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## G = D5×C42order 420 = 22·3·5·7

### Direct product of C42 and D5

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

 Derived series C1 — C5 — D5×C42
 Chief series C1 — C5 — C35 — C105 — D5×C21 — D5×C42
 Lower central C5 — D5×C42
 Upper central C1 — C42

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

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

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

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

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

168 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 6A 6B 6C 6D 6E 6F 7A ··· 7F 10A 10B 14A ··· 14F 14G ··· 14R 15A 15B 15C 15D 21A ··· 21L 30A 30B 30C 30D 35A ··· 35L 42A ··· 42L 42M ··· 42AJ 70A ··· 70L 105A ··· 105X 210A ··· 210X order 1 2 2 2 3 3 5 5 6 6 6 6 6 6 7 ··· 7 10 10 14 ··· 14 14 ··· 14 15 15 15 15 21 ··· 21 30 30 30 30 35 ··· 35 42 ··· 42 42 ··· 42 70 ··· 70 105 ··· 105 210 ··· 210 size 1 1 5 5 1 1 2 2 1 1 5 5 5 5 1 ··· 1 2 2 1 ··· 1 5 ··· 5 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

168 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C7 C14 C14 C21 C42 C42 D5 D10 C3×D5 C6×D5 C7×D5 D5×C14 D5×C21 D5×C42 kernel D5×C42 D5×C21 C210 D5×C14 C7×D5 C70 C6×D5 C3×D5 C30 D10 D5 C10 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 12 24 12 2 2 4 4 12 12 24 24

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

 14 0 0 0 88 0 0 0 88
,
 1 0 0 0 0 1 0 210 32
,
 210 0 0 0 0 210 0 210 0
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

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