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## G = C4.D52order 416 = 25·13

### 5th non-split extension by C4 of D52 acting via D52/C52=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C4.D52
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — D26⋊C4 — C4.D52
 Lower central C13 — C2×C26 — C4.D52
 Upper central C1 — C22 — C42

Generators and relations for C4.D52
G = < a,b,c | a4=b52=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 632 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C13, C42, C22⋊C4 [×4], C2×D4, C2×Q8, D13 [×2], C26, C26 [×2], C4.4D4, Dic13 [×2], C52 [×2], C52 [×2], D26 [×6], C2×C26, Dic26 [×2], D52 [×2], C2×Dic13 [×2], C2×C52, C2×C52 [×2], C22×D13 [×2], D26⋊C4 [×4], C4×C52, C2×Dic26, C2×D52, C4.D52
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], D13, C4.4D4, D26 [×3], D52 [×2], C22×D13, C2×D52, D525C2 [×2], C4.D52

Smallest permutation representation of C4.D52
On 208 points
Generators in S208
```(1 64 127 161)(2 65 128 162)(3 66 129 163)(4 67 130 164)(5 68 131 165)(6 69 132 166)(7 70 133 167)(8 71 134 168)(9 72 135 169)(10 73 136 170)(11 74 137 171)(12 75 138 172)(13 76 139 173)(14 77 140 174)(15 78 141 175)(16 79 142 176)(17 80 143 177)(18 81 144 178)(19 82 145 179)(20 83 146 180)(21 84 147 181)(22 85 148 182)(23 86 149 183)(24 87 150 184)(25 88 151 185)(26 89 152 186)(27 90 153 187)(28 91 154 188)(29 92 155 189)(30 93 156 190)(31 94 105 191)(32 95 106 192)(33 96 107 193)(34 97 108 194)(35 98 109 195)(36 99 110 196)(37 100 111 197)(38 101 112 198)(39 102 113 199)(40 103 114 200)(41 104 115 201)(42 53 116 202)(43 54 117 203)(44 55 118 204)(45 56 119 205)(46 57 120 206)(47 58 121 207)(48 59 122 208)(49 60 123 157)(50 61 124 158)(51 62 125 159)(52 63 126 160)
(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)
(1 186 127 89)(2 88 128 185)(3 184 129 87)(4 86 130 183)(5 182 131 85)(6 84 132 181)(7 180 133 83)(8 82 134 179)(9 178 135 81)(10 80 136 177)(11 176 137 79)(12 78 138 175)(13 174 139 77)(14 76 140 173)(15 172 141 75)(16 74 142 171)(17 170 143 73)(18 72 144 169)(19 168 145 71)(20 70 146 167)(21 166 147 69)(22 68 148 165)(23 164 149 67)(24 66 150 163)(25 162 151 65)(26 64 152 161)(27 160 153 63)(28 62 154 159)(29 158 155 61)(30 60 156 157)(31 208 105 59)(32 58 106 207)(33 206 107 57)(34 56 108 205)(35 204 109 55)(36 54 110 203)(37 202 111 53)(38 104 112 201)(39 200 113 103)(40 102 114 199)(41 198 115 101)(42 100 116 197)(43 196 117 99)(44 98 118 195)(45 194 119 97)(46 96 120 193)(47 192 121 95)(48 94 122 191)(49 190 123 93)(50 92 124 189)(51 188 125 91)(52 90 126 187)```

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

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

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

110 conjugacy classes

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

110 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 C4○D4 D13 D26 D52 D52⋊5C2 kernel C4.D52 D26⋊C4 C4×C52 C2×Dic26 C2×D52 C52 C26 C42 C2×C4 C4 C2 # reps 1 4 1 1 1 2 4 6 18 24 48

Matrix representation of C4.D52 in GL4(𝔽53) generated by

 15 4 0 0 23 38 0 0 0 0 10 19 0 0 17 43
,
 27 39 0 0 52 26 0 0 0 0 11 4 0 0 51 4
,
 23 23 0 0 0 30 0 0 0 0 51 48 0 0 1 2
`G:=sub<GL(4,GF(53))| [15,23,0,0,4,38,0,0,0,0,10,17,0,0,19,43],[27,52,0,0,39,26,0,0,0,0,11,51,0,0,4,4],[23,0,0,0,23,30,0,0,0,0,51,1,0,0,48,2] >;`

C4.D52 in GAP, Magma, Sage, TeX

`C_4.D_{52}`
`% in TeX`

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

`G:=SmallGroup(416,96);`
`// by ID`

`G=gap.SmallGroup(416,96);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,55,218,86,13829]);`
`// Polycyclic`

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

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