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## G = D52.C4order 416 = 25·13

### The non-split extension by D52 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D52.C4
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — D13⋊C8 — D52.C4
 Lower central C13 — C26 — D52.C4
 Upper central C1 — C2 — Q8

Generators and relations for D52.C4
G = < a,b,c | a52=b2=1, c4=a26, bab=a-1, cac-1=a5, cbc-1=a4b >

Subgroups: 404 in 62 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2 [×3], C4 [×3], C4, C22 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C13, C2×C8 [×3], M4(2) [×3], C4○D4, D13 [×3], C26, C8○D4, Dic13, C52 [×3], D26 [×3], C13⋊C8, C13⋊C8 [×3], C4×D13 [×3], D52 [×3], Q8×C13, D13⋊C8 [×3], C52.C4 [×3], D52⋊C2, D52.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C8○D4, C13⋊C4, C2×C13⋊C4 [×3], C22×C13⋊C4, D52.C4

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

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E ··· 8J 13A 13B 13C 26A 26B 26C 52A ··· 52I order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 ··· 8 13 13 13 26 26 26 52 ··· 52 size 1 1 26 26 26 2 2 2 13 13 13 13 13 13 26 ··· 26 4 4 4 4 4 4 8 ··· 8

35 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 8 type + + + + + + + image C1 C2 C2 C2 C4 C4 C8○D4 C13⋊C4 C2×C13⋊C4 D52.C4 kernel D52.C4 D13⋊C8 C52.C4 D52⋊C2 D52 Q8×C13 C13 Q8 C4 C1 # reps 1 3 3 1 6 2 4 3 9 3

Matrix representation of D52.C4 in GL6(𝔽313)

 0 1 0 0 0 0 312 0 0 0 0 0 0 0 0 0 312 1 0 0 71 31 281 242 0 0 281 61 40 2 0 0 71 30 311 212
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 312 0 0 0 1 0 312 0 0 0 0 0 312 0 0 0 282 31 70 1
,
 308 0 0 0 0 0 0 308 0 0 0 0 0 0 88 205 108 225 0 0 132 290 150 163 0 0 114 119 308 67 0 0 269 290 13 253

`G:=sub<GL(6,GF(313))| [0,312,0,0,0,0,1,0,0,0,0,0,0,0,0,71,281,71,0,0,0,31,61,30,0,0,312,281,40,311,0,0,1,242,2,212],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,282,0,0,1,0,0,31,0,0,312,312,312,70,0,0,0,0,0,1],[308,0,0,0,0,0,0,308,0,0,0,0,0,0,88,132,114,269,0,0,205,290,119,290,0,0,108,150,308,13,0,0,225,163,67,253] >;`

D52.C4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(416,207);`
`// by ID`

`G=gap.SmallGroup(416,207);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,188,86,69,9221,1751]);`
`// 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^5,c*b*c^-1=a^4*b>;`
`// generators/relations`

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