metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19⋊F5, C95⋊1C4, C5⋊Dic19, D5.D19, (D5×C19).1C2, SmallGroup(380,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C95 — C19⋊F5 |
Generators and relations for C19⋊F5
G = < a,b,c | a19=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Smallest permutation representation of C19⋊F5
►On 95 points
Generators in S95
(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)
(1 31 47 59 81)(2 32 48 60 82)(3 33 49 61 83)(4 34 50 62 84)(5 35 51 63 85)(6 36 52 64 86)(7 37 53 65 87)(8 38 54 66 88)(9 20 55 67 89)(10 21 56 68 90)(11 22 57 69 91)(12 23 39 70 92)(13 24 40 71 93)(14 25 41 72 94)(15 26 42 73 95)(16 27 43 74 77)(17 28 44 75 78)(18 29 45 76 79)(19 30 46 58 80)
(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(20 39 89 70)(21 57 90 69)(22 56 91 68)(23 55 92 67)(24 54 93 66)(25 53 94 65)(26 52 95 64)(27 51 77 63)(28 50 78 62)(29 49 79 61)(30 48 80 60)(31 47 81 59)(32 46 82 58)(33 45 83 76)(34 44 84 75)(35 43 85 74)(36 42 86 73)(37 41 87 72)(38 40 88 71)
G:=sub<Sym(95)| (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), (1,31,47,59,81)(2,32,48,60,82)(3,33,49,61,83)(4,34,50,62,84)(5,35,51,63,85)(6,36,52,64,86)(7,37,53,65,87)(8,38,54,66,88)(9,20,55,67,89)(10,21,56,68,90)(11,22,57,69,91)(12,23,39,70,92)(13,24,40,71,93)(14,25,41,72,94)(15,26,42,73,95)(16,27,43,74,77)(17,28,44,75,78)(18,29,45,76,79)(19,30,46,58,80), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(20,39,89,70)(21,57,90,69)(22,56,91,68)(23,55,92,67)(24,54,93,66)(25,53,94,65)(26,52,95,64)(27,51,77,63)(28,50,78,62)(29,49,79,61)(30,48,80,60)(31,47,81,59)(32,46,82,58)(33,45,83,76)(34,44,84,75)(35,43,85,74)(36,42,86,73)(37,41,87,72)(38,40,88,71)>;
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), (1,31,47,59,81)(2,32,48,60,82)(3,33,49,61,83)(4,34,50,62,84)(5,35,51,63,85)(6,36,52,64,86)(7,37,53,65,87)(8,38,54,66,88)(9,20,55,67,89)(10,21,56,68,90)(11,22,57,69,91)(12,23,39,70,92)(13,24,40,71,93)(14,25,41,72,94)(15,26,42,73,95)(16,27,43,74,77)(17,28,44,75,78)(18,29,45,76,79)(19,30,46,58,80), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(20,39,89,70)(21,57,90,69)(22,56,91,68)(23,55,92,67)(24,54,93,66)(25,53,94,65)(26,52,95,64)(27,51,77,63)(28,50,78,62)(29,49,79,61)(30,48,80,60)(31,47,81,59)(32,46,82,58)(33,45,83,76)(34,44,84,75)(35,43,85,74)(36,42,86,73)(37,41,87,72)(38,40,88,71) );
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)], [(1,31,47,59,81),(2,32,48,60,82),(3,33,49,61,83),(4,34,50,62,84),(5,35,51,63,85),(6,36,52,64,86),(7,37,53,65,87),(8,38,54,66,88),(9,20,55,67,89),(10,21,56,68,90),(11,22,57,69,91),(12,23,39,70,92),(13,24,40,71,93),(14,25,41,72,94),(15,26,42,73,95),(16,27,43,74,77),(17,28,44,75,78),(18,29,45,76,79),(19,30,46,58,80)], [(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(20,39,89,70),(21,57,90,69),(22,56,91,68),(23,55,92,67),(24,54,93,66),(25,53,94,65),(26,52,95,64),(27,51,77,63),(28,50,78,62),(29,49,79,61),(30,48,80,60),(31,47,81,59),(32,46,82,58),(33,45,83,76),(34,44,84,75),(35,43,85,74),(36,42,86,73),(37,41,87,72),(38,40,88,71)]])
41 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 19A | ··· | 19I | 38A | ··· | 38I | 95A | ··· | 95R |
| order | 1 | 2 | 4 | 4 | 5 | 19 | ··· | 19 | 38 | ··· | 38 | 95 | ··· | 95 |
| size | 1 | 5 | 95 | 95 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D19 | Dic19 | F5 | C19⋊F5 |
| kernel | C19⋊F5 | D5×C19 | C95 | D5 | C5 | C19 | C1 |
| # reps | 1 | 1 | 2 | 9 | 9 | 1 | 18 |
Matrix representation of C19⋊F5 ►in GL4(𝔽761) generated by
| 50 | 328 | 0 | 0 |
| 760 | 100 | 0 | 0 |
| 0 | 0 | 50 | 328 |
| 0 | 0 | 760 | 100 |
| 0 | 0 | 760 | 0 |
| 0 | 0 | 0 | 760 |
| 1 | 0 | 712 | 545 |
| 0 | 1 | 409 | 48 |
| 323 | 127 | 438 | 634 |
| 335 | 438 | 426 | 323 |
| 0 | 0 | 89 | 382 |
| 0 | 0 | 553 | 672 |
G:=sub<GL(4,GF(761))| [50,760,0,0,328,100,0,0,0,0,50,760,0,0,328,100],[0,0,1,0,0,0,0,1,760,0,712,409,0,760,545,48],[323,335,0,0,127,438,0,0,438,426,89,553,634,323,382,672] >;
C19⋊F5 in GAP, Magma, Sage, TeX
C_{19}\rtimes F_5 % in TeX
G:=Group("C19:F5"); // GroupNames label
G:=SmallGroup(380,6);
// by ID
G=gap.SmallGroup(380,6);
# by ID
G:=PCGroup([4,-2,-2,-5,-19,8,146,102,5763]);
// Polycyclic
G:=Group<a,b,c|a^19=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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