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## G = D52.2C4order 416 = 25·13

### The non-split extension by D52 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D52.2C4
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D52⋊5C2 — D52.2C4
 Lower central C13 — C26 — D52.2C4
 Upper central C1 — C4 — M4(2)

Generators and relations for D52.2C4
G = < a,b,c | a52=b2=1, c4=a26, bab=a-1, cac-1=a27, cbc-1=a26b >

Subgroups: 344 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C13, C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, D13 [×2], C26, C26, C8○D4, Dic13 [×2], C52 [×2], D26 [×2], C2×C26, C132C8 [×2], C104 [×2], Dic26, C4×D13 [×2], D52, C13⋊D4 [×2], C2×C52, C8×D13 [×2], C8⋊D13 [×2], C2×C132C8, C13×M4(2), D525C2, D52.2C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, D13, C8○D4, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D52.2C4

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 13A ··· 13F 26A ··· 26F 26G ··· 26L 52A ··· 52L 52M ··· 52R 104A ··· 104X order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 26 26 1 1 2 26 26 2 2 2 2 13 13 13 13 26 26 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D13 C8○D4 D26 D26 C4×D13 C4×D13 D52.2C4 kernel D52.2C4 C8×D13 C8⋊D13 C2×C13⋊2C8 C13×M4(2) D52⋊5C2 Dic26 D52 C13⋊D4 M4(2) C13 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 6 4 12 6 12 12 12

Matrix representation of D52.2C4 in GL4(𝔽313) generated by

 293 78 0 0 81 44 0 0 0 0 131 196 0 0 37 182
,
 312 0 0 0 256 1 0 0 0 0 131 196 0 0 251 182
,
 312 0 0 0 0 312 0 0 0 0 125 0 0 0 98 188
`G:=sub<GL(4,GF(313))| [293,81,0,0,78,44,0,0,0,0,131,37,0,0,196,182],[312,256,0,0,0,1,0,0,0,0,131,251,0,0,196,182],[312,0,0,0,0,312,0,0,0,0,125,98,0,0,0,188] >;`

D52.2C4 in GAP, Magma, Sage, TeX

`D_{52}._2C_4`
`% in TeX`

`G:=Group("D52.2C4");`
`// GroupNames label`

`G:=SmallGroup(416,128);`
`// by ID`

`G=gap.SmallGroup(416,128);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,188,50,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=b^2=1,c^4=a^26,b*a*b=a^-1,c*a*c^-1=a^27,c*b*c^-1=a^26*b>;`
`// generators/relations`

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