direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C15×C7⋊C3, C105⋊C3, C21⋊C15, C35⋊C32, C7⋊(C3×C15), SmallGroup(315,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C15×C7⋊C3 |
Generators and relations for C15×C7⋊C3
G = < a,b,c | a15=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C15×C7⋊C3
►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 59 102 20 35 74 81)(2 60 103 21 36 75 82)(3 46 104 22 37 61 83)(4 47 105 23 38 62 84)(5 48 91 24 39 63 85)(6 49 92 25 40 64 86)(7 50 93 26 41 65 87)(8 51 94 27 42 66 88)(9 52 95 28 43 67 89)(10 53 96 29 44 68 90)(11 54 97 30 45 69 76)(12 55 98 16 31 70 77)(13 56 99 17 32 71 78)(14 57 100 18 33 72 79)(15 58 101 19 34 73 80)
(16 77 70)(17 78 71)(18 79 72)(19 80 73)(20 81 74)(21 82 75)(22 83 61)(23 84 62)(24 85 63)(25 86 64)(26 87 65)(27 88 66)(28 89 67)(29 90 68)(30 76 69)(31 55 98)(32 56 99)(33 57 100)(34 58 101)(35 59 102)(36 60 103)(37 46 104)(38 47 105)(39 48 91)(40 49 92)(41 50 93)(42 51 94)(43 52 95)(44 53 96)(45 54 97)
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), (1,59,102,20,35,74,81)(2,60,103,21,36,75,82)(3,46,104,22,37,61,83)(4,47,105,23,38,62,84)(5,48,91,24,39,63,85)(6,49,92,25,40,64,86)(7,50,93,26,41,65,87)(8,51,94,27,42,66,88)(9,52,95,28,43,67,89)(10,53,96,29,44,68,90)(11,54,97,30,45,69,76)(12,55,98,16,31,70,77)(13,56,99,17,32,71,78)(14,57,100,18,33,72,79)(15,58,101,19,34,73,80), (16,77,70)(17,78,71)(18,79,72)(19,80,73)(20,81,74)(21,82,75)(22,83,61)(23,84,62)(24,85,63)(25,86,64)(26,87,65)(27,88,66)(28,89,67)(29,90,68)(30,76,69)(31,55,98)(32,56,99)(33,57,100)(34,58,101)(35,59,102)(36,60,103)(37,46,104)(38,47,105)(39,48,91)(40,49,92)(41,50,93)(42,51,94)(43,52,95)(44,53,96)(45,54,97)>;
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), (1,59,102,20,35,74,81)(2,60,103,21,36,75,82)(3,46,104,22,37,61,83)(4,47,105,23,38,62,84)(5,48,91,24,39,63,85)(6,49,92,25,40,64,86)(7,50,93,26,41,65,87)(8,51,94,27,42,66,88)(9,52,95,28,43,67,89)(10,53,96,29,44,68,90)(11,54,97,30,45,69,76)(12,55,98,16,31,70,77)(13,56,99,17,32,71,78)(14,57,100,18,33,72,79)(15,58,101,19,34,73,80), (16,77,70)(17,78,71)(18,79,72)(19,80,73)(20,81,74)(21,82,75)(22,83,61)(23,84,62)(24,85,63)(25,86,64)(26,87,65)(27,88,66)(28,89,67)(29,90,68)(30,76,69)(31,55,98)(32,56,99)(33,57,100)(34,58,101)(35,59,102)(36,60,103)(37,46,104)(38,47,105)(39,48,91)(40,49,92)(41,50,93)(42,51,94)(43,52,95)(44,53,96)(45,54,97) );
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)], [(1,59,102,20,35,74,81),(2,60,103,21,36,75,82),(3,46,104,22,37,61,83),(4,47,105,23,38,62,84),(5,48,91,24,39,63,85),(6,49,92,25,40,64,86),(7,50,93,26,41,65,87),(8,51,94,27,42,66,88),(9,52,95,28,43,67,89),(10,53,96,29,44,68,90),(11,54,97,30,45,69,76),(12,55,98,16,31,70,77),(13,56,99,17,32,71,78),(14,57,100,18,33,72,79),(15,58,101,19,34,73,80)], [(16,77,70),(17,78,71),(18,79,72),(19,80,73),(20,81,74),(21,82,75),(22,83,61),(23,84,62),(24,85,63),(25,86,64),(26,87,65),(27,88,66),(28,89,67),(29,90,68),(30,76,69),(31,55,98),(32,56,99),(33,57,100),(34,58,101),(35,59,102),(36,60,103),(37,46,104),(38,47,105),(39,48,91),(40,49,92),(41,50,93),(42,51,94),(43,52,95),(44,53,96),(45,54,97)]])
75 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 5A | 5B | 5C | 5D | 7A | 7B | 15A | ··· | 15H | 15I | ··· | 15AF | 21A | 21B | 21C | 21D | 35A | ··· | 35H | 105A | ··· | 105P |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 7 | 7 | 15 | ··· | 15 | 15 | ··· | 15 | 21 | 21 | 21 | 21 | 35 | ··· | 35 | 105 | ··· | 105 |
| size | 1 | 1 | 1 | 7 | ··· | 7 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C3 | C5 | C15 | C15 | C7⋊C3 | C3×C7⋊C3 | C5×C7⋊C3 | C15×C7⋊C3 |
| kernel | C15×C7⋊C3 | C5×C7⋊C3 | C105 | C3×C7⋊C3 | C7⋊C3 | C21 | C15 | C5 | C3 | C1 |
| # reps | 1 | 6 | 2 | 4 | 24 | 8 | 2 | 4 | 8 | 16 |
Matrix representation of C15×C7⋊C3 ►in GL4(𝔽211) generated by
| 14 | 0 | 0 | 0 |
| 0 | 188 | 0 | 0 |
| 0 | 0 | 188 | 0 |
| 0 | 0 | 0 | 188 |
| 1 | 0 | 0 | 0 |
| 0 | 210 | 20 | 1 |
| 0 | 0 | 20 | 1 |
| 0 | 210 | 21 | 1 |
| 14 | 0 | 0 | 0 |
| 0 | 21 | 1 | 191 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 190 |
G:=sub<GL(4,GF(211))| [14,0,0,0,0,188,0,0,0,0,188,0,0,0,0,188],[1,0,0,0,0,210,0,210,0,20,20,21,0,1,1,1],[14,0,0,0,0,21,1,1,0,1,0,1,0,191,0,190] >;
C15×C7⋊C3 in GAP, Magma, Sage, TeX
C_{15}\times C_7\rtimes C_3 % in TeX
G:=Group("C15xC7:C3"); // GroupNames label
G:=SmallGroup(315,3);
// by ID
G=gap.SmallGroup(315,3);
# by ID
G:=PCGroup([4,-3,-3,-5,-7,1443]);
// Polycyclic
G:=Group<a,b,c|a^15=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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