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G = C2×C52.4C4order 416 = 25·13

Direct product of C2 and C52.4C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C52.4C4
 Chief series C1 — C13 — C26 — C52 — C13⋊2C8 — C2×C13⋊2C8 — C2×C52.4C4
 Lower central C13 — C26 — C2×C52.4C4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C52.4C4
G = < a,b,c | a2=b52=1, c4=b26, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 208 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C13, C2×C8 [×2], M4(2) [×4], C22×C4, C26, C26 [×2], C26 [×2], C2×M4(2), C52 [×2], C52 [×2], C2×C26, C2×C26 [×2], C2×C26 [×2], C132C8 [×4], C2×C52 [×2], C2×C52 [×4], C22×C26, C2×C132C8 [×2], C52.4C4 [×4], C22×C52, C2×C52.4C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, D13, C2×M4(2), Dic13 [×4], D26 [×3], C2×Dic13 [×6], C22×D13, C52.4C4 [×2], C22×Dic13, C2×C52.4C4

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

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

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

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

116 conjugacy classes

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

116 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 M4(2) D13 Dic13 D26 Dic13 C52.4C4 kernel C2×C52.4C4 C2×C13⋊2C8 C52.4C4 C22×C52 C2×C52 C22×C26 C26 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 4 6 18 18 6 48

Matrix representation of C2×C52.4C4 in GL3(𝔽313) generated by

 312 0 0 0 312 0 0 0 312
,
 312 0 0 0 151 0 0 0 199
,
 25 0 0 0 0 1 0 288 0
G:=sub<GL(3,GF(313))| [312,0,0,0,312,0,0,0,312],[312,0,0,0,151,0,0,0,199],[25,0,0,0,0,288,0,1,0] >;

C2×C52.4C4 in GAP, Magma, Sage, TeX

C_2\times C_{52}._4C_4
% in TeX

G:=Group("C2xC52.4C4");
// GroupNames label

G:=SmallGroup(416,142);
// by ID

G=gap.SmallGroup(416,142);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,69,13829]);
// 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^-1>;
// generators/relations

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