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## G = Q16⋊D13order 416 = 25·13

### 2nd semidirect product of Q16 and D13 acting via D13/C13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Q16⋊D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — Q8×D13 — Q16⋊D13
 Lower central C13 — C26 — C52 — Q16⋊D13
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊D13
G = < a,b,c,d | a8=c13=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 456 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, Q8, C13, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, D13, C26, C8.C22, Dic13, Dic13, C52, C52, D26, D26, C132C8, C104, Dic26, Dic26, C4×D13, C4×D13, D52, D52, Q8×C13, C8⋊D13, C104⋊C2, Q8⋊D13, C13⋊Q16, C13×Q16, Q8×D13, D52⋊C2, Q16⋊D13
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8.C22, D26, C22×D13, D4×D13, Q16⋊D13

Smallest permutation representation of Q16⋊D13
On 208 points
Generators in S208
```(1 193 34 177 24 205 46 163)(2 194 35 178 25 206 47 164)(3 195 36 179 26 207 48 165)(4 183 37 180 14 208 49 166)(5 184 38 181 15 196 50 167)(6 185 39 182 16 197 51 168)(7 186 27 170 17 198 52 169)(8 187 28 171 18 199 40 157)(9 188 29 172 19 200 41 158)(10 189 30 173 20 201 42 159)(11 190 31 174 21 202 43 160)(12 191 32 175 22 203 44 161)(13 192 33 176 23 204 45 162)(53 140 101 116 76 152 91 124)(54 141 102 117 77 153 79 125)(55 142 103 105 78 154 80 126)(56 143 104 106 66 155 81 127)(57 131 92 107 67 156 82 128)(58 132 93 108 68 144 83 129)(59 133 94 109 69 145 84 130)(60 134 95 110 70 146 85 118)(61 135 96 111 71 147 86 119)(62 136 97 112 72 148 87 120)(63 137 98 113 73 149 88 121)(64 138 99 114 74 150 89 122)(65 139 100 115 75 151 90 123)
(1 69 24 59)(2 70 25 60)(3 71 26 61)(4 72 14 62)(5 73 15 63)(6 74 16 64)(7 75 17 65)(8 76 18 53)(9 77 19 54)(10 78 20 55)(11 66 21 56)(12 67 22 57)(13 68 23 58)(27 100 52 90)(28 101 40 91)(29 102 41 79)(30 103 42 80)(31 104 43 81)(32 92 44 82)(33 93 45 83)(34 94 46 84)(35 95 47 85)(36 96 48 86)(37 97 49 87)(38 98 50 88)(39 99 51 89)(105 201 126 189)(106 202 127 190)(107 203 128 191)(108 204 129 192)(109 205 130 193)(110 206 118 194)(111 207 119 195)(112 208 120 183)(113 196 121 184)(114 197 122 185)(115 198 123 186)(116 199 124 187)(117 200 125 188)(131 161 156 175)(132 162 144 176)(133 163 145 177)(134 164 146 178)(135 165 147 179)(136 166 148 180)(137 167 149 181)(138 168 150 182)(139 169 151 170)(140 157 152 171)(141 158 153 172)(142 159 154 173)(143 160 155 174)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 20)(15 19)(16 18)(21 26)(22 25)(23 24)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 78)(63 77)(64 76)(65 75)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(105 112)(106 111)(107 110)(108 109)(113 117)(114 116)(118 128)(119 127)(120 126)(121 125)(122 124)(129 130)(131 134)(132 133)(135 143)(136 142)(137 141)(138 140)(144 145)(146 156)(147 155)(148 154)(149 153)(150 152)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(183 201)(184 200)(185 199)(186 198)(187 197)(188 196)(189 208)(190 207)(191 206)(192 205)(193 204)(194 203)(195 202)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 13A ··· 13F 26A ··· 26F 52A ··· 52F 52G ··· 52R 104A ··· 104L order 1 2 2 2 4 4 4 4 4 8 8 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 26 52 2 4 4 26 52 4 52 2 ··· 2 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 C8.C22 D4×D13 Q16⋊D13 kernel Q16⋊D13 C8⋊D13 C104⋊C2 Q8⋊D13 C13⋊Q16 C13×Q16 Q8×D13 D52⋊C2 Dic13 D26 Q16 C8 Q8 C13 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 6 6 12 1 6 12

Matrix representation of Q16⋊D13 in GL4(𝔽313) generated by

 0 0 86 45 0 0 81 83 255 220 225 117 83 136 196 88
,
 213 168 244 84 109 100 26 301 140 41 14 145 199 179 168 299
,
 115 1 0 0 200 146 0 0 0 0 0 1 0 0 312 261
,
 146 312 0 0 31 167 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(313))| [0,0,255,83,0,0,220,136,86,81,225,196,45,83,117,88],[213,109,140,199,168,100,41,179,244,26,14,168,84,301,145,299],[115,200,0,0,1,146,0,0,0,0,0,312,0,0,1,261],[146,31,0,0,312,167,0,0,0,0,0,1,0,0,1,0] >;`

Q16⋊D13 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_{13}`
`% in TeX`

`G:=Group("Q16:D13");`
`// GroupNames label`

`G:=SmallGroup(416,139);`
`// by ID`

`G=gap.SmallGroup(416,139);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,116,86,297,159,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^13=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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