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## G = Dic13⋊4D4order 416 = 25·13

### 1st semidirect product of Dic13 and D4 acting through Inn(Dic13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — Dic13⋊4D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C13⋊D4 — Dic13⋊4D4
 Lower central C13 — C26 — Dic13⋊4D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic134D4
G = < a,b,c,d | a26=c4=d2=1, b2=a13, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 592 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22, C22 [×2], C22 [×6], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C13, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, D13 [×2], C26 [×3], C26 [×2], C4×D4, Dic13 [×4], Dic13, C52 [×2], D26 [×2], D26 [×2], C2×C26, C2×C26 [×2], C2×C26 [×2], C4×D13 [×2], C2×Dic13 [×3], C2×Dic13 [×2], C13⋊D4 [×4], C2×C52 [×2], C22×D13, C22×C26, C4×Dic13, C26.D4, D26⋊C4, C13×C22⋊C4, C2×C4×D13, C22×Dic13, C2×C13⋊D4, Dic134D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D4×D13, D42D13, Dic134D4

Smallest permutation representation of Dic134D4
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 98 14 85)(2 97 15 84)(3 96 16 83)(4 95 17 82)(5 94 18 81)(6 93 19 80)(7 92 20 79)(8 91 21 104)(9 90 22 103)(10 89 23 102)(11 88 24 101)(12 87 25 100)(13 86 26 99)(27 179 40 166)(28 178 41 165)(29 177 42 164)(30 176 43 163)(31 175 44 162)(32 174 45 161)(33 173 46 160)(34 172 47 159)(35 171 48 158)(36 170 49 157)(37 169 50 182)(38 168 51 181)(39 167 52 180)(53 196 66 183)(54 195 67 208)(55 194 68 207)(56 193 69 206)(57 192 70 205)(58 191 71 204)(59 190 72 203)(60 189 73 202)(61 188 74 201)(62 187 75 200)(63 186 76 199)(64 185 77 198)(65 184 78 197)(105 141 118 154)(106 140 119 153)(107 139 120 152)(108 138 121 151)(109 137 122 150)(110 136 123 149)(111 135 124 148)(112 134 125 147)(113 133 126 146)(114 132 127 145)(115 131 128 144)(116 156 129 143)(117 155 130 142)
(1 40 202 152)(2 39 203 151)(3 38 204 150)(4 37 205 149)(5 36 206 148)(6 35 207 147)(7 34 208 146)(8 33 183 145)(9 32 184 144)(10 31 185 143)(11 30 186 142)(12 29 187 141)(13 28 188 140)(14 27 189 139)(15 52 190 138)(16 51 191 137)(17 50 192 136)(18 49 193 135)(19 48 194 134)(20 47 195 133)(21 46 196 132)(22 45 197 131)(23 44 198 156)(24 43 199 155)(25 42 200 154)(26 41 201 153)(53 114 91 173)(54 113 92 172)(55 112 93 171)(56 111 94 170)(57 110 95 169)(58 109 96 168)(59 108 97 167)(60 107 98 166)(61 106 99 165)(62 105 100 164)(63 130 101 163)(64 129 102 162)(65 128 103 161)(66 127 104 160)(67 126 79 159)(68 125 80 158)(69 124 81 157)(70 123 82 182)(71 122 83 181)(72 121 84 180)(73 120 85 179)(74 119 86 178)(75 118 87 177)(76 117 88 176)(77 116 89 175)(78 115 90 174)
(1 152)(2 153)(3 154)(4 155)(5 156)(6 131)(7 132)(8 133)(9 134)(10 135)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 144)(20 145)(21 146)(22 147)(23 148)(24 149)(25 150)(26 151)(27 189)(28 190)(29 191)(30 192)(31 193)(32 194)(33 195)(34 196)(35 197)(36 198)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 183)(48 184)(49 185)(50 186)(51 187)(52 188)(53 159)(54 160)(55 161)(56 162)(57 163)(58 164)(59 165)(60 166)(61 167)(62 168)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 177)(72 178)(73 179)(74 180)(75 181)(76 182)(77 157)(78 158)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 105)(97 106)(98 107)(99 108)(100 109)(101 110)(102 111)(103 112)(104 113)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 13A ··· 13F 26A ··· 26R 26S ··· 26AD 52A ··· 52X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 26 26 2 2 2 2 13 13 13 13 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 D13 D26 D26 C4×D13 D4×D13 D4⋊2D13 kernel Dic13⋊4D4 C4×Dic13 C26.D4 D26⋊C4 C13×C22⋊C4 C2×C4×D13 C22×Dic13 C2×C13⋊D4 C13⋊D4 Dic13 C26 C22⋊C4 C2×C4 C23 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 6 12 6 24 6 6

Matrix representation of Dic134D4 in GL4(𝔽53) generated by

 0 52 0 0 1 27 0 0 0 0 1 0 0 0 0 1
,
 23 0 0 0 15 30 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 26 52 0 0 0 0 21 9 0 0 51 32
,
 1 0 0 0 0 1 0 0 0 0 21 9 0 0 10 32
G:=sub<GL(4,GF(53))| [0,1,0,0,52,27,0,0,0,0,1,0,0,0,0,1],[23,15,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[1,26,0,0,0,52,0,0,0,0,21,51,0,0,9,32],[1,0,0,0,0,1,0,0,0,0,21,10,0,0,9,32] >;

Dic134D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{13}\rtimes_4D_4
% in TeX

G:=Group("Dic13:4D4");
// GroupNames label

G:=SmallGroup(416,102);
// by ID

G=gap.SmallGroup(416,102);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,188,50,13829]);
// Polycyclic

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

׿
×
𝔽