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G = C7×D15order 210 = 2·3·5·7

Direct product of C7 and D15

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

Aliases: C7×D15, C353S3, C213D5, C151C14, C1054C2, C5⋊(S3×C7), C3⋊(C7×D5), SmallGroup(210,10)

Series: Derived Chief Lower central Upper central

C1C15 — C7×D15
C1C5C15C105 — C7×D15
C15 — C7×D15
C1C7

Generators and relations for C7×D15
 G = < a,b,c | a7=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
5S3
3D5
15C14
5S3×C7
3C7×D5

Smallest permutation representation of C7×D15
On 105 points
Generators in S105
(1 91 88 65 47 35 16)(2 92 89 66 48 36 17)(3 93 90 67 49 37 18)(4 94 76 68 50 38 19)(5 95 77 69 51 39 20)(6 96 78 70 52 40 21)(7 97 79 71 53 41 22)(8 98 80 72 54 42 23)(9 99 81 73 55 43 24)(10 100 82 74 56 44 25)(11 101 83 75 57 45 26)(12 102 84 61 58 31 27)(13 103 85 62 59 32 28)(14 104 86 63 60 33 29)(15 105 87 64 46 34 30)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 47)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 68)(62 67)(63 66)(64 65)(69 75)(70 74)(71 73)(76 84)(77 83)(78 82)(79 81)(85 90)(86 89)(87 88)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)

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

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

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

C7×D15 is a maximal subgroup of   S3×C7×D5  D15⋊D7

63 conjugacy classes

class 1  2  3 5A5B7A···7F14A···14F15A15B15C15D21A···21F35A···35L105A···105X
order123557···714···141515151521···2135···35105···105
size1152221···115···1522222···22···22···2

63 irreducible representations

dim1111222222
type+++++
imageC1C2C7C14S3D5D15S3×C7C7×D5C7×D15
kernelC7×D15C105D15C15C35C21C7C5C3C1
# reps116612461224

Matrix representation of C7×D15 in GL2(𝔽29) generated by

70
07
,
116
1622
,
223
137
G:=sub<GL(2,GF(29))| [7,0,0,7],[11,16,6,22],[22,13,3,7] >;

C7×D15 in GAP, Magma, Sage, TeX

C_7\times D_{15}
% in TeX

G:=Group("C7xD15");
// GroupNames label

G:=SmallGroup(210,10);
// by ID

G=gap.SmallGroup(210,10);
# by ID

G:=PCGroup([4,-2,-7,-3,-5,338,2691]);
// Polycyclic

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

Export

Subgroup lattice of C7×D15 in TeX

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