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## G = D7×D15order 420 = 22·3·5·7

### Direct product of D7 and D15

Aliases: D7×D15, C71D30, C351D6, C211D10, C154D14, D1053C2, C1053C22, (C5×D7)⋊S3, (C3×D7)⋊D5, C32(D5×D7), C52(S3×D7), (C7×D15)⋊1C2, (D7×C15)⋊1C2, SmallGroup(420,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105 — D7×D15
 Chief series C1 — C7 — C35 — C105 — D7×C15 — D7×D15
 Lower central C105 — D7×D15
 Upper central C1

Generators and relations for D7×D15
G = < a,b,c,d | a7=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

7C2
15C2
105C2
105C22
5S3
7C6
35S3
3D5
7C10
21D5
15D7
15C14
35D6
21D10
15D14
7D15
7C30
5D21
3D35
7D30

Smallest permutation representation of D7×D15
On 105 points
Generators in S105
(1 56 75 102 84 35 26)(2 57 61 103 85 36 27)(3 58 62 104 86 37 28)(4 59 63 105 87 38 29)(5 60 64 91 88 39 30)(6 46 65 92 89 40 16)(7 47 66 93 90 41 17)(8 48 67 94 76 42 18)(9 49 68 95 77 43 19)(10 50 69 96 78 44 20)(11 51 70 97 79 45 21)(12 52 71 98 80 31 22)(13 53 72 99 81 32 23)(14 54 73 100 82 33 24)(15 55 74 101 83 34 25)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)
(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 20)(17 19)(21 30)(22 29)(23 28)(24 27)(25 26)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 75)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 97)(92 96)(93 95)(98 105)(99 104)(100 103)(101 102)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 6 7A 7B 7C 10A 10B 14A 14B 14C 15A 15B 15C 15D 21A 21B 21C 30A 30B 30C 30D 35A ··· 35F 105A ··· 105L order 1 2 2 2 3 5 5 6 7 7 7 10 10 14 14 14 15 15 15 15 21 21 21 30 30 30 30 35 ··· 35 105 ··· 105 size 1 7 15 105 2 2 2 14 2 2 2 14 14 30 30 30 2 2 2 2 4 4 4 14 14 14 14 4 ··· 4 4 ··· 4

45 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 D15 D30 S3×D7 D5×D7 D7×D15 kernel D7×D15 D7×C15 C7×D15 D105 C5×D7 C3×D7 C35 D15 C21 C15 D7 C7 C5 C3 C1 # reps 1 1 1 1 1 2 1 3 2 3 4 4 3 6 12

Matrix representation of D7×D15 in GL4(𝔽211) generated by

 0 1 0 0 210 18 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 5 88 0 0 123 55
,
 1 0 0 0 0 1 0 0 0 0 210 178 0 0 0 1
G:=sub<GL(4,GF(211))| [0,210,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,5,123,0,0,88,55],[1,0,0,0,0,1,0,0,0,0,210,0,0,0,178,1] >;

D7×D15 in GAP, Magma, Sage, TeX

D_7\times D_{15}
% in TeX

G:=Group("D7xD15");
// GroupNames label

G:=SmallGroup(420,26);
// by ID

G=gap.SmallGroup(420,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,67,488,9004]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^15=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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