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## G = C52.23D4order 416 = 25·13

### 23rd non-split extension by C52 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52.23D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×D52 — C52.23D4
 Lower central C13 — C2×C26 — C52.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C52.23D4
G = < a,b,c | a52=b4=c2=1, bab-1=a25, cac=a-1, cbc=a26b-1 >

Subgroups: 584 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, D52 [×2], C2×Dic13 [×2], C2×C52, C2×C52 [×2], Q8×C13 [×2], C22×D13 [×2], C4×Dic13, D26⋊C4 [×4], C2×D52, Q8×C26, C52.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], D13, C4.4D4, D26 [×3], C13⋊D4 [×2], C22×D13, D52⋊C2 [×2], C2×C13⋊D4, C52.23D4

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

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 C4○D4 D13 D26 C13⋊D4 D52⋊C2 kernel C52.23D4 C4×Dic13 D26⋊C4 C2×D52 Q8×C26 C52 C26 C2×Q8 C2×C4 C4 C2 # reps 1 1 4 1 1 2 4 6 18 24 12

Matrix representation of C52.23D4 in GL4(𝔽53) generated by

 0 52 0 0 1 0 0 0 0 0 50 12 0 0 44 18
,
 0 30 0 0 23 0 0 0 0 0 1 40 0 0 45 52
,
 0 1 0 0 1 0 0 0 0 0 18 23 0 0 9 35
`G:=sub<GL(4,GF(53))| [0,1,0,0,52,0,0,0,0,0,50,44,0,0,12,18],[0,23,0,0,30,0,0,0,0,0,1,45,0,0,40,52],[0,1,0,0,1,0,0,0,0,0,18,9,0,0,23,35] >;`

C52.23D4 in GAP, Magma, Sage, TeX

`C_{52}._{23}D_4`
`% in TeX`

`G:=Group("C52.23D4");`
`// GroupNames label`

`G:=SmallGroup(416,168);`
`// by ID`

`G=gap.SmallGroup(416,168);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,218,188,86,13829]);`
`// Polycyclic`

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

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