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## G = D5×D21order 420 = 22·3·5·7

### Direct product of D5 and D21

Aliases: D5×D21, C51D42, C352D6, C151D14, C214D10, D1052C2, C1052C22, (C7×D5)⋊S3, (C3×D5)⋊D7, C72(S3×D5), C31(D5×D7), (D5×C21)⋊1C2, (C5×D21)⋊1C2, SmallGroup(420,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105 — D5×D21
 Chief series C1 — C7 — C35 — C105 — C5×D21 — D5×D21
 Lower central C105 — D5×D21
 Upper central C1

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

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