Copied to
clipboard

## G = C13×Q16order 208 = 24·13

### Direct product of C13 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×Q16, C8.C26, Q8.C26, C104.3C2, C26.16D4, C52.19C22, C4.3(C2×C26), C2.5(D4×C13), (Q8×C13).2C2, SmallGroup(208,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×Q16
 Chief series C1 — C2 — C4 — C52 — Q8×C13 — C13×Q16
 Lower central C1 — C2 — C4 — C13×Q16
 Upper central C1 — C26 — C52 — C13×Q16

Generators and relations for C13×Q16
G = < a,b,c | a13=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C13×Q16
Regular action 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 82 151 182 134 50 98 55)(2 83 152 170 135 51 99 56)(3 84 153 171 136 52 100 57)(4 85 154 172 137 40 101 58)(5 86 155 173 138 41 102 59)(6 87 156 174 139 42 103 60)(7 88 144 175 140 43 104 61)(8 89 145 176 141 44 92 62)(9 90 146 177 142 45 93 63)(10 91 147 178 143 46 94 64)(11 79 148 179 131 47 95 65)(12 80 149 180 132 48 96 53)(13 81 150 181 133 49 97 54)(14 76 107 196 37 190 160 120)(15 77 108 197 38 191 161 121)(16 78 109 198 39 192 162 122)(17 66 110 199 27 193 163 123)(18 67 111 200 28 194 164 124)(19 68 112 201 29 195 165 125)(20 69 113 202 30 183 166 126)(21 70 114 203 31 184 167 127)(22 71 115 204 32 185 168 128)(23 72 116 205 33 186 169 129)(24 73 117 206 34 187 157 130)(25 74 105 207 35 188 158 118)(26 75 106 208 36 189 159 119)
(1 200 134 124)(2 201 135 125)(3 202 136 126)(4 203 137 127)(5 204 138 128)(6 205 139 129)(7 206 140 130)(8 207 141 118)(9 208 142 119)(10 196 143 120)(11 197 131 121)(12 198 132 122)(13 199 133 123)(14 64 37 178)(15 65 38 179)(16 53 39 180)(17 54 27 181)(18 55 28 182)(19 56 29 170)(20 57 30 171)(21 58 31 172)(22 59 32 173)(23 60 33 174)(24 61 34 175)(25 62 35 176)(26 63 36 177)(40 167 85 114)(41 168 86 115)(42 169 87 116)(43 157 88 117)(44 158 89 105)(45 159 90 106)(46 160 91 107)(47 161 79 108)(48 162 80 109)(49 163 81 110)(50 164 82 111)(51 165 83 112)(52 166 84 113)(66 97 193 150)(67 98 194 151)(68 99 195 152)(69 100 183 153)(70 101 184 154)(71 102 185 155)(72 103 186 156)(73 104 187 144)(74 92 188 145)(75 93 189 146)(76 94 190 147)(77 95 191 148)(78 96 192 149)

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

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

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

C13×Q16 is a maximal subgroup of   C8.6D26  C13⋊Q32  Q16⋊D13  D104⋊C2

91 conjugacy classes

 class 1 2 4A 4B 4C 8A 8B 13A ··· 13L 26A ··· 26L 52A ··· 52L 52M ··· 52AJ 104A ··· 104X order 1 2 4 4 4 8 8 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 4 4 2 2 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

91 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C13 C26 C26 D4 Q16 D4×C13 C13×Q16 kernel C13×Q16 C104 Q8×C13 Q16 C8 Q8 C26 C13 C2 C1 # reps 1 1 2 12 12 24 1 2 12 24

Matrix representation of C13×Q16 in GL2(𝔽313) generated by

 294 0 0 294
,
 0 120 253 120
,
 284 118 30 29
G:=sub<GL(2,GF(313))| [294,0,0,294],[0,253,120,120],[284,30,118,29] >;

C13×Q16 in GAP, Magma, Sage, TeX

C_{13}\times Q_{16}
% in TeX

G:=Group("C13xQ16");
// GroupNames label

G:=SmallGroup(208,27);
// by ID

G=gap.SmallGroup(208,27);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-2,520,541,526,3123,1568,58]);
// Polycyclic

G:=Group<a,b,c|a^13=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

׿
×
𝔽