Copied to
clipboard

G = C5×D41order 410 = 2·5·41

Direct product of C5 and D41

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

Aliases: C5×D41, C2052C2, C413C10, SmallGroup(410,4)

Series: Derived Chief Lower central Upper central

C1C41 — C5×D41
C1C41C205 — C5×D41
C41 — C5×D41
C1C5

Generators and relations for C5×D41
 G = < a,b,c | a5=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >

41C2
41C10

Smallest permutation representation of C5×D41
On 205 points
Generators in S205
(1 185 156 111 51)(2 186 157 112 52)(3 187 158 113 53)(4 188 159 114 54)(5 189 160 115 55)(6 190 161 116 56)(7 191 162 117 57)(8 192 163 118 58)(9 193 164 119 59)(10 194 124 120 60)(11 195 125 121 61)(12 196 126 122 62)(13 197 127 123 63)(14 198 128 83 64)(15 199 129 84 65)(16 200 130 85 66)(17 201 131 86 67)(18 202 132 87 68)(19 203 133 88 69)(20 204 134 89 70)(21 205 135 90 71)(22 165 136 91 72)(23 166 137 92 73)(24 167 138 93 74)(25 168 139 94 75)(26 169 140 95 76)(27 170 141 96 77)(28 171 142 97 78)(29 172 143 98 79)(30 173 144 99 80)(31 174 145 100 81)(32 175 146 101 82)(33 176 147 102 42)(34 177 148 103 43)(35 178 149 104 44)(36 179 150 105 45)(37 180 151 106 46)(38 181 152 107 47)(39 182 153 108 48)(40 183 154 109 49)(41 184 155 110 50)
(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 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(98 123)(99 122)(100 121)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 156)(165 204)(166 203)(167 202)(168 201)(169 200)(170 199)(171 198)(172 197)(173 196)(174 195)(175 194)(176 193)(177 192)(178 191)(179 190)(180 189)(181 188)(182 187)(183 186)(184 185)

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

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

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

110 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D41A···41T205A···205CB
order1255551010101041···41205···205
size1411111414141412···22···2

110 irreducible representations

dim111122
type+++
imageC1C2C5C10D41C5×D41
kernelC5×D41C205D41C41C5C1
# reps11442080

Matrix representation of C5×D41 in GL2(𝔽821) generated by

1610
0161
,
3491
8200
,
01
10
G:=sub<GL(2,GF(821))| [161,0,0,161],[349,820,1,0],[0,1,1,0] >;

C5×D41 in GAP, Magma, Sage, TeX

C_5\times D_{41}
% in TeX

G:=Group("C5xD41");
// GroupNames label

G:=SmallGroup(410,4);
// by ID

G=gap.SmallGroup(410,4);
# by ID

G:=PCGroup([3,-2,-5,-41,3602]);
// Polycyclic

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

Export

Subgroup lattice of C5×D41 in TeX

׿
×
𝔽