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## G = D52.3C4order 416 = 25·13

### The non-split extension by D52 of C4 acting through Inn(D52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D52.3C4
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D52⋊5C2 — D52.3C4
 Lower central C13 — C26 — D52.3C4
 Upper central C1 — C8 — C2×C8

Generators and relations for D52.3C4
G = < a,b,c | a52=b2=1, c4=a26, bab=a-1, ac=ca, bc=cb >

Subgroups: 344 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C13, C2×C8, C2×C8, M4(2), C4○D4, D13, C26, C26, C8○D4, Dic13, C52, D26, C2×C26, C132C8, C104, Dic26, C4×D13, D52, C13⋊D4, C2×C52, C8×D13, C8⋊D13, C52.4C4, C2×C104, D525C2, D52.3C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D13, C8○D4, D26, C4×D13, C22×D13, C2×C4×D13, D52.3C4

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

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

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

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

116 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 ··· 26R 52A ··· 52X 104A ··· 104AV 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 52 ··· 52 104 ··· 104 size 1 1 2 26 26 1 1 2 26 26 1 1 1 1 2 2 26 26 26 26 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D13 C8○D4 D26 D26 C4×D13 C4×D13 D52.3C4 kernel D52.3C4 C8×D13 C8⋊D13 C52.4C4 C2×C104 D52⋊5C2 Dic26 D52 C13⋊D4 C2×C8 C13 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 6 4 12 6 12 12 48

Matrix representation of D52.3C4 in GL2(𝔽313) generated by

 142 69 244 134
,
 142 69 48 171
,
 188 0 0 188
`G:=sub<GL(2,GF(313))| [142,244,69,134],[142,48,69,171],[188,0,0,188] >;`

D52.3C4 in GAP, Magma, Sage, TeX

`D_{52}._3C_4`
`% in TeX`

`G:=Group("D52.3C4");`
`// GroupNames label`

`G:=SmallGroup(416,122);`
`// by ID`

`G=gap.SmallGroup(416,122);`
`# by ID`

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

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

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