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## G = D4⋊Dic13order 416 = 25·13

### 1st semidirect product of D4 and Dic13 acting via Dic13/C26=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D4⋊Dic13
 Chief series C1 — C13 — C26 — C2×C26 — C2×C52 — C52⋊3C4 — D4⋊Dic13
 Lower central C13 — C26 — C52 — D4⋊Dic13
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for D4⋊Dic13
G = < a,b,c,d | a4=b2=c26=1, d2=c13, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4⋊Dic13
On 208 points
Generators in S208
(1 205 81 117)(2 206 82 118)(3 207 83 119)(4 208 84 120)(5 183 85 121)(6 184 86 122)(7 185 87 123)(8 186 88 124)(9 187 89 125)(10 188 90 126)(11 189 91 127)(12 190 92 128)(13 191 93 129)(14 192 94 130)(15 193 95 105)(16 194 96 106)(17 195 97 107)(18 196 98 108)(19 197 99 109)(20 198 100 110)(21 199 101 111)(22 200 102 112)(23 201 103 113)(24 202 104 114)(25 203 79 115)(26 204 80 116)(27 64 156 167)(28 65 131 168)(29 66 132 169)(30 67 133 170)(31 68 134 171)(32 69 135 172)(33 70 136 173)(34 71 137 174)(35 72 138 175)(36 73 139 176)(37 74 140 177)(38 75 141 178)(39 76 142 179)(40 77 143 180)(41 78 144 181)(42 53 145 182)(43 54 146 157)(44 55 147 158)(45 56 148 159)(46 57 149 160)(47 58 150 161)(48 59 151 162)(49 60 152 163)(50 61 153 164)(51 62 154 165)(52 63 155 166)
(1 117)(2 118)(3 119)(4 120)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(53 182)(54 157)(55 158)(56 159)(57 160)(58 161)(59 162)(60 163)(61 164)(62 165)(63 166)(64 167)(65 168)(66 169)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 176)(74 177)(75 178)(76 179)(77 180)(78 181)(79 203)(80 204)(81 205)(82 206)(83 207)(84 208)(85 183)(86 184)(87 185)(88 186)(89 187)(90 188)(91 189)(92 190)(93 191)(94 192)(95 193)(96 194)(97 195)(98 196)(99 197)(100 198)(101 199)(102 200)(103 201)(104 202)
(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 62 14 75)(2 61 15 74)(3 60 16 73)(4 59 17 72)(5 58 18 71)(6 57 19 70)(7 56 20 69)(8 55 21 68)(9 54 22 67)(10 53 23 66)(11 78 24 65)(12 77 25 64)(13 76 26 63)(27 190 40 203)(28 189 41 202)(29 188 42 201)(30 187 43 200)(31 186 44 199)(32 185 45 198)(33 184 46 197)(34 183 47 196)(35 208 48 195)(36 207 49 194)(37 206 50 193)(38 205 51 192)(39 204 52 191)(79 167 92 180)(80 166 93 179)(81 165 94 178)(82 164 95 177)(83 163 96 176)(84 162 97 175)(85 161 98 174)(86 160 99 173)(87 159 100 172)(88 158 101 171)(89 157 102 170)(90 182 103 169)(91 181 104 168)(105 140 118 153)(106 139 119 152)(107 138 120 151)(108 137 121 150)(109 136 122 149)(110 135 123 148)(111 134 124 147)(112 133 125 146)(113 132 126 145)(114 131 127 144)(115 156 128 143)(116 155 129 142)(117 154 130 141)

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C4 D4 D4 D8 SD16 D13 D26 Dic13 C13⋊D4 C13⋊D4 D4⋊D13 D4.D13 kernel D4⋊Dic13 C2×C13⋊2C8 C52⋊3C4 D4×C26 D4×C13 C52 C2×C26 C26 C26 C2×D4 C2×C4 D4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 6 6 12 12 12 6 6

Matrix representation of D4⋊Dic13 in GL4(𝔽313) generated by

 1 0 0 0 0 1 0 0 0 0 57 44 0 0 168 256
,
 1 0 0 0 0 1 0 0 0 0 57 44 0 0 40 256
,
 201 312 0 0 208 27 0 0 0 0 312 0 0 0 0 312
,
 3 117 0 0 131 310 0 0 0 0 197 43 0 0 51 116
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,57,168,0,0,44,256],[1,0,0,0,0,1,0,0,0,0,57,40,0,0,44,256],[201,208,0,0,312,27,0,0,0,0,312,0,0,0,0,312],[3,131,0,0,117,310,0,0,0,0,197,51,0,0,43,116] >;

D4⋊Dic13 in GAP, Magma, Sage, TeX

D_4\rtimes {\rm Dic}_{13}
% in TeX

G:=Group("D4:Dic13");
// GroupNames label

G:=SmallGroup(416,39);
// by ID

G=gap.SmallGroup(416,39);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,579,297,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^26=1,d^2=c^13,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

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