direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C15, C7⋊3C30, C35⋊6C6, C105⋊5C2, C21⋊2C10, SmallGroup(210,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×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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