direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C20×C7⋊C3, C28⋊C15, C140⋊C3, C7⋊2C60, C35⋊8C12, C70.4C6, C14.2C30, C2.(C10×C7⋊C3), C10.2(C2×C7⋊C3), (C10×C7⋊C3).4C2, (C2×C7⋊C3).2C10, SmallGroup(420,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C20×C7⋊C3 |
Generators and relations for C20×C7⋊C3
G = < a,b,c | a20=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C20×C7⋊C3
►On 140 points
Generators in S140
(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 64 107 84 48 34 133)(2 65 108 85 49 35 134)(3 66 109 86 50 36 135)(4 67 110 87 51 37 136)(5 68 111 88 52 38 137)(6 69 112 89 53 39 138)(7 70 113 90 54 40 139)(8 71 114 91 55 21 140)(9 72 115 92 56 22 121)(10 73 116 93 57 23 122)(11 74 117 94 58 24 123)(12 75 118 95 59 25 124)(13 76 119 96 60 26 125)(14 77 120 97 41 27 126)(15 78 101 98 42 28 127)(16 79 102 99 43 29 128)(17 80 103 100 44 30 129)(18 61 104 81 45 31 130)(19 62 105 82 46 32 131)(20 63 106 83 47 33 132)
(21 91 140)(22 92 121)(23 93 122)(24 94 123)(25 95 124)(26 96 125)(27 97 126)(28 98 127)(29 99 128)(30 100 129)(31 81 130)(32 82 131)(33 83 132)(34 84 133)(35 85 134)(36 86 135)(37 87 136)(38 88 137)(39 89 138)(40 90 139)(41 77 120)(42 78 101)(43 79 102)(44 80 103)(45 61 104)(46 62 105)(47 63 106)(48 64 107)(49 65 108)(50 66 109)(51 67 110)(52 68 111)(53 69 112)(54 70 113)(55 71 114)(56 72 115)(57 73 116)(58 74 117)(59 75 118)(60 76 119)
G:=sub<Sym(140)| (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), (1,64,107,84,48,34,133)(2,65,108,85,49,35,134)(3,66,109,86,50,36,135)(4,67,110,87,51,37,136)(5,68,111,88,52,38,137)(6,69,112,89,53,39,138)(7,70,113,90,54,40,139)(8,71,114,91,55,21,140)(9,72,115,92,56,22,121)(10,73,116,93,57,23,122)(11,74,117,94,58,24,123)(12,75,118,95,59,25,124)(13,76,119,96,60,26,125)(14,77,120,97,41,27,126)(15,78,101,98,42,28,127)(16,79,102,99,43,29,128)(17,80,103,100,44,30,129)(18,61,104,81,45,31,130)(19,62,105,82,46,32,131)(20,63,106,83,47,33,132), (21,91,140)(22,92,121)(23,93,122)(24,94,123)(25,95,124)(26,96,125)(27,97,126)(28,98,127)(29,99,128)(30,100,129)(31,81,130)(32,82,131)(33,83,132)(34,84,133)(35,85,134)(36,86,135)(37,87,136)(38,88,137)(39,89,138)(40,90,139)(41,77,120)(42,78,101)(43,79,102)(44,80,103)(45,61,104)(46,62,105)(47,63,106)(48,64,107)(49,65,108)(50,66,109)(51,67,110)(52,68,111)(53,69,112)(54,70,113)(55,71,114)(56,72,115)(57,73,116)(58,74,117)(59,75,118)(60,76,119)>;
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), (1,64,107,84,48,34,133)(2,65,108,85,49,35,134)(3,66,109,86,50,36,135)(4,67,110,87,51,37,136)(5,68,111,88,52,38,137)(6,69,112,89,53,39,138)(7,70,113,90,54,40,139)(8,71,114,91,55,21,140)(9,72,115,92,56,22,121)(10,73,116,93,57,23,122)(11,74,117,94,58,24,123)(12,75,118,95,59,25,124)(13,76,119,96,60,26,125)(14,77,120,97,41,27,126)(15,78,101,98,42,28,127)(16,79,102,99,43,29,128)(17,80,103,100,44,30,129)(18,61,104,81,45,31,130)(19,62,105,82,46,32,131)(20,63,106,83,47,33,132), (21,91,140)(22,92,121)(23,93,122)(24,94,123)(25,95,124)(26,96,125)(27,97,126)(28,98,127)(29,99,128)(30,100,129)(31,81,130)(32,82,131)(33,83,132)(34,84,133)(35,85,134)(36,86,135)(37,87,136)(38,88,137)(39,89,138)(40,90,139)(41,77,120)(42,78,101)(43,79,102)(44,80,103)(45,61,104)(46,62,105)(47,63,106)(48,64,107)(49,65,108)(50,66,109)(51,67,110)(52,68,111)(53,69,112)(54,70,113)(55,71,114)(56,72,115)(57,73,116)(58,74,117)(59,75,118)(60,76,119) );
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)], [(1,64,107,84,48,34,133),(2,65,108,85,49,35,134),(3,66,109,86,50,36,135),(4,67,110,87,51,37,136),(5,68,111,88,52,38,137),(6,69,112,89,53,39,138),(7,70,113,90,54,40,139),(8,71,114,91,55,21,140),(9,72,115,92,56,22,121),(10,73,116,93,57,23,122),(11,74,117,94,58,24,123),(12,75,118,95,59,25,124),(13,76,119,96,60,26,125),(14,77,120,97,41,27,126),(15,78,101,98,42,28,127),(16,79,102,99,43,29,128),(17,80,103,100,44,30,129),(18,61,104,81,45,31,130),(19,62,105,82,46,32,131),(20,63,106,83,47,33,132)], [(21,91,140),(22,92,121),(23,93,122),(24,94,123),(25,95,124),(26,96,125),(27,97,126),(28,98,127),(29,99,128),(30,100,129),(31,81,130),(32,82,131),(33,83,132),(34,84,133),(35,85,134),(36,86,135),(37,87,136),(38,88,137),(39,89,138),(40,90,139),(41,77,120),(42,78,101),(43,79,102),(44,80,103),(45,61,104),(46,62,105),(47,63,106),(48,64,107),(49,65,108),(50,66,109),(51,67,110),(52,68,111),(53,69,112),(54,70,113),(55,71,114),(56,72,115),(57,73,116),(58,74,117),(59,75,118),(60,76,119)]])
100 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 7A | 7B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 14A | 14B | 15A | ··· | 15H | 20A | ··· | 20H | 28A | 28B | 28C | 28D | 30A | ··· | 30H | 35A | ··· | 35H | 60A | ··· | 60P | 70A | ··· | 70H | 140A | ··· | 140P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 14 | 14 | 15 | ··· | 15 | 20 | ··· | 20 | 28 | 28 | 28 | 28 | 30 | ··· | 30 | 35 | ··· | 35 | 60 | ··· | 60 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 7 | 3 | 3 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 3 | 3 | 7 | ··· | 7 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | C7⋊C3 | C2×C7⋊C3 | C4×C7⋊C3 | C5×C7⋊C3 | C10×C7⋊C3 | C20×C7⋊C3 |
| kernel | C20×C7⋊C3 | C10×C7⋊C3 | C140 | C5×C7⋊C3 | C4×C7⋊C3 | C70 | C2×C7⋊C3 | C35 | C28 | C7⋊C3 | C14 | C7 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of C20×C7⋊C3 ►in GL3(𝔽421) generated by
| 408 | 0 | 0 |
| 0 | 408 | 0 |
| 0 | 0 | 408 |
| 176 | 177 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 244 | 420 | 420 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(421))| [408,0,0,0,408,0,0,0,408],[176,1,0,177,0,1,1,0,0],[1,244,0,0,420,1,0,420,0] >;
C20×C7⋊C3 in GAP, Magma, Sage, TeX
C_{20}\times C_7\rtimes C_3 % in TeX
G:=Group("C20xC7:C3"); // GroupNames label
G:=SmallGroup(420,4);
// by ID
G=gap.SmallGroup(420,4);
# by ID
G:=PCGroup([5,-2,-3,-5,-2,-7,150,1509]);
// Polycyclic
G:=Group<a,b,c|a^20=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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