direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×D15, C7⋊1D30, C35⋊1D6, C21⋊1D10, C15⋊4D14, D105⋊3C2, C105⋊3C22, (C5×D7)⋊S3, (C3×D7)⋊D5, C3⋊2(D5×D7), C5⋊2(S3×D7), (C7×D15)⋊1C2, (D7×C15)⋊1C2, SmallGroup(420,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — D7×D15 |
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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