direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D15, C35⋊3S3, C21⋊3D5, C15⋊1C14, C105⋊4C2, C5⋊(S3×C7), C3⋊(C7×D5), SmallGroup(210,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C7×D15 |
Generators and relations for C7×D15
G = < a,b,c | a7=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D15
►On 105 points
Generators in S105
(1 92 77 69 58 31 17)(2 93 78 70 59 32 18)(3 94 79 71 60 33 19)(4 95 80 72 46 34 20)(5 96 81 73 47 35 21)(6 97 82 74 48 36 22)(7 98 83 75 49 37 23)(8 99 84 61 50 38 24)(9 100 85 62 51 39 25)(10 101 86 63 52 40 26)(11 102 87 64 53 41 27)(12 103 88 65 54 42 28)(13 104 89 66 55 43 29)(14 105 90 67 56 44 30)(15 91 76 68 57 45 16)
(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 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(76 77)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(91 92)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
G:=sub<Sym(105)| (1,92,77,69,58,31,17)(2,93,78,70,59,32,18)(3,94,79,71,60,33,19)(4,95,80,72,46,34,20)(5,96,81,73,47,35,21)(6,97,82,74,48,36,22)(7,98,83,75,49,37,23)(8,99,84,61,50,38,24)(9,100,85,62,51,39,25)(10,101,86,63,52,40,26)(11,102,87,64,53,41,27)(12,103,88,65,54,42,28)(13,104,89,66,55,43,29)(14,105,90,67,56,44,30)(15,91,76,68,57,45,16), (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)(16,17)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(76,77)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(91,92)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)>;
G:=Group( (1,92,77,69,58,31,17)(2,93,78,70,59,32,18)(3,94,79,71,60,33,19)(4,95,80,72,46,34,20)(5,96,81,73,47,35,21)(6,97,82,74,48,36,22)(7,98,83,75,49,37,23)(8,99,84,61,50,38,24)(9,100,85,62,51,39,25)(10,101,86,63,52,40,26)(11,102,87,64,53,41,27)(12,103,88,65,54,42,28)(13,104,89,66,55,43,29)(14,105,90,67,56,44,30)(15,91,76,68,57,45,16), (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)(16,17)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(76,77)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(91,92)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100) );
G=PermutationGroup([[(1,92,77,69,58,31,17),(2,93,78,70,59,32,18),(3,94,79,71,60,33,19),(4,95,80,72,46,34,20),(5,96,81,73,47,35,21),(6,97,82,74,48,36,22),(7,98,83,75,49,37,23),(8,99,84,61,50,38,24),(9,100,85,62,51,39,25),(10,101,86,63,52,40,26),(11,102,87,64,53,41,27),(12,103,88,65,54,42,28),(13,104,89,66,55,43,29),(14,105,90,67,56,44,30),(15,91,76,68,57,45,16)], [(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),(16,17),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(76,77),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(91,92),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)]])
C7×D15 is a maximal subgroup of
S3×C7×D5 D15⋊D7
63 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 7A | ··· | 7F | 14A | ··· | 14F | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 35A | ··· | 35L | 105A | ··· | 105X |
| order | 1 | 2 | 3 | 5 | 5 | 7 | ··· | 7 | 14 | ··· | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 105 | ··· | 105 |
| size | 1 | 15 | 2 | 2 | 2 | 1 | ··· | 1 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C7 | C14 | S3 | D5 | D15 | S3×C7 | C7×D5 | C7×D15 |
| kernel | C7×D15 | C105 | D15 | C15 | C35 | C21 | C7 | C5 | C3 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 2 | 4 | 6 | 12 | 24 |
Matrix representation of C7×D15 ►in GL2(𝔽29) generated by
| 7 | 0 |
| 0 | 7 |
| 11 | 6 |
| 16 | 22 |
| 22 | 3 |
| 13 | 7 |
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
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