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G = D5×C21order 210 = 2·3·5·7

Direct product of C21 and D5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D5×C21, C5⋊C42, C357C6, C152C14, C1056C2, SmallGroup(210,6)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C21
C1C5C35C105 — D5×C21
C5 — D5×C21
C1C21

Generators and relations for D5×C21
 G = < a,b,c | a21=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C6
5C14
5C42

Smallest permutation representation of D5×C21
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 76 100 59 42)(2 77 101 60 22)(3 78 102 61 23)(4 79 103 62 24)(5 80 104 63 25)(6 81 105 43 26)(7 82 85 44 27)(8 83 86 45 28)(9 84 87 46 29)(10 64 88 47 30)(11 65 89 48 31)(12 66 90 49 32)(13 67 91 50 33)(14 68 92 51 34)(15 69 93 52 35)(16 70 94 53 36)(17 71 95 54 37)(18 72 96 55 38)(19 73 97 56 39)(20 74 98 57 40)(21 75 99 58 41)
(1 42)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(43 81)(44 82)(45 83)(46 84)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)

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), (1,76,100,59,42)(2,77,101,60,22)(3,78,102,61,23)(4,79,103,62,24)(5,80,104,63,25)(6,81,105,43,26)(7,82,85,44,27)(8,83,86,45,28)(9,84,87,46,29)(10,64,88,47,30)(11,65,89,48,31)(12,66,90,49,32)(13,67,91,50,33)(14,68,92,51,34)(15,69,93,52,35)(16,70,94,53,36)(17,71,95,54,37)(18,72,96,55,38)(19,73,97,56,39)(20,74,98,57,40)(21,75,99,58,41), (1,42)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(43,81)(44,82)(45,83)(46,84)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)>;

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), (1,76,100,59,42)(2,77,101,60,22)(3,78,102,61,23)(4,79,103,62,24)(5,80,104,63,25)(6,81,105,43,26)(7,82,85,44,27)(8,83,86,45,28)(9,84,87,46,29)(10,64,88,47,30)(11,65,89,48,31)(12,66,90,49,32)(13,67,91,50,33)(14,68,92,51,34)(15,69,93,52,35)(16,70,94,53,36)(17,71,95,54,37)(18,72,96,55,38)(19,73,97,56,39)(20,74,98,57,40)(21,75,99,58,41), (1,42)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(43,81)(44,82)(45,83)(46,84)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80) );

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)], [(1,76,100,59,42),(2,77,101,60,22),(3,78,102,61,23),(4,79,103,62,24),(5,80,104,63,25),(6,81,105,43,26),(7,82,85,44,27),(8,83,86,45,28),(9,84,87,46,29),(10,64,88,47,30),(11,65,89,48,31),(12,66,90,49,32),(13,67,91,50,33),(14,68,92,51,34),(15,69,93,52,35),(16,70,94,53,36),(17,71,95,54,37),(18,72,96,55,38),(19,73,97,56,39),(20,74,98,57,40),(21,75,99,58,41)], [(1,42),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(43,81),(44,82),(45,83),(46,84),(47,64),(48,65),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80)])

D5×C21 is a maximal subgroup of   C5⋊Dic21

84 conjugacy classes

class 1  2 3A3B5A5B6A6B7A···7F14A···14F15A15B15C15D21A···21L35A···35L42A···42L105A···105X
order123355667···714···141515151521···2135···3542···42105···105
size151122551···15···522221···12···25···52···2

84 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C7C14C21C42D5C3×D5C7×D5D5×C21
kernelD5×C21C105C7×D5C35C3×D5C15D5C5C21C7C3C1
# reps1122661212241224

Matrix representation of D5×C21 in GL3(𝔽211) generated by

5800
0140
0014
,
100
001
0210178
,
21000
001
010
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] >;

D5×C21 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

Subgroup lattice of D5×C21 in TeX

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