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G = C13⋊Q16order 208 = 24·13

The semidirect product of C13 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C132Q16, Q8.D13, C4.4D26, C26.10D4, C52.4C22, Dic26.2C2, C132C8.1C2, (Q8×C13).1C2, C2.7(C13⋊D4), SmallGroup(208,18)

Series: Derived Chief Lower central Upper central

C1C52 — C13⋊Q16
C1C13C26C52Dic26 — C13⋊Q16
C13C26C52 — C13⋊Q16
C1C2C4Q8

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

2C4
26C4
13C8
13Q8
2Dic13
2C52
13Q16

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

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

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

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

C13⋊Q16 is a maximal subgroup of   D4.D26  D26.6D4  Q16×D13  Q16⋊D13  Q8.D26  C52.C23  D4.9D26
C13⋊Q16 is a maximal quotient of   C26.D8  C26.Q16  Q8⋊Dic13

37 conjugacy classes

class 1  2 4A4B4C8A8B13A···13F26A···26F52A···52R
order124448813···1326···2652···52
size11245226262···22···24···4

37 irreducible representations

dim1111222224
type+++++-++-
imageC1C2C2C2D4Q16D13D26C13⋊D4C13⋊Q16
kernelC13⋊Q16C132C8Dic26Q8×C13C26C13Q8C4C2C1
# reps11111266126

Matrix representation of C13⋊Q16 in GL4(𝔽313) generated by

0100
31229300
0010
0001
,
649800
7024900
00061
00118120
,
312000
031200
00256270
0024357
G:=sub<GL(4,GF(313))| [0,312,0,0,1,293,0,0,0,0,1,0,0,0,0,1],[64,70,0,0,98,249,0,0,0,0,0,118,0,0,61,120],[312,0,0,0,0,312,0,0,0,0,256,243,0,0,270,57] >;

C13⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C13:Q16");
// GroupNames label

G:=SmallGroup(208,18);
// by ID

G=gap.SmallGroup(208,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,46,182,97,42,4804]);
// Polycyclic

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

Export

Subgroup lattice of C13⋊Q16 in TeX

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