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G = C52.C23order 416 = 25·13

18th non-split extension by C52 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52.C23
 Chief series C1 — C13 — C26 — C52 — D52 — D52⋊5C2 — C52.C23
 Lower central C13 — C26 — C52 — C52.C23
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for C52.C23
G = < a,b,c,d | a52=b2=d2=1, c2=a26, bab=a-1, ac=ca, dad=a27, bc=cb, dbd=a13b, cd=dc >

Subgroups: 368 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C13, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, D13, C26, C26, C4○D8, Dic13, C52, C52, D26, C2×C26, C2×C26, C132C8, Dic26, C4×D13, D52, C13⋊D4, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C2×C132C8, D4⋊D13, D4.D13, Q8⋊D13, C13⋊Q16, D525C2, C13×C4○D4, C52.C23
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4○D8, D26, C13⋊D4, C22×D13, C2×C13⋊D4, C52.C23

Smallest permutation representation of C52.C23
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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 156)(13 155)(14 154)(15 153)(16 152)(17 151)(18 150)(19 149)(20 148)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 140)(29 139)(30 138)(31 137)(32 136)(33 135)(34 134)(35 133)(36 132)(37 131)(38 130)(39 129)(40 128)(41 127)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 193)(54 192)(55 191)(56 190)(57 189)(58 188)(59 187)(60 186)(61 185)(62 184)(63 183)(64 182)(65 181)(66 180)(67 179)(68 178)(69 177)(70 176)(71 175)(72 174)(73 173)(74 172)(75 171)(76 170)(77 169)(78 168)(79 167)(80 166)(81 165)(82 164)(83 163)(84 162)(85 161)(86 160)(87 159)(88 158)(89 157)(90 208)(91 207)(92 206)(93 205)(94 204)(95 203)(96 202)(97 201)(98 200)(99 199)(100 198)(101 197)(102 196)(103 195)(104 194)
(1 40 27 14)(2 41 28 15)(3 42 29 16)(4 43 30 17)(5 44 31 18)(6 45 32 19)(7 46 33 20)(8 47 34 21)(9 48 35 22)(10 49 36 23)(11 50 37 24)(12 51 38 25)(13 52 39 26)(53 66 79 92)(54 67 80 93)(55 68 81 94)(56 69 82 95)(57 70 83 96)(58 71 84 97)(59 72 85 98)(60 73 86 99)(61 74 87 100)(62 75 88 101)(63 76 89 102)(64 77 90 103)(65 78 91 104)(105 118 131 144)(106 119 132 145)(107 120 133 146)(108 121 134 147)(109 122 135 148)(110 123 136 149)(111 124 137 150)(112 125 138 151)(113 126 139 152)(114 127 140 153)(115 128 141 154)(116 129 142 155)(117 130 143 156)(157 196 183 170)(158 197 184 171)(159 198 185 172)(160 199 186 173)(161 200 187 174)(162 201 188 175)(163 202 189 176)(164 203 190 177)(165 204 191 178)(166 205 192 179)(167 206 193 180)(168 207 194 181)(169 208 195 182)
(1 65)(2 92)(3 67)(4 94)(5 69)(6 96)(7 71)(8 98)(9 73)(10 100)(11 75)(12 102)(13 77)(14 104)(15 79)(16 54)(17 81)(18 56)(19 83)(20 58)(21 85)(22 60)(23 87)(24 62)(25 89)(26 64)(27 91)(28 66)(29 93)(30 68)(31 95)(32 70)(33 97)(34 72)(35 99)(36 74)(37 101)(38 76)(39 103)(40 78)(41 53)(42 80)(43 55)(44 82)(45 57)(46 84)(47 59)(48 86)(49 61)(50 88)(51 63)(52 90)(105 158)(106 185)(107 160)(108 187)(109 162)(110 189)(111 164)(112 191)(113 166)(114 193)(115 168)(116 195)(117 170)(118 197)(119 172)(120 199)(121 174)(122 201)(123 176)(124 203)(125 178)(126 205)(127 180)(128 207)(129 182)(130 157)(131 184)(132 159)(133 186)(134 161)(135 188)(136 163)(137 190)(138 165)(139 192)(140 167)(141 194)(142 169)(143 196)(144 171)(145 198)(146 173)(147 200)(148 175)(149 202)(150 177)(151 204)(152 179)(153 206)(154 181)(155 208)(156 183)```

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 13A ··· 13F 26A ··· 26F 26G ··· 26X 52A ··· 52L 52M ··· 52AD order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 2 4 52 1 1 2 4 52 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D13 C4○D8 D26 D26 D26 C13⋊D4 C13⋊D4 C52.C23 kernel C52.C23 C2×C13⋊2C8 D4⋊D13 D4.D13 Q8⋊D13 C13⋊Q16 D52⋊5C2 C13×C4○D4 C52 C2×C26 C4○D4 C13 C2×C4 D4 Q8 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 6 4 6 6 6 12 12 12

Matrix representation of C52.C23 in GL4(𝔽313) generated by

 151 288 0 0 134 239 0 0 0 0 25 0 0 0 18 288
,
 146 165 0 0 70 167 0 0 0 0 5 160 0 0 172 308
,
 1 0 0 0 0 1 0 0 0 0 288 0 0 0 0 288
,
 1 0 0 0 0 1 0 0 0 0 312 281 0 0 0 1
`G:=sub<GL(4,GF(313))| [151,134,0,0,288,239,0,0,0,0,25,18,0,0,0,288],[146,70,0,0,165,167,0,0,0,0,5,172,0,0,160,308],[1,0,0,0,0,1,0,0,0,0,288,0,0,0,0,288],[1,0,0,0,0,1,0,0,0,0,312,0,0,0,281,1] >;`

C52.C23 in GAP, Magma, Sage, TeX

`C_{52}.C_2^3`
`% in TeX`

`G:=Group("C52.C2^3");`
`// GroupNames label`

`G:=SmallGroup(416,171);`
`// by ID`

`G=gap.SmallGroup(416,171);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,579,159,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^52=b^2=d^2=1,c^2=a^26,b*a*b=a^-1,a*c=c*a,d*a*d=a^27,b*c=c*b,d*b*d=a^13*b,c*d=d*c>;`
`// generators/relations`

׿
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