Copied to
clipboard

## 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 [×3], C2 [×4], C4 [×5], C22, C22 [×2], C22 [×8], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], D13, C26 [×3], C26 [×3], C4⋊D4, Dic13 [×2], Dic13 [×2], C52, D26 [×3], C2×C26, C2×C26 [×2], C2×C26 [×5], C2×Dic13 [×3], C2×Dic13 [×2], C13⋊D4 [×4], C2×C52, D4×C13 [×2], C22×D13, C22×C26 [×2], C26.D4, D26⋊C4, C23.D13, C22×Dic13, C2×C13⋊D4 [×2], D4×C26, Dic13⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, D13, C4⋊D4, D26 [×3], C13⋊D4 [×2], 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)
(1 119 14 106)(2 118 15 105)(3 117 16 130)(4 116 17 129)(5 115 18 128)(6 114 19 127)(7 113 20 126)(8 112 21 125)(9 111 22 124)(10 110 23 123)(11 109 24 122)(12 108 25 121)(13 107 26 120)(27 160 40 173)(28 159 41 172)(29 158 42 171)(30 157 43 170)(31 182 44 169)(32 181 45 168)(33 180 46 167)(34 179 47 166)(35 178 48 165)(36 177 49 164)(37 176 50 163)(38 175 51 162)(39 174 52 161)(53 93 66 80)(54 92 67 79)(55 91 68 104)(56 90 69 103)(57 89 70 102)(58 88 71 101)(59 87 72 100)(60 86 73 99)(61 85 74 98)(62 84 75 97)(63 83 76 96)(64 82 77 95)(65 81 78 94)(131 203 144 190)(132 202 145 189)(133 201 146 188)(134 200 147 187)(135 199 148 186)(136 198 149 185)(137 197 150 184)(138 196 151 183)(139 195 152 208)(140 194 153 207)(141 193 154 206)(142 192 155 205)(143 191 156 204)
(1 77 39 144)(2 78 40 145)(3 53 41 146)(4 54 42 147)(5 55 43 148)(6 56 44 149)(7 57 45 150)(8 58 46 151)(9 59 47 152)(10 60 48 153)(11 61 49 154)(12 62 50 155)(13 63 51 156)(14 64 52 131)(15 65 27 132)(16 66 28 133)(17 67 29 134)(18 68 30 135)(19 69 31 136)(20 70 32 137)(21 71 33 138)(22 72 34 139)(23 73 35 140)(24 74 36 141)(25 75 37 142)(26 76 38 143)(79 171 200 116)(80 172 201 117)(81 173 202 118)(82 174 203 119)(83 175 204 120)(84 176 205 121)(85 177 206 122)(86 178 207 123)(87 179 208 124)(88 180 183 125)(89 181 184 126)(90 182 185 127)(91 157 186 128)(92 158 187 129)(93 159 188 130)(94 160 189 105)(95 161 190 106)(96 162 191 107)(97 163 192 108)(98 164 193 109)(99 165 194 110)(100 166 195 111)(101 167 196 112)(102 168 197 113)(103 169 198 114)(104 170 199 115)
(53 146)(54 147)(55 148)(56 149)(57 150)(58 151)(59 152)(60 153)(61 154)(62 155)(63 156)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 144)(78 145)(79 200)(80 201)(81 202)(82 203)(83 204)(84 205)(85 206)(86 207)(87 208)(88 183)(89 184)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(97 192)(98 193)(99 194)(100 195)(101 196)(102 197)(103 198)(104 199)

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

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

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

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

׿
×
𝔽