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G = C4⋊2D52order 416 = 25·13

The semidirect product of C4 and D52 acting via D52/D26=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C4⋊2D52
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — C4⋊2D52
 Lower central C13 — C2×C26 — C4⋊2D52
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C42D52
G = < a,b,c | a52=b4=c2=1, bab-1=a27, cac=a-1, cbc=b-1 >

Subgroups: 856 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C4⋊D4, Dic13, C52, C52, D26, D26, C2×C26, C4×D13, D52, C2×Dic13, C2×C52, C2×C52, C22×D13, C22×D13, D26⋊C4, C13×C4⋊C4, C2×C4×D13, C2×D52, C2×D52, C42D52
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4⋊D4, D26, D52, C22×D13, C2×D52, D4×D13, D52⋊C2, C42D52

Smallest permutation representation of C42D52
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)
(1 183 100 155)(2 158 101 130)(3 185 102 105)(4 160 103 132)(5 187 104 107)(6 162 53 134)(7 189 54 109)(8 164 55 136)(9 191 56 111)(10 166 57 138)(11 193 58 113)(12 168 59 140)(13 195 60 115)(14 170 61 142)(15 197 62 117)(16 172 63 144)(17 199 64 119)(18 174 65 146)(19 201 66 121)(20 176 67 148)(21 203 68 123)(22 178 69 150)(23 205 70 125)(24 180 71 152)(25 207 72 127)(26 182 73 154)(27 157 74 129)(28 184 75 156)(29 159 76 131)(30 186 77 106)(31 161 78 133)(32 188 79 108)(33 163 80 135)(34 190 81 110)(35 165 82 137)(36 192 83 112)(37 167 84 139)(38 194 85 114)(39 169 86 141)(40 196 87 116)(41 171 88 143)(42 198 89 118)(43 173 90 145)(44 200 91 120)(45 175 92 147)(46 202 93 122)(47 177 94 149)(48 204 95 124)(49 179 96 151)(50 206 97 126)(51 181 98 153)(52 208 99 128)
(1 155)(2 154)(3 153)(4 152)(5 151)(6 150)(7 149)(8 148)(9 147)(10 146)(11 145)(12 144)(13 143)(14 142)(15 141)(16 140)(17 139)(18 138)(19 137)(20 136)(21 135)(22 134)(23 133)(24 132)(25 131)(26 130)(27 129)(28 128)(29 127)(30 126)(31 125)(32 124)(33 123)(34 122)(35 121)(36 120)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 156)(53 178)(54 177)(55 176)(56 175)(57 174)(58 173)(59 172)(60 171)(61 170)(62 169)(63 168)(64 167)(65 166)(66 165)(67 164)(68 163)(69 162)(70 161)(71 160)(72 159)(73 158)(74 157)(75 208)(76 207)(77 206)(78 205)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 195)(89 194)(90 193)(91 192)(92 191)(93 190)(94 189)(95 188)(96 187)(97 186)(98 185)(99 184)(100 183)(101 182)(102 181)(103 180)(104 179)```

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 C4○D4 D13 D26 D52 D4×D13 D52⋊C2 kernel C4⋊2D52 D26⋊C4 C13×C4⋊C4 C2×C4×D13 C2×D52 C52 D26 C26 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 2 2 2 6 18 24 6 6

Matrix representation of C42D52 in GL4(𝔽53) generated by

 48 44 0 0 24 43 0 0 0 0 30 0 0 0 29 23
,
 3 21 0 0 50 50 0 0 0 0 37 13 0 0 21 16
,
 2 8 0 0 46 51 0 0 0 0 37 13 0 0 13 16
`G:=sub<GL(4,GF(53))| [48,24,0,0,44,43,0,0,0,0,30,29,0,0,0,23],[3,50,0,0,21,50,0,0,0,0,37,21,0,0,13,16],[2,46,0,0,8,51,0,0,0,0,37,13,0,0,13,16] >;`

C42D52 in GAP, Magma, Sage, TeX

`C_4\rtimes_2D_{52}`
`% in TeX`

`G:=Group("C4:2D52");`
`// GroupNames label`

`G:=SmallGroup(416,116);`
`// by ID`

`G=gap.SmallGroup(416,116);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,50,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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