direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5xC21, C5:C42, C35:7C6, C15:2C14, C105:6C2, SmallGroup(210,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5xC21 |
Generators and relations for D5xC21
G = < a,b,c | a21=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C6, C7, D5, C14, C21, C3xD5, C42, C7xD5, D5xC21
Smallest permutation representation of D5xC21
►On 105 points
Generators in S105
(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)
(1 64 96 31 44)(2 65 97 32 45)(3 66 98 33 46)(4 67 99 34 47)(5 68 100 35 48)(6 69 101 36 49)(7 70 102 37 50)(8 71 103 38 51)(9 72 104 39 52)(10 73 105 40 53)(11 74 85 41 54)(12 75 86 42 55)(13 76 87 22 56)(14 77 88 23 57)(15 78 89 24 58)(16 79 90 25 59)(17 80 91 26 60)(18 81 92 27 61)(19 82 93 28 62)(20 83 94 29 63)(21 84 95 30 43)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 43)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)
G:=sub<Sym(105)| (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), (1,64,96,31,44)(2,65,97,32,45)(3,66,98,33,46)(4,67,99,34,47)(5,68,100,35,48)(6,69,101,36,49)(7,70,102,37,50)(8,71,103,38,51)(9,72,104,39,52)(10,73,105,40,53)(11,74,85,41,54)(12,75,86,42,55)(13,76,87,22,56)(14,77,88,23,57)(15,78,89,24,58)(16,79,90,25,59)(17,80,91,26,60)(18,81,92,27,61)(19,82,93,28,62)(20,83,94,29,63)(21,84,95,30,43), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,43)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)>;
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), (1,64,96,31,44)(2,65,97,32,45)(3,66,98,33,46)(4,67,99,34,47)(5,68,100,35,48)(6,69,101,36,49)(7,70,102,37,50)(8,71,103,38,51)(9,72,104,39,52)(10,73,105,40,53)(11,74,85,41,54)(12,75,86,42,55)(13,76,87,22,56)(14,77,88,23,57)(15,78,89,24,58)(16,79,90,25,59)(17,80,91,26,60)(18,81,92,27,61)(19,82,93,28,62)(20,83,94,29,63)(21,84,95,30,43), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,43)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75) );
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)], [(1,64,96,31,44),(2,65,97,32,45),(3,66,98,33,46),(4,67,99,34,47),(5,68,100,35,48),(6,69,101,36,49),(7,70,102,37,50),(8,71,103,38,51),(9,72,104,39,52),(10,73,105,40,53),(11,74,85,41,54),(12,75,86,42,55),(13,76,87,22,56),(14,77,88,23,57),(15,78,89,24,58),(16,79,90,25,59),(17,80,91,26,60),(18,81,92,27,61),(19,82,93,28,62),(20,83,94,29,63),(21,84,95,30,43)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,43),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75)]])
D5xC21 is a maximal subgroup of
C5:Dic21
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 15A | 15B | 15C | 15D | 21A | ··· | 21L | 35A | ··· | 35L | 42A | ··· | 42L | 105A | ··· | 105X |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 42 | ··· | 42 | 105 | ··· | 105 |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 5 | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | D5 | C3xD5 | C7xD5 | D5xC21 |
| kernel | D5xC21 | C105 | C7xD5 | C35 | C3xD5 | C15 | D5 | C5 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 4 | 12 | 24 |
Matrix representation of D5xC21 ►in GL3(F211) generated by
| 58 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 210 | 178 |
| 210 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(211))| [58,0,0,0,14,0,0,0,14],[1,0,0,0,0,210,0,1,178],[210,0,0,0,0,1,0,1,0] >;
D5xC21 in GAP, Magma, Sage, TeX
D_5\times C_{21} % in TeX
G:=Group("D5xC21"); // GroupNames label
G:=SmallGroup(210,6);
// by ID
G=gap.SmallGroup(210,6);
# by ID
G:=PCGroup([4,-2,-3,-7,-5,2691]);
// Polycyclic
G:=Group<a,b,c|a^21=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export