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## G = C52⋊2D4order 416 = 25·13

### 2nd semidirect product of C52 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52⋊2D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — C52⋊2D4
 Lower central C13 — C2×C26 — C52⋊2D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C522D4
G = < a,b,c | a52=b4=c2=1, bab-1=a-1, cac=a25, cbc=b-1 >

Subgroups: 616 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], D13 [×2], C26 [×3], C26 [×2], C4⋊D4, Dic13 [×3], C52 [×2], D26 [×2], D26 [×2], C2×C26, C2×C26 [×6], C4×D13 [×2], C2×Dic13, C2×Dic13 [×2], C13⋊D4 [×4], C2×C52, D4×C13 [×2], C22×D13, C22×C26 [×2], C523C4, C23.D13 [×2], C2×C4×D13, C2×C13⋊D4 [×2], D4×C26, C522D4
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, C522D4

Smallest permutation representation of C522D4
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)
(1 145 76 188)(2 144 77 187)(3 143 78 186)(4 142 79 185)(5 141 80 184)(6 140 81 183)(7 139 82 182)(8 138 83 181)(9 137 84 180)(10 136 85 179)(11 135 86 178)(12 134 87 177)(13 133 88 176)(14 132 89 175)(15 131 90 174)(16 130 91 173)(17 129 92 172)(18 128 93 171)(19 127 94 170)(20 126 95 169)(21 125 96 168)(22 124 97 167)(23 123 98 166)(24 122 99 165)(25 121 100 164)(26 120 101 163)(27 119 102 162)(28 118 103 161)(29 117 104 160)(30 116 53 159)(31 115 54 158)(32 114 55 157)(33 113 56 208)(34 112 57 207)(35 111 58 206)(36 110 59 205)(37 109 60 204)(38 108 61 203)(39 107 62 202)(40 106 63 201)(41 105 64 200)(42 156 65 199)(43 155 66 198)(44 154 67 197)(45 153 68 196)(46 152 69 195)(47 151 70 194)(48 150 71 193)(49 149 72 192)(50 148 73 191)(51 147 74 190)(52 146 75 189)
(2 26)(3 51)(4 24)(5 49)(6 22)(7 47)(8 20)(9 45)(10 18)(11 43)(12 16)(13 41)(15 39)(17 37)(19 35)(21 33)(23 31)(25 29)(28 52)(30 50)(32 48)(34 46)(36 44)(38 42)(53 73)(54 98)(55 71)(56 96)(57 69)(58 94)(59 67)(60 92)(61 65)(62 90)(64 88)(66 86)(68 84)(70 82)(72 80)(74 78)(75 103)(77 101)(79 99)(81 97)(83 95)(85 93)(87 91)(100 104)(105 176)(106 201)(107 174)(108 199)(109 172)(110 197)(111 170)(112 195)(113 168)(114 193)(115 166)(116 191)(117 164)(118 189)(119 162)(120 187)(121 160)(122 185)(123 158)(124 183)(125 208)(126 181)(127 206)(128 179)(129 204)(130 177)(131 202)(132 175)(133 200)(134 173)(135 198)(136 171)(137 196)(138 169)(139 194)(140 167)(141 192)(142 165)(143 190)(144 163)(145 188)(146 161)(147 186)(148 159)(149 184)(150 157)(151 182)(152 207)(153 180)(154 205)(155 178)(156 203)```

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

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

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

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 4 4 26 26 2 2 26 26 52 52 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 C4○D4 D13 D26 D26 C13⋊D4 D4×D13 D4⋊2D13 kernel C52⋊2D4 C52⋊3C4 C23.D13 C2×C4×D13 C2×C13⋊D4 D4×C26 C52 D26 C26 C2×D4 C2×C4 C23 C4 C2 C2 # reps 1 1 2 1 2 1 2 2 2 6 6 12 24 6 6

Matrix representation of C522D4 in GL6(𝔽53)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 39 39 0 0 0 0 6 40 0 0 0 0 0 0 29 45 0 0 0 0 39 24
,
 39 46 0 0 0 0 13 14 0 0 0 0 0 0 45 11 0 0 0 0 28 8 0 0 0 0 0 0 41 17 0 0 0 0 29 12
,
 1 0 0 0 0 0 49 52 0 0 0 0 0 0 8 42 0 0 0 0 25 45 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(53))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,39,6,0,0,0,0,39,40,0,0,0,0,0,0,29,39,0,0,0,0,45,24],[39,13,0,0,0,0,46,14,0,0,0,0,0,0,45,28,0,0,0,0,11,8,0,0,0,0,0,0,41,29,0,0,0,0,17,12],[1,49,0,0,0,0,0,52,0,0,0,0,0,0,8,25,0,0,0,0,42,45,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C522D4 in GAP, Magma, Sage, TeX

`C_{52}\rtimes_2D_4`
`% in TeX`

`G:=Group("C52:2D4");`
`// GroupNames label`

`G:=SmallGroup(416,159);`
`// by ID`

`G=gap.SmallGroup(416,159);`
`# by ID`

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

`G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^25,c*b*c=b^-1>;`
`// generators/relations`

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