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## G = Q16×D13order 416 = 25·13

### Direct product of Q16 and D13

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, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 400 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×2], C4, C4 [×5], C22, C8, C8, C2×C4 [×3], Q8 [×2], Q8 [×4], C13, C2×C8, Q16, Q16 [×3], C2×Q8 [×2], D13 [×2], C26, C2×Q16, Dic13, Dic13 [×2], C52, C52 [×2], D26, C132C8, C104, Dic26 [×2], Dic26 [×2], C4×D13, C4×D13 [×2], Q8×C13 [×2], C8×D13, Dic52, C13⋊Q16 [×2], C13×Q16, Q8×D13 [×2], Q16×D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, Q16 [×2], C2×D4, D13, C2×Q16, D26 [×3], C22×D13, D4×D13, Q16×D13

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 Q16 D13 D26 D26 D4×D13 Q16×D13 kernel Q16×D13 C8×D13 Dic52 C13⋊Q16 C13×Q16 Q8×D13 Dic13 D26 D13 Q16 C8 Q8 C2 C1 # reps 1 1 1 2 1 2 1 1 4 6 6 12 6 12

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

 1 0 0 0 0 1 0 0 0 0 0 179 0 0 306 120
,
 1 0 0 0 0 1 0 0 0 0 154 284 0 0 235 159
,
 198 1 0 0 312 0 0 0 0 0 1 0 0 0 0 1
,
 91 235 0 0 58 222 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,0,306,0,0,179,120],[1,0,0,0,0,1,0,0,0,0,154,235,0,0,284,159],[198,312,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[91,58,0,0,235,222,0,0,0,0,1,0,0,0,0,1] >;

Q16×D13 in GAP, Magma, Sage, TeX

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

G:=Group("Q16xD13");
// GroupNames label

G:=SmallGroup(416,138);
// by ID

G=gap.SmallGroup(416,138);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,116,86,297,159,69,13829]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
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