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G = D5×C43order 430 = 2·5·43

Direct product of C43 and D5

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

Aliases: D5×C43, C5⋊C86, C2153C2, SmallGroup(430,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C43
C1C5C215 — D5×C43
C5 — D5×C43
C1C43

Generators and relations for D5×C43
 G = < a,b,c | a43=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C86

Smallest permutation representation of D5×C43
On 215 points
Generators in S215
(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)
(1 156 91 181 71)(2 157 92 182 72)(3 158 93 183 73)(4 159 94 184 74)(5 160 95 185 75)(6 161 96 186 76)(7 162 97 187 77)(8 163 98 188 78)(9 164 99 189 79)(10 165 100 190 80)(11 166 101 191 81)(12 167 102 192 82)(13 168 103 193 83)(14 169 104 194 84)(15 170 105 195 85)(16 171 106 196 86)(17 172 107 197 44)(18 130 108 198 45)(19 131 109 199 46)(20 132 110 200 47)(21 133 111 201 48)(22 134 112 202 49)(23 135 113 203 50)(24 136 114 204 51)(25 137 115 205 52)(26 138 116 206 53)(27 139 117 207 54)(28 140 118 208 55)(29 141 119 209 56)(30 142 120 210 57)(31 143 121 211 58)(32 144 122 212 59)(33 145 123 213 60)(34 146 124 214 61)(35 147 125 215 62)(36 148 126 173 63)(37 149 127 174 64)(38 150 128 175 65)(39 151 129 176 66)(40 152 87 177 67)(41 153 88 178 68)(42 154 89 179 69)(43 155 90 180 70)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(130 198)(131 199)(132 200)(133 201)(134 202)(135 203)(136 204)(137 205)(138 206)(139 207)(140 208)(141 209)(142 210)(143 211)(144 212)(145 213)(146 214)(147 215)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 181)(157 182)(158 183)(159 184)(160 185)(161 186)(162 187)(163 188)(164 189)(165 190)(166 191)(167 192)(168 193)(169 194)(170 195)(171 196)(172 197)

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

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

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

172 conjugacy classes

class 1  2 5A5B43A···43AP86A···86AP215A···215CF
order125543···4386···86215···215
size15221···15···52···2

172 irreducible representations

dim111122
type+++
imageC1C2C43C86D5D5×C43
kernelD5×C43C215D5C5C43C1
# reps114242284

Matrix representation of D5×C43 in GL2(𝔽431) generated by

2200
0220
,
901
4300
,
01
10
G:=sub<GL(2,GF(431))| [220,0,0,220],[90,430,1,0],[0,1,1,0] >;

D5×C43 in GAP, Magma, Sage, TeX

D_5\times C_{43}
% in TeX

G:=Group("D5xC43");
// GroupNames label

G:=SmallGroup(430,1);
// by ID

G=gap.SmallGroup(430,1);
# by ID

G:=PCGroup([3,-2,-43,-5,3098]);
// Polycyclic

G:=Group<a,b,c|a^43=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×C43 in TeX

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