direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C5×C19⋊C3, C95⋊C3, C19⋊C15, SmallGroup(285,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C5×C19⋊C3 |
Generators and relations for C5×C19⋊C3
G = < a,b,c | a5=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >
Smallest permutation representation of C5×C19⋊C3
►On 95 points
Generators in S95
(1 77 58 39 20)(2 78 59 40 21)(3 79 60 41 22)(4 80 61 42 23)(5 81 62 43 24)(6 82 63 44 25)(7 83 64 45 26)(8 84 65 46 27)(9 85 66 47 28)(10 86 67 48 29)(11 87 68 49 30)(12 88 69 50 31)(13 89 70 51 32)(14 90 71 52 33)(15 91 72 53 34)(16 92 73 54 35)(17 93 74 55 36)(18 94 75 56 37)(19 95 76 57 38)
(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)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)(78 84 88)(79 91 80)(81 86 83)(82 93 94)(85 95 89)(87 90 92)
G:=sub<Sym(95)| (1,77,58,39,20)(2,78,59,40,21)(3,79,60,41,22)(4,80,61,42,23)(5,81,62,43,24)(6,82,63,44,25)(7,83,64,45,26)(8,84,65,46,27)(9,85,66,47,28)(10,86,67,48,29)(11,87,68,49,30)(12,88,69,50,31)(13,89,70,51,32)(14,90,71,52,33)(15,91,72,53,34)(16,92,73,54,35)(17,93,74,55,36)(18,94,75,56,37)(19,95,76,57,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)>;
G:=Group( (1,77,58,39,20)(2,78,59,40,21)(3,79,60,41,22)(4,80,61,42,23)(5,81,62,43,24)(6,82,63,44,25)(7,83,64,45,26)(8,84,65,46,27)(9,85,66,47,28)(10,86,67,48,29)(11,87,68,49,30)(12,88,69,50,31)(13,89,70,51,32)(14,90,71,52,33)(15,91,72,53,34)(16,92,73,54,35)(17,93,74,55,36)(18,94,75,56,37)(19,95,76,57,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92) );
G=PermutationGroup([[(1,77,58,39,20),(2,78,59,40,21),(3,79,60,41,22),(4,80,61,42,23),(5,81,62,43,24),(6,82,63,44,25),(7,83,64,45,26),(8,84,65,46,27),(9,85,66,47,28),(10,86,67,48,29),(11,87,68,49,30),(12,88,69,50,31),(13,89,70,51,32),(14,90,71,52,33),(15,91,72,53,34),(16,92,73,54,35),(17,93,74,55,36),(18,94,75,56,37),(19,95,76,57,38)], [(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)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54),(59,65,69),(60,72,61),(62,67,64),(63,74,75),(66,76,70),(68,71,73),(78,84,88),(79,91,80),(81,86,83),(82,93,94),(85,95,89),(87,90,92)]])
45 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 15A | ··· | 15H | 19A | ··· | 19F | 95A | ··· | 95X |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 19 | ··· | 19 | 95 | ··· | 95 |
| size | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 3 | ··· | 3 | 3 | ··· | 3 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C19⋊C3 | C5×C19⋊C3 |
| kernel | C5×C19⋊C3 | C95 | C19⋊C3 | C19 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 6 | 24 |
Matrix representation of C5×C19⋊C3 ►in GL3(𝔽11) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 3 | 9 |
| 0 | 8 | 7 |
| 3 | 2 | 5 |
| 1 | 3 | 8 |
| 0 | 6 | 5 |
| 0 | 9 | 4 |
G:=sub<GL(3,GF(11))| [4,0,0,0,4,0,0,0,4],[0,0,3,3,8,2,9,7,5],[1,0,0,3,6,9,8,5,4] >;
C5×C19⋊C3 in GAP, Magma, Sage, TeX
C_5\times C_{19}\rtimes C_3 % in TeX
G:=Group("C5xC19:C3"); // GroupNames label
G:=SmallGroup(285,1);
// by ID
G=gap.SmallGroup(285,1);
# by ID
G:=PCGroup([3,-3,-5,-19,947]);
// Polycyclic
G:=Group<a,b,c|a^5=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations
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