζ1391371361341311131013313213111310133132139137136134139137136134131213813513131213813513    symplectic lifted from C13⋊C8, Schur index 2
ρ234-4-4i4i0000ζ131213813513ζ139137136134ζ131113101331320000000013111310133132139137136134131213813513ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131143ζ131043ζ13343ζ132    complex faithful, Schur index 4
ρ244-44i-4i0000ζ139137136134ζ13111310133132ζ1312138135130000000013121381351313111310133132139137136134ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13124ζ1384ζ1354ζ13    complex faithful, Schur index 4
ρ254-44i-4i0000ζ13111310133132ζ131213813513ζ1391371361340000000013913713613413121381351313111310133132ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ1394ζ1374ζ1364ζ134    complex faithful, Schur index 4
ρ264-4-4i4i0000ζ13111310133132ζ131213813513ζ1391371361340000000013913713613413121381351313111310133132ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ13943ζ13743ζ13643ζ134    complex faithful, Schur index 4
ρ274-44i-4i0000ζ131213813513ζ139137136134ζ131113101331320000000013111310133132139137136134131213813513ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13114ζ13104ζ1334ζ132    complex faithful, Schur index 4
ρ284-4-4i4i0000ζ139137136134ζ13111310133132ζ1312138135130000000013121381351313111310133132139137136134ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ131243ζ13843ζ13543ζ13    complex faithful, Schur index 4

Smallest permutation representation of C13⋊C16
Regular action on 208 points
Generators in S208
(1 136 111 168 20 195 73 189 51 154 39 124 87)(2 196 40 112 190 88 21 155 137 74 125 169 52)(3 89 126 41 156 53 191 75 197 22 170 97 138)(4 54 171 127 76 139 157 23 90 192 98 42 198)(5 140 99 172 24 199 77 177 55 158 43 128 91)(6 200 44 100 178 92 25 159 141 78 113 173 56)(7 93 114 45 160 57 179 79 201 26 174 101 142)(8 58 175 115 80 143 145 27 94 180 102 46 202)(9 144 103 176 28 203 65 181 59 146 47 116 95)(10 204 48 104 182 96 29 147 129 66 117 161 60)(11 81 118 33 148 61 183 67 205 30 162 105 130)(12 62 163 119 68 131 149 31 82 184 106 34 206)(13 132 107 164 32 207 69 185 63 150 35 120 83)(14 208 36 108 186 84 17 151 133 70 121 165 64)(15 85 122 37 152 49 187 71 193 18 166 109 134)(16 50 167 123 72 135 153 19 86 188 110 38 194)
(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)| (1,136,111,168,20,195,73,189,51,154,39,124,87)(2,196,40,112,190,88,21,155,137,74,125,169,52)(3,89,126,41,156,53,191,75,197,22,170,97,138)(4,54,171,127,76,139,157,23,90,192,98,42,198)(5,140,99,172,24,199,77,177,55,158,43,128,91)(6,200,44,100,178,92,25,159,141,78,113,173,56)(7,93,114,45,160,57,179,79,201,26,174,101,142)(8,58,175,115,80,143,145,27,94,180,102,46,202)(9,144,103,176,28,203,65,181,59,146,47,116,95)(10,204,48,104,182,96,29,147,129,66,117,161,60)(11,81,118,33,148,61,183,67,205,30,162,105,130)(12,62,163,119,68,131,149,31,82,184,106,34,206)(13,132,107,164,32,207,69,185,63,150,35,120,83)(14,208,36,108,186,84,17,151,133,70,121,165,64)(15,85,122,37,152,49,187,71,193,18,166,109,134)(16,50,167,123,72,135,153,19,86,188,110,38,194), (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( (1,136,111,168,20,195,73,189,51,154,39,124,87)(2,196,40,112,190,88,21,155,137,74,125,169,52)(3,89,126,41,156,53,191,75,197,22,170,97,138)(4,54,171,127,76,139,157,23,90,192,98,42,198)(5,140,99,172,24,199,77,177,55,158,43,128,91)(6,200,44,100,178,92,25,159,141,78,113,173,56)(7,93,114,45,160,57,179,79,201,26,174,101,142)(8,58,175,115,80,143,145,27,94,180,102,46,202)(9,144,103,176,28,203,65,181,59,146,47,116,95)(10,204,48,104,182,96,29,147,129,66,117,161,60)(11,81,118,33,148,61,183,67,205,30,162,105,130)(12,62,163,119,68,131,149,31,82,184,106,34,206)(13,132,107,164,32,207,69,185,63,150,35,120,83)(14,208,36,108,186,84,17,151,133,70,121,165,64)(15,85,122,37,152,49,187,71,193,18,166,109,134)(16,50,167,123,72,135,153,19,86,188,110,38,194), (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([(1,136,111,168,20,195,73,189,51,154,39,124,87),(2,196,40,112,190,88,21,155,137,74,125,169,52),(3,89,126,41,156,53,191,75,197,22,170,97,138),(4,54,171,127,76,139,157,23,90,192,98,42,198),(5,140,99,172,24,199,77,177,55,158,43,128,91),(6,200,44,100,178,92,25,159,141,78,113,173,56),(7,93,114,45,160,57,179,79,201,26,174,101,142),(8,58,175,115,80,143,145,27,94,180,102,46,202),(9,144,103,176,28,203,65,181,59,146,47,116,95),(10,204,48,104,182,96,29,147,129,66,117,161,60),(11,81,118,33,148,61,183,67,205,30,162,105,130),(12,62,163,119,68,131,149,31,82,184,106,34,206),(13,132,107,164,32,207,69,185,63,150,35,120,83),(14,208,36,108,186,84,17,151,133,70,121,165,64),(15,85,122,37,152,49,187,71,193,18,166,109,134),(16,50,167,123,72,135,153,19,86,188,110,38,194)], [(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)])

C13⋊C16 is a maximal subgroup of   D13⋊C16  D26.C8  C52.C8
C13⋊C16 is a maximal quotient of   C13⋊C32

Matrix representation of C13⋊C16 in GL4(𝔽5) generated by

2032
3123
1023
4424
,
0210
3022
4000
1020
G:=sub<GL(4,GF(5))| [2,3,1,4,0,1,0,4,3,2,2,2,2,3,3,4],[0,3,4,1,2,0,0,0,1,2,0,2,0,2,0,0] >;

C13⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C13:C16");
// GroupNames label

G:=SmallGroup(208,3);
// by ID

G=gap.SmallGroup(208,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,10,26,42,3204,2409]);
// Polycyclic

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

Export

Subgroup lattice of C13⋊C16 in TeX
Character table of C13⋊C16 in TeX

׿
×
𝔽