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G = D5×C41order 410 = 2·5·41

Direct product of C41 and D5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D5×C41, C5⋊C82, C2053C2, SmallGroup(410,3)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C41
C1C5C205 — D5×C41
C5 — D5×C41
C1C41

Generators and relations for D5×C41
 G = < a,b,c | a41=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C82

Smallest permutation representation of D5×C41
On 205 points
Generators in S205
(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)
(1 82 86 132 198)(2 42 87 133 199)(3 43 88 134 200)(4 44 89 135 201)(5 45 90 136 202)(6 46 91 137 203)(7 47 92 138 204)(8 48 93 139 205)(9 49 94 140 165)(10 50 95 141 166)(11 51 96 142 167)(12 52 97 143 168)(13 53 98 144 169)(14 54 99 145 170)(15 55 100 146 171)(16 56 101 147 172)(17 57 102 148 173)(18 58 103 149 174)(19 59 104 150 175)(20 60 105 151 176)(21 61 106 152 177)(22 62 107 153 178)(23 63 108 154 179)(24 64 109 155 180)(25 65 110 156 181)(26 66 111 157 182)(27 67 112 158 183)(28 68 113 159 184)(29 69 114 160 185)(30 70 115 161 186)(31 71 116 162 187)(32 72 117 163 188)(33 73 118 164 189)(34 74 119 124 190)(35 75 120 125 191)(36 76 121 126 192)(37 77 122 127 193)(38 78 123 128 194)(39 79 83 129 195)(40 80 84 130 196)(41 81 85 131 197)
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 151)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 162)(72 163)(73 164)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)

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

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

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

164 conjugacy classes

class 1  2 5A5B41A···41AN82A···82AN205A···205CB
order125541···4182···82205···205
size15221···15···52···2

164 irreducible representations

dim111122
type+++
imageC1C2C41C82D5D5×C41
kernelD5×C41C205D5C5C41C1
# reps114040280

Matrix representation of D5×C41 in GL2(𝔽821) generated by

5660
0566
,
01
820212
,
01
10
G:=sub<GL(2,GF(821))| [566,0,0,566],[0,820,1,212],[0,1,1,0] >;

D5×C41 in GAP, Magma, Sage, TeX

D_5\times C_{41}
% in TeX

G:=Group("D5xC41");
// GroupNames label

G:=SmallGroup(410,3);
// by ID

G=gap.SmallGroup(410,3);
# by ID

G:=PCGroup([3,-2,-41,-5,2954]);
// Polycyclic

G:=Group<a,b,c|a^41=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C41 in TeX

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