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## G = C2×C52.C4order 416 = 25·13

### Direct product of C2 and C52.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C52.C4
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — C2×C13⋊C8 — C2×C52.C4
 Lower central C13 — C26 — C2×C52.C4
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C52.C4
G = < a,b,c | a2=b52=1, c4=b26, ab=ba, ac=ca, cbc-1=b31 >

Subgroups: 436 in 68 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, C13, C2×C8 [×2], M4(2) [×4], C22×C4, D13 [×2], C26, C26 [×2], C2×M4(2), Dic13 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, C13⋊C8 [×4], C4×D13 [×4], C2×Dic13, C2×C52, C22×D13, C52.C4 [×4], C2×C13⋊C8 [×2], C2×C4×D13, C2×C52.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C13⋊C4, C2×C13⋊C4 [×3], C52.C4 [×2], C22×C13⋊C4, C2×C52.C4

Smallest permutation representation of C2×C52.C4
On 208 points
Generators in S208
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 196)(143 197)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 157)(156 158)
(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)
(1 204 86 111 27 178 60 137)(2 199 59 142 28 173 85 116)(3 194 84 121 29 168 58 147)(4 189 57 152 30 163 83 126)(5 184 82 131 31 158 56 105)(6 179 55 110 32 205 81 136)(7 174 80 141 33 200 54 115)(8 169 53 120 34 195 79 146)(9 164 78 151 35 190 104 125)(10 159 103 130 36 185 77 156)(11 206 76 109 37 180 102 135)(12 201 101 140 38 175 75 114)(13 196 74 119 39 170 100 145)(14 191 99 150 40 165 73 124)(15 186 72 129 41 160 98 155)(16 181 97 108 42 207 71 134)(17 176 70 139 43 202 96 113)(18 171 95 118 44 197 69 144)(19 166 68 149 45 192 94 123)(20 161 93 128 46 187 67 154)(21 208 66 107 47 182 92 133)(22 203 91 138 48 177 65 112)(23 198 64 117 49 172 90 143)(24 193 89 148 50 167 63 122)(25 188 62 127 51 162 88 153)(26 183 87 106 52 157 61 132)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 M4(2) C13⋊C4 C2×C13⋊C4 C2×C13⋊C4 C52.C4 kernel C2×C52.C4 C52.C4 C2×C13⋊C8 C2×C4×D13 C4×D13 C2×C52 C22×D13 C26 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 4 3 6 3 12

Matrix representation of C2×C52.C4 in GL6(𝔽313)

 312 0 0 0 0 0 0 312 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 25 0 0 0 0 0 0 288 0 0 0 0 0 0 101 272 92 82 0 0 2 90 62 73 0 0 266 67 62 52 0 0 289 134 44 300
,
 0 1 0 0 0 0 25 0 0 0 0 0 0 0 6 216 255 3 0 0 169 229 259 269 0 0 175 195 257 103 0 0 163 89 226 134

G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,0,0,0,0,0,0,288,0,0,0,0,0,0,101,2,266,289,0,0,272,90,67,134,0,0,92,62,62,44,0,0,82,73,52,300],[0,25,0,0,0,0,1,0,0,0,0,0,0,0,6,169,175,163,0,0,216,229,195,89,0,0,255,259,257,226,0,0,3,269,103,134] >;

C2×C52.C4 in GAP, Magma, Sage, TeX

C_2\times C_{52}.C_4
% in TeX

G:=Group("C2xC52.C4");
// GroupNames label

G:=SmallGroup(416,200);
// by ID

G=gap.SmallGroup(416,200);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,86,69,9221,1751]);
// Polycyclic

G:=Group<a,b,c|a^2=b^52=1,c^4=b^26,a*b=b*a,a*c=c*a,c*b*c^-1=b^31>;
// generators/relations

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