direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×D21, C5⋊1D42, C35⋊2D6, C15⋊1D14, C21⋊4D10, D105⋊2C2, C105⋊2C22, (C7×D5)⋊S3, (C3×D5)⋊D7, C7⋊2(S3×D5), C3⋊1(D5×D7), (D5×C21)⋊1C2, (C5×D21)⋊1C2, SmallGroup(420,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — D5×D21 |
Generators and relations for D5×D21
G = < a,b,c,d | a5=b2=c21=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
5C2
21C2
105C2
105C22
5C6
7S3
35S3
21C10
21D5
3D7
5C14
15D7
35D6
21D10
15D14
7D15
5D21
5C42
3D35
5D42
Smallest permutation representation of D5×D21
►On 105 points
Generators in S105
(1 53 40 90 72)(2 54 41 91 73)(3 55 42 92 74)(4 56 22 93 75)(5 57 23 94 76)(6 58 24 95 77)(7 59 25 96 78)(8 60 26 97 79)(9 61 27 98 80)(10 62 28 99 81)(11 63 29 100 82)(12 43 30 101 83)(13 44 31 102 84)(14 45 32 103 64)(15 46 33 104 65)(16 47 34 105 66)(17 48 35 85 67)(18 49 36 86 68)(19 50 37 87 69)(20 51 38 88 70)(21 52 39 89 71)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(43 101)(44 102)(45 103)(46 104)(47 105)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)(63 100)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(37 42)(38 41)(39 40)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(80 84)(81 83)(85 94)(86 93)(87 92)(88 91)(89 90)(95 105)(96 104)(97 103)(98 102)(99 101)
G:=sub<Sym(105)| (1,53,40,90,72)(2,54,41,91,73)(3,55,42,92,74)(4,56,22,93,75)(5,57,23,94,76)(6,58,24,95,77)(7,59,25,96,78)(8,60,26,97,79)(9,61,27,98,80)(10,62,28,99,81)(11,63,29,100,82)(12,43,30,101,83)(13,44,31,102,84)(14,45,32,103,64)(15,46,33,104,65)(16,47,34,105,66)(17,48,35,85,67)(18,49,36,86,68)(19,50,37,87,69)(20,51,38,88,70)(21,52,39,89,71), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(43,101)(44,102)(45,103)(46,104)(47,105)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99)(63,100), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,42)(38,41)(39,40)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(80,84)(81,83)(85,94)(86,93)(87,92)(88,91)(89,90)(95,105)(96,104)(97,103)(98,102)(99,101)>;
G:=Group( (1,53,40,90,72)(2,54,41,91,73)(3,55,42,92,74)(4,56,22,93,75)(5,57,23,94,76)(6,58,24,95,77)(7,59,25,96,78)(8,60,26,97,79)(9,61,27,98,80)(10,62,28,99,81)(11,63,29,100,82)(12,43,30,101,83)(13,44,31,102,84)(14,45,32,103,64)(15,46,33,104,65)(16,47,34,105,66)(17,48,35,85,67)(18,49,36,86,68)(19,50,37,87,69)(20,51,38,88,70)(21,52,39,89,71), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(43,101)(44,102)(45,103)(46,104)(47,105)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99)(63,100), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,42)(38,41)(39,40)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(80,84)(81,83)(85,94)(86,93)(87,92)(88,91)(89,90)(95,105)(96,104)(97,103)(98,102)(99,101) );
G=PermutationGroup([[(1,53,40,90,72),(2,54,41,91,73),(3,55,42,92,74),(4,56,22,93,75),(5,57,23,94,76),(6,58,24,95,77),(7,59,25,96,78),(8,60,26,97,79),(9,61,27,98,80),(10,62,28,99,81),(11,63,29,100,82),(12,43,30,101,83),(13,44,31,102,84),(14,45,32,103,64),(15,46,33,104,65),(16,47,34,105,66),(17,48,35,85,67),(18,49,36,86,68),(19,50,37,87,69),(20,51,38,88,70),(21,52,39,89,71)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(43,101),(44,102),(45,103),(46,104),(47,105),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,97),(61,98),(62,99),(63,100)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(37,42),(38,41),(39,40),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(80,84),(81,83),(85,94),(86,93),(87,92),(88,91),(89,90),(95,105),(96,104),(97,103),(98,102),(99,101)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 15A | 15B | 21A | ··· | 21F | 35A | ··· | 35F | 42A | ··· | 42F | 105A | ··· | 105L |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 42 | ··· | 42 | 105 | ··· | 105 |
| size | 1 | 5 | 21 | 105 | 2 | 2 | 2 | 10 | 2 | 2 | 2 | 42 | 42 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | D7 | D10 | D14 | D21 | D42 | S3×D5 | D5×D7 | D5×D21 |
| kernel | D5×D21 | D5×C21 | C5×D21 | D105 | C7×D5 | D21 | C35 | C3×D5 | C21 | C15 | D5 | C5 | C7 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 6 | 6 | 2 | 6 | 12 |
Matrix representation of D5×D21 ►in GL4(𝔽211) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 132 |
| 0 | 0 | 186 | 177 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 177 | 79 |
| 0 | 0 | 167 | 34 |
| 185 | 132 | 0 | 0 |
| 79 | 29 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 185 | 132 | 0 | 0 |
| 86 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(211))| [1,0,0,0,0,1,0,0,0,0,66,186,0,0,132,177],[1,0,0,0,0,1,0,0,0,0,177,167,0,0,79,34],[185,79,0,0,132,29,0,0,0,0,1,0,0,0,0,1],[185,86,0,0,132,26,0,0,0,0,1,0,0,0,0,1] >;
D5×D21 in GAP, Magma, Sage, TeX
D_5\times D_{21} % in TeX
G:=Group("D5xD21"); // GroupNames label
G:=SmallGroup(420,28);
// by ID
G=gap.SmallGroup(420,28);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,122,488,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^21=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export