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## G = D7×C15order 210 = 2·3·5·7

### Direct product of C15 and D7

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C15
 Chief series C1 — C7 — C35 — C105 — D7×C15
 Lower central C7 — D7×C15
 Upper central C1 — C15

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

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

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

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

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

75 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 6A 6B 7A 7B 7C 10A 10B 10C 10D 15A ··· 15H 21A ··· 21F 30A ··· 30H 35A ··· 35L 105A ··· 105X order 1 2 3 3 5 5 5 5 6 6 7 7 7 10 10 10 10 15 ··· 15 21 ··· 21 30 ··· 30 35 ··· 35 105 ··· 105 size 1 7 1 1 1 1 1 1 7 7 2 2 2 7 7 7 7 1 ··· 1 2 ··· 2 7 ··· 7 2 ··· 2 2 ··· 2

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C5 C6 C10 C15 C30 D7 C3×D7 C5×D7 D7×C15 kernel D7×C15 C105 C5×D7 C3×D7 C35 C21 D7 C7 C15 C5 C3 C1 # reps 1 1 2 4 2 4 8 8 3 6 12 24

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

 19 0 0 19
,
 0 1 210 81
,
 0 1 1 0
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

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