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G = D5×C42order 420 = 22·3·5·7

Direct product of C42 and D5

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

Aliases: D5×C42, C10⋊C42, C707C6, C2106C2, C302C14, C10513C22, C5⋊(C2×C42), C359(C2×C6), C153(C2×C14), SmallGroup(420,35)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C42
C1C5C35C105D5×C21 — D5×C42
C5 — D5×C42
C1C42

Generators and relations for D5×C42
 G = < a,b,c | a42=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C6
5C6
5C14
5C14
5C2×C6
5C2×C14
5C42
5C42
5C2×C42

Smallest permutation representation of D5×C42
On 210 points
Generators in S210
(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)
(1 174 162 65 112)(2 175 163 66 113)(3 176 164 67 114)(4 177 165 68 115)(5 178 166 69 116)(6 179 167 70 117)(7 180 168 71 118)(8 181 127 72 119)(9 182 128 73 120)(10 183 129 74 121)(11 184 130 75 122)(12 185 131 76 123)(13 186 132 77 124)(14 187 133 78 125)(15 188 134 79 126)(16 189 135 80 85)(17 190 136 81 86)(18 191 137 82 87)(19 192 138 83 88)(20 193 139 84 89)(21 194 140 43 90)(22 195 141 44 91)(23 196 142 45 92)(24 197 143 46 93)(25 198 144 47 94)(26 199 145 48 95)(27 200 146 49 96)(28 201 147 50 97)(29 202 148 51 98)(30 203 149 52 99)(31 204 150 53 100)(32 205 151 54 101)(33 206 152 55 102)(34 207 153 56 103)(35 208 154 57 104)(36 209 155 58 105)(37 210 156 59 106)(38 169 157 60 107)(39 170 158 61 108)(40 171 159 62 109)(41 172 160 63 110)(42 173 161 64 111)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 193)(64 194)(65 195)(66 196)(67 197)(68 198)(69 199)(70 200)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 169)(82 170)(83 171)(84 172)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)

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

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

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

168 conjugacy classes

class 1 2A2B2C3A3B5A5B6A6B6C6D6E6F7A···7F10A10B14A···14F14G···14R15A15B15C15D21A···21L30A30B30C30D35A···35L42A···42L42M···42AJ70A···70L105A···105X210A···210X
order122233556666667···7101014···1414···141515151521···213030303035···3542···4242···4270···70105···105210···210
size115511221155551···1221···15···522221···122222···21···15···52···22···22···2

168 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C6C6C7C14C14C21C42C42D5D10C3×D5C6×D5C7×D5D5×C14D5×C21D5×C42
kernelD5×C42D5×C21C210D5×C14C7×D5C70C6×D5C3×D5C30D10D5C10C42C21C14C7C6C3C2C1
# reps1212426126122412224412122424

Matrix representation of D5×C42 in GL3(𝔽211) generated by

1400
0880
0088
,
100
001
021032
,
21000
00210
02100
G:=sub<GL(3,GF(211))| [14,0,0,0,88,0,0,0,88],[1,0,0,0,0,210,0,1,32],[210,0,0,0,0,210,0,210,0] >;

D5×C42 in GAP, Magma, Sage, TeX

D_5\times C_{42}
% in TeX

G:=Group("D5xC42");
// GroupNames label

G:=SmallGroup(420,35);
// by ID

G=gap.SmallGroup(420,35);
# by ID

G:=PCGroup([5,-2,-2,-3,-7,-5,8404]);
// Polycyclic

G:=Group<a,b,c|a^42=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×C42 in TeX

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