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G = D7×C15order 210 = 2·3·5·7

Direct product of C15 and D7

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

Aliases: D7×C15, C73C30, C356C6, C1055C2, C212C10, SmallGroup(210,5)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C15
C1C7C35C105 — D7×C15
C7 — D7×C15
C1C15

Generators and relations for D7×C15
 G = < a,b,c | a15=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C6
7C10
7C30

Smallest permutation representation of D7×C15
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 97 26 87 73 54 36)(2 98 27 88 74 55 37)(3 99 28 89 75 56 38)(4 100 29 90 61 57 39)(5 101 30 76 62 58 40)(6 102 16 77 63 59 41)(7 103 17 78 64 60 42)(8 104 18 79 65 46 43)(9 105 19 80 66 47 44)(10 91 20 81 67 48 45)(11 92 21 82 68 49 31)(12 93 22 83 69 50 32)(13 94 23 84 70 51 33)(14 95 24 85 71 52 34)(15 96 25 86 72 53 35)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 31)(12 32)(13 33)(14 34)(15 35)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 61)(30 62)(46 104)(47 105)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)

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

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

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

75 conjugacy classes

class 1  2 3A3B5A5B5C5D6A6B7A7B7C10A10B10C10D15A···15H21A···21F30A···30H35A···35L105A···105X
order12335555667771010101015···1521···2130···3035···35105···105
size171111117722277771···12···27···72···22···2

75 irreducible representations

dim111111112222
type+++
imageC1C2C3C5C6C10C15C30D7C3×D7C5×D7D7×C15
kernelD7×C15C105C5×D7C3×D7C35C21D7C7C15C5C3C1
# reps11242488361224

Matrix representation of D7×C15 in GL2(𝔽211) generated by

190
019
,
01
21081
,
01
10
G:=sub<GL(2,GF(211))| [19,0,0,19],[0,210,1,81],[0,1,1,0] >;

D7×C15 in GAP, Magma, Sage, TeX

D_7\times C_{15}
% in TeX

G:=Group("D7xC15");
// GroupNames label

G:=SmallGroup(210,5);
// by ID

G=gap.SmallGroup(210,5);
# by ID

G:=PCGroup([4,-2,-3,-5,-7,2883]);
// Polycyclic

G:=Group<a,b,c|a^15=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D7×C15 in TeX

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