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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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C13, C2×C8, M4(2), C22×C4, D13, C26, C26, C2×M4(2), Dic13, C52, D26, D26, C2×C26, C13⋊C8, C4×D13, C2×Dic13, C2×C52, C22×D13, C52.C4, C2×C13⋊C8, C2×C4×D13, C2×C52.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C2×M4(2), C13⋊C4, C2×C13⋊C4, C52.C4, C22×C13⋊C4, C2×C52.C4

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

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

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

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

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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