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G = C52.10D4order 416 = 25·13

10th non-split extension by C52 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52.10D4
 Chief series C1 — C13 — C26 — C52 — C2×C52 — C52.4C4 — C52.10D4
 Lower central C13 — C26 — C2×C26 — C52.10D4
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C52.10D4
G = < a,b,c | a52=1, b4=a26, c2=a13, bab-1=a-1, cac-1=a25, cbc-1=a13b3 >

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

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

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

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

71 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 2 2 2 4 4 52 52 52 52 2 ··· 2 2 ··· 2 4 ··· 4

71 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + - image C1 C2 C2 C4 D4 D13 Dic13 D26 C13⋊D4 C4.10D4 C52.10D4 kernel C52.10D4 C52.4C4 Q8×C26 C2×C52 C52 C2×Q8 C2×C4 C2×C4 C4 C13 C1 # reps 1 2 1 4 2 6 12 6 24 1 12

Matrix representation of C52.10D4 in GL4(𝔽313) generated by

 0 277 0 0 36 0 0 0 0 0 0 113 0 0 200 0
,
 0 0 55 170 0 0 170 258 143 55 0 0 55 170 0 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 312 0 0 0
`G:=sub<GL(4,GF(313))| [0,36,0,0,277,0,0,0,0,0,0,200,0,0,113,0],[0,0,143,55,0,0,55,170,55,170,0,0,170,258,0,0],[0,0,0,312,0,0,1,0,1,0,0,0,0,1,0,0] >;`

C52.10D4 in GAP, Magma, Sage, TeX

`C_{52}._{10}D_4`
`% in TeX`

`G:=Group("C52.10D4");`
`// GroupNames label`

`G:=SmallGroup(416,43);`
`// by ID`

`G=gap.SmallGroup(416,43);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,188,86,579,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^13,b*a*b^-1=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^13*b^3>;`
`// generators/relations`

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