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G = C5×D43order 430 = 2·5·43

Direct product of C5 and D43

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

Aliases: C5×D43, C43⋊C10, C2152C2, SmallGroup(430,2)

Series: Derived Chief Lower central Upper central

C1C43 — C5×D43
C1C43C215 — C5×D43
C43 — C5×D43
C1C5

Generators and relations for C5×D43
 G = < a,b,c | a5=b43=c2=1, ab=ba, ac=ca, cbc=b-1 >

43C2
43C10

Smallest permutation representation of C5×D43
On 215 points
Generators in S215
(1 213 154 94 86)(2 214 155 95 44)(3 215 156 96 45)(4 173 157 97 46)(5 174 158 98 47)(6 175 159 99 48)(7 176 160 100 49)(8 177 161 101 50)(9 178 162 102 51)(10 179 163 103 52)(11 180 164 104 53)(12 181 165 105 54)(13 182 166 106 55)(14 183 167 107 56)(15 184 168 108 57)(16 185 169 109 58)(17 186 170 110 59)(18 187 171 111 60)(19 188 172 112 61)(20 189 130 113 62)(21 190 131 114 63)(22 191 132 115 64)(23 192 133 116 65)(24 193 134 117 66)(25 194 135 118 67)(26 195 136 119 68)(27 196 137 120 69)(28 197 138 121 70)(29 198 139 122 71)(30 199 140 123 72)(31 200 141 124 73)(32 201 142 125 74)(33 202 143 126 75)(34 203 144 127 76)(35 204 145 128 77)(36 205 146 129 78)(37 206 147 87 79)(38 207 148 88 80)(39 208 149 89 81)(40 209 150 90 82)(41 210 151 91 83)(42 211 152 92 84)(43 212 153 93 85)
(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)(173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215)
(1 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(85 86)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 129)(102 128)(103 127)(104 126)(105 125)(106 124)(107 123)(108 122)(109 121)(110 120)(111 119)(112 118)(113 117)(114 116)(130 134)(131 133)(135 172)(136 171)(137 170)(138 169)(139 168)(140 167)(141 166)(142 165)(143 164)(144 163)(145 162)(146 161)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)(173 209)(174 208)(175 207)(176 206)(177 205)(178 204)(179 203)(180 202)(181 201)(182 200)(183 199)(184 198)(185 197)(186 196)(187 195)(188 194)(189 193)(190 192)(210 215)(211 214)(212 213)

G:=sub<Sym(215)| (1,213,154,94,86)(2,214,155,95,44)(3,215,156,96,45)(4,173,157,97,46)(5,174,158,98,47)(6,175,159,99,48)(7,176,160,100,49)(8,177,161,101,50)(9,178,162,102,51)(10,179,163,103,52)(11,180,164,104,53)(12,181,165,105,54)(13,182,166,106,55)(14,183,167,107,56)(15,184,168,108,57)(16,185,169,109,58)(17,186,170,110,59)(18,187,171,111,60)(19,188,172,112,61)(20,189,130,113,62)(21,190,131,114,63)(22,191,132,115,64)(23,192,133,116,65)(24,193,134,117,66)(25,194,135,118,67)(26,195,136,119,68)(27,196,137,120,69)(28,197,138,121,70)(29,198,139,122,71)(30,199,140,123,72)(31,200,141,124,73)(32,201,142,125,74)(33,202,143,126,75)(34,203,144,127,76)(35,204,145,128,77)(36,205,146,129,78)(37,206,147,87,79)(38,207,148,88,80)(39,208,149,89,81)(40,209,150,90,82)(41,210,151,91,83)(42,211,152,92,84)(43,212,153,93,85), (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)(173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(85,86)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)(109,121)(110,120)(111,119)(112,118)(113,117)(114,116)(130,134)(131,133)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(173,209)(174,208)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,201)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(210,215)(211,214)(212,213)>;

G:=Group( (1,213,154,94,86)(2,214,155,95,44)(3,215,156,96,45)(4,173,157,97,46)(5,174,158,98,47)(6,175,159,99,48)(7,176,160,100,49)(8,177,161,101,50)(9,178,162,102,51)(10,179,163,103,52)(11,180,164,104,53)(12,181,165,105,54)(13,182,166,106,55)(14,183,167,107,56)(15,184,168,108,57)(16,185,169,109,58)(17,186,170,110,59)(18,187,171,111,60)(19,188,172,112,61)(20,189,130,113,62)(21,190,131,114,63)(22,191,132,115,64)(23,192,133,116,65)(24,193,134,117,66)(25,194,135,118,67)(26,195,136,119,68)(27,196,137,120,69)(28,197,138,121,70)(29,198,139,122,71)(30,199,140,123,72)(31,200,141,124,73)(32,201,142,125,74)(33,202,143,126,75)(34,203,144,127,76)(35,204,145,128,77)(36,205,146,129,78)(37,206,147,87,79)(38,207,148,88,80)(39,208,149,89,81)(40,209,150,90,82)(41,210,151,91,83)(42,211,152,92,84)(43,212,153,93,85), (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)(173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(85,86)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)(109,121)(110,120)(111,119)(112,118)(113,117)(114,116)(130,134)(131,133)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(173,209)(174,208)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,201)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(210,215)(211,214)(212,213) );

G=PermutationGroup([(1,213,154,94,86),(2,214,155,95,44),(3,215,156,96,45),(4,173,157,97,46),(5,174,158,98,47),(6,175,159,99,48),(7,176,160,100,49),(8,177,161,101,50),(9,178,162,102,51),(10,179,163,103,52),(11,180,164,104,53),(12,181,165,105,54),(13,182,166,106,55),(14,183,167,107,56),(15,184,168,108,57),(16,185,169,109,58),(17,186,170,110,59),(18,187,171,111,60),(19,188,172,112,61),(20,189,130,113,62),(21,190,131,114,63),(22,191,132,115,64),(23,192,133,116,65),(24,193,134,117,66),(25,194,135,118,67),(26,195,136,119,68),(27,196,137,120,69),(28,197,138,121,70),(29,198,139,122,71),(30,199,140,123,72),(31,200,141,124,73),(32,201,142,125,74),(33,202,143,126,75),(34,203,144,127,76),(35,204,145,128,77),(36,205,146,129,78),(37,206,147,87,79),(38,207,148,88,80),(39,208,149,89,81),(40,209,150,90,82),(41,210,151,91,83),(42,211,152,92,84),(43,212,153,93,85)], [(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),(173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(44,84),(45,83),(46,82),(47,81),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(85,86),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,129),(102,128),(103,127),(104,126),(105,125),(106,124),(107,123),(108,122),(109,121),(110,120),(111,119),(112,118),(113,117),(114,116),(130,134),(131,133),(135,172),(136,171),(137,170),(138,169),(139,168),(140,167),(141,166),(142,165),(143,164),(144,163),(145,162),(146,161),(147,160),(148,159),(149,158),(150,157),(151,156),(152,155),(153,154),(173,209),(174,208),(175,207),(176,206),(177,205),(178,204),(179,203),(180,202),(181,201),(182,200),(183,199),(184,198),(185,197),(186,196),(187,195),(188,194),(189,193),(190,192),(210,215),(211,214),(212,213)])

115 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D43A···43U215A···215CF
order1255551010101043···43215···215
size1431111434343432···22···2

115 irreducible representations

dim111122
type+++
imageC1C2C5C10D43C5×D43
kernelC5×D43C215D43C43C5C1
# reps11442184

Matrix representation of C5×D43 in GL2(𝔽431) generated by

4050
0405
,
01
430147
,
01
10
G:=sub<GL(2,GF(431))| [405,0,0,405],[0,430,1,147],[0,1,1,0] >;

C5×D43 in GAP, Magma, Sage, TeX

C_5\times D_{43}
% in TeX

G:=Group("C5xD43");
// GroupNames label

G:=SmallGroup(430,2);
// by ID

G=gap.SmallGroup(430,2);
# by ID

G:=PCGroup([3,-2,-5,-43,3782]);
// Polycyclic

G:=Group<a,b,c|a^5=b^43=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D43 in TeX

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