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G = C13×SD32order 416 = 25·13

Direct product of C13 and SD32

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

Aliases: C13×SD32, D8.C26, C162C26, C2086C2, Q161C26, C26.16D8, C52.37D4, C104.25C22, C8.3(C2×C26), C4.2(D4×C13), C2.4(C13×D8), (C13×Q16)⋊5C2, (C13×D8).2C2, SmallGroup(416,62)

Series: Derived Chief Lower central Upper central

C1C8 — C13×SD32
C1C2C4C8C104C13×Q16 — C13×SD32
C1C2C4C8 — C13×SD32
C1C26C52C104 — C13×SD32

Generators and relations for C13×SD32
 G = < a,b,c | a13=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

8C2
4C22
4C4
8C26
2D4
2Q8
4C52
4C2×C26
2Q8×C13
2D4×C13

Smallest permutation representation of C13×SD32
On 208 points
Generators in S208
(1 105 131 151 180 17 42 126 92 62 200 72 171)(2 106 132 152 181 18 43 127 93 63 201 73 172)(3 107 133 153 182 19 44 128 94 64 202 74 173)(4 108 134 154 183 20 45 113 95 49 203 75 174)(5 109 135 155 184 21 46 114 96 50 204 76 175)(6 110 136 156 185 22 47 115 81 51 205 77 176)(7 111 137 157 186 23 48 116 82 52 206 78 161)(8 112 138 158 187 24 33 117 83 53 207 79 162)(9 97 139 159 188 25 34 118 84 54 208 80 163)(10 98 140 160 189 26 35 119 85 55 193 65 164)(11 99 141 145 190 27 36 120 86 56 194 66 165)(12 100 142 146 191 28 37 121 87 57 195 67 166)(13 101 143 147 192 29 38 122 88 58 196 68 167)(14 102 144 148 177 30 39 123 89 59 197 69 168)(15 103 129 149 178 31 40 124 90 60 198 70 169)(16 104 130 150 179 32 41 125 91 61 199 71 170)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 43)(35 41)(36 48)(37 39)(38 46)(40 44)(45 47)(49 51)(50 58)(52 56)(53 63)(55 61)(57 59)(60 64)(65 71)(66 78)(67 69)(68 76)(70 74)(73 79)(75 77)(81 95)(82 86)(83 93)(85 91)(87 89)(88 96)(90 94)(98 104)(99 111)(100 102)(101 109)(103 107)(106 112)(108 110)(113 115)(114 122)(116 120)(117 127)(119 125)(121 123)(124 128)(129 133)(130 140)(132 138)(134 136)(135 143)(137 141)(142 144)(145 157)(146 148)(147 155)(149 153)(150 160)(152 158)(154 156)(161 165)(162 172)(164 170)(166 168)(167 175)(169 173)(174 176)(177 191)(178 182)(179 189)(181 187)(183 185)(184 192)(186 190)(193 199)(194 206)(195 197)(196 204)(198 202)(201 207)(203 205)

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

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

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

143 conjugacy classes

class 1 2A2B4A4B8A8B13A···13L16A16B16C16D26A···26L26M···26X52A···52L52M···52X104A···104X208A···208AV
order122448813···131616161626···2626···2652···5252···52104···104208···208
size11828221···122221···18···82···28···82···22···2

143 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C13C26C26C26D4D8SD32D4×C13C13×D8C13×SD32
kernelC13×SD32C208C13×D8C13×Q16SD32C16D8Q16C52C26C13C4C2C1
# reps111112121212124122448

Matrix representation of C13×SD32 in GL2(𝔽1249) generated by

2400
0240
,
2761034
215276
,
10
01248
G:=sub<GL(2,GF(1249))| [240,0,0,240],[276,215,1034,276],[1,0,0,1248] >;

C13×SD32 in GAP, Magma, Sage, TeX

C_{13}\times {\rm SD}_{32}
% in TeX

G:=Group("C13xSD32");
// GroupNames label

G:=SmallGroup(416,62);
// by ID

G=gap.SmallGroup(416,62);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,1248,649,3747,1881,165,9364,4690,88]);
// 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^7>;
// generators/relations

Export

Subgroup lattice of C13×SD32 in TeX

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