direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×D20, C35⋊6D4, C28⋊3D5, C20⋊1C14, C140⋊4C2, D10⋊1C14, C14.15D10, C70.20C22, C4⋊(C7×D5), C5⋊1(C7×D4), (D5×C14)⋊4C2, C2.4(D5×C14), C10.3(C2×C14), SmallGroup(280,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D20
G = < a,b,c | a7=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D20
►On 140 points
Generators in S140
(1 79 124 105 60 26 94)(2 80 125 106 41 27 95)(3 61 126 107 42 28 96)(4 62 127 108 43 29 97)(5 63 128 109 44 30 98)(6 64 129 110 45 31 99)(7 65 130 111 46 32 100)(8 66 131 112 47 33 81)(9 67 132 113 48 34 82)(10 68 133 114 49 35 83)(11 69 134 115 50 36 84)(12 70 135 116 51 37 85)(13 71 136 117 52 38 86)(14 72 137 118 53 39 87)(15 73 138 119 54 40 88)(16 74 139 120 55 21 89)(17 75 140 101 56 22 90)(18 76 121 102 57 23 91)(19 77 122 103 58 24 92)(20 78 123 104 59 25 93)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)
G:=sub<Sym(140)| (1,79,124,105,60,26,94)(2,80,125,106,41,27,95)(3,61,126,107,42,28,96)(4,62,127,108,43,29,97)(5,63,128,109,44,30,98)(6,64,129,110,45,31,99)(7,65,130,111,46,32,100)(8,66,131,112,47,33,81)(9,67,132,113,48,34,82)(10,68,133,114,49,35,83)(11,69,134,115,50,36,84)(12,70,135,116,51,37,85)(13,71,136,117,52,38,86)(14,72,137,118,53,39,87)(15,73,138,119,54,40,88)(16,74,139,120,55,21,89)(17,75,140,101,56,22,90)(18,76,121,102,57,23,91)(19,77,122,103,58,24,92)(20,78,123,104,59,25,93), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)>;
G:=Group( (1,79,124,105,60,26,94)(2,80,125,106,41,27,95)(3,61,126,107,42,28,96)(4,62,127,108,43,29,97)(5,63,128,109,44,30,98)(6,64,129,110,45,31,99)(7,65,130,111,46,32,100)(8,66,131,112,47,33,81)(9,67,132,113,48,34,82)(10,68,133,114,49,35,83)(11,69,134,115,50,36,84)(12,70,135,116,51,37,85)(13,71,136,117,52,38,86)(14,72,137,118,53,39,87)(15,73,138,119,54,40,88)(16,74,139,120,55,21,89)(17,75,140,101,56,22,90)(18,76,121,102,57,23,91)(19,77,122,103,58,24,92)(20,78,123,104,59,25,93), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134) );
G=PermutationGroup([[(1,79,124,105,60,26,94),(2,80,125,106,41,27,95),(3,61,126,107,42,28,96),(4,62,127,108,43,29,97),(5,63,128,109,44,30,98),(6,64,129,110,45,31,99),(7,65,130,111,46,32,100),(8,66,131,112,47,33,81),(9,67,132,113,48,34,82),(10,68,133,114,49,35,83),(11,69,134,115,50,36,84),(12,70,135,116,51,37,85),(13,71,136,117,52,38,86),(14,72,137,118,53,39,87),(15,73,138,119,54,40,88),(16,74,139,120,55,21,89),(17,75,140,101,56,22,90),(18,76,121,102,57,23,91),(19,77,122,103,58,24,92),(20,78,123,104,59,25,93)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,108),(102,107),(103,106),(104,105),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(121,126),(122,125),(123,124),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134)]])
91 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 7A | ··· | 7F | 10A | 10B | 14A | ··· | 14F | 14G | ··· | 14R | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | D5 | D10 | D20 | C7×D4 | C7×D5 | D5×C14 | C7×D20 |
| kernel | C7×D20 | C140 | D5×C14 | D20 | C20 | D10 | C35 | C28 | C14 | C7 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 2 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C7×D20 ►in GL2(𝔽281) generated by
| 181 | 0 |
| 0 | 181 |
| 134 | 50 |
| 231 | 17 |
| 134 | 50 |
| 231 | 147 |
G:=sub<GL(2,GF(281))| [181,0,0,181],[134,231,50,17],[134,231,50,147] >;
C7×D20 in GAP, Magma, Sage, TeX
C_7\times D_{20} % in TeX
G:=Group("C7xD20"); // GroupNames label
G:=SmallGroup(280,21);
// by ID
G=gap.SmallGroup(280,21);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-5,301,146,5604]);
// Polycyclic
G:=Group<a,b,c|a^7=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export