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## G = C42⋊D13order 416 = 25·13

### 1st semidirect product of C42 and D13 acting via D13/C13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C42⋊D13
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — C42⋊D13
 Lower central C13 — C26 — C42⋊D13
 Upper central C1 — C2×C4 — C42

Generators and relations for C42⋊D13
G = < a,b,c,d | a4=b4=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 464 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C13, C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, D13 [×2], C26, C26 [×2], C42⋊C2, Dic13 [×2], Dic13 [×2], C52 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, C4×D13 [×4], C2×Dic13, C2×Dic13 [×2], C2×C52, C2×C52 [×2], C22×D13, C4×Dic13, C26.D4 [×2], D26⋊C4 [×2], C4×C52, C2×C4×D13, C42⋊D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], D13, C42⋊C2, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D525C2 [×2], C42⋊D13

Smallest permutation representation of C42⋊D13
On 208 points
Generators in S208
```(1 119 17 111)(2 120 18 112)(3 121 19 113)(4 122 20 114)(5 123 21 115)(6 124 22 116)(7 125 23 117)(8 126 24 105)(9 127 25 106)(10 128 26 107)(11 129 14 108)(12 130 15 109)(13 118 16 110)(27 147 42 133)(28 148 43 134)(29 149 44 135)(30 150 45 136)(31 151 46 137)(32 152 47 138)(33 153 48 139)(34 154 49 140)(35 155 50 141)(36 156 51 142)(37 144 52 143)(38 145 40 131)(39 146 41 132)(53 175 67 160)(54 176 68 161)(55 177 69 162)(56 178 70 163)(57 179 71 164)(58 180 72 165)(59 181 73 166)(60 182 74 167)(61 170 75 168)(62 171 76 169)(63 172 77 157)(64 173 78 158)(65 174 66 159)(79 198 97 195)(80 199 98 183)(81 200 99 184)(82 201 100 185)(83 202 101 186)(84 203 102 187)(85 204 103 188)(86 205 104 189)(87 206 92 190)(88 207 93 191)(89 208 94 192)(90 196 95 193)(91 197 96 194)
(1 84 28 57)(2 85 29 58)(3 86 30 59)(4 87 31 60)(5 88 32 61)(6 89 33 62)(7 90 34 63)(8 91 35 64)(9 79 36 65)(10 80 37 53)(11 81 38 54)(12 82 39 55)(13 83 27 56)(14 99 40 68)(15 100 41 69)(16 101 42 70)(17 102 43 71)(18 103 44 72)(19 104 45 73)(20 92 46 74)(21 93 47 75)(22 94 48 76)(23 95 49 77)(24 96 50 78)(25 97 51 66)(26 98 52 67)(105 194 141 158)(106 195 142 159)(107 183 143 160)(108 184 131 161)(109 185 132 162)(110 186 133 163)(111 187 134 164)(112 188 135 165)(113 189 136 166)(114 190 137 167)(115 191 138 168)(116 192 139 169)(117 193 140 157)(118 202 147 178)(119 203 148 179)(120 204 149 180)(121 205 150 181)(122 206 151 182)(123 207 152 170)(124 208 153 171)(125 196 154 172)(126 197 155 173)(127 198 156 174)(128 199 144 175)(129 200 145 176)(130 201 146 177)
(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 27)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(53 92)(54 104)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 91)(77 90)(78 89)(105 139)(106 138)(107 137)(108 136)(109 135)(110 134)(111 133)(112 132)(113 131)(114 143)(115 142)(116 141)(117 140)(118 148)(119 147)(120 146)(121 145)(122 144)(123 156)(124 155)(125 154)(126 153)(127 152)(128 151)(129 150)(130 149)(157 196)(158 208)(159 207)(160 206)(161 205)(162 204)(163 203)(164 202)(165 201)(166 200)(167 199)(168 198)(169 197)(170 195)(171 194)(172 193)(173 192)(174 191)(175 190)(176 189)(177 188)(178 187)(179 186)(180 185)(181 184)(182 183)```

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

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

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

116 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 13A ··· 13F 26A ··· 26R 52A ··· 52BT order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 26 26 1 1 1 1 2 2 2 2 26 ··· 26 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4○D4 D13 D26 C4×D13 D52⋊5C2 kernel C42⋊D13 C4×Dic13 C26.D4 D26⋊C4 C4×C52 C2×C4×D13 C4×D13 C26 C42 C2×C4 C4 C2 # reps 1 1 2 2 1 1 8 4 6 18 24 48

Matrix representation of C42⋊D13 in GL5(𝔽53)

 1 0 0 0 0 0 52 0 0 0 0 0 52 0 0 0 0 0 30 0 0 0 0 0 30
,
 23 0 0 0 0 0 52 0 0 0 0 0 52 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 52 1 0 0 0 25 27 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 52 0 0 0 0 25 1 0 0 0 0 0 1 0 0 0 0 0 52

`G:=sub<GL(5,GF(53))| [1,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,30,0,0,0,0,0,30],[23,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,52,25,0,0,0,1,27,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,52,25,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,52] >;`

C42⋊D13 in GAP, Magma, Sage, TeX

`C_4^2\rtimes D_{13}`
`% in TeX`

`G:=Group("C4^2:D13");`
`// GroupNames label`

`G:=SmallGroup(416,93);`
`// by ID`

`G=gap.SmallGroup(416,93);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,50,13829]);`
`// Polycyclic`

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

׿
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