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## G = Dic13⋊D4order 416 = 25·13

### 2nd semidirect product of Dic13 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — Dic13⋊D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C13⋊D4 — Dic13⋊D4
 Lower central C13 — C2×C26 — Dic13⋊D4
 Upper central C1 — C22 — C2×D4

Generators and relations for Dic13⋊D4
G = < a,b,c,d | a26=c4=d2=1, b2=a13, bab-1=a-1, ac=ca, ad=da, cbc-1=a13b, bd=db, dcd=c-1 >

Subgroups: 616 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D13, C26, C26, C4⋊D4, Dic13, Dic13, C52, D26, C2×C26, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C13⋊D4, C2×C52, D4×C13, C22×D13, C22×C26, C26.D4, D26⋊C4, C23.D13, C22×Dic13, C2×C13⋊D4, D4×C26, Dic13⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4⋊D4, D26, C13⋊D4, C22×D13, D4×D13, D42D13, C2×C13⋊D4, Dic13⋊D4

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

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D13 D26 D26 C13⋊D4 D4×D13 D4⋊2D13 kernel Dic13⋊D4 C26.D4 D26⋊C4 C23.D13 C22×Dic13 C2×C13⋊D4 D4×C26 Dic13 C2×C26 C26 C2×D4 C2×C4 C23 C22 C2 C2 # reps 1 1 1 1 1 2 1 2 2 2 6 6 12 24 6 6

Matrix representation of Dic13⋊D4 in GL4(𝔽53) generated by

 1 52 0 0 15 39 0 0 0 0 1 0 0 0 0 1
,
 8 6 0 0 51 45 0 0 0 0 52 0 0 0 0 52
,
 45 47 0 0 37 8 0 0 0 0 20 7 0 0 26 33
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 17 52
G:=sub<GL(4,GF(53))| [1,15,0,0,52,39,0,0,0,0,1,0,0,0,0,1],[8,51,0,0,6,45,0,0,0,0,52,0,0,0,0,52],[45,37,0,0,47,8,0,0,0,0,20,26,0,0,7,33],[1,0,0,0,0,1,0,0,0,0,1,17,0,0,0,52] >;

Dic13⋊D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,160);
// by ID

G=gap.SmallGroup(416,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,218,188,13829]);
// Polycyclic

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

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