direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C19, C5⋊C38, C95⋊3C2, SmallGroup(190,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C19 |
Generators and relations for D5×C19
G = < a,b,c | a19=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C19
►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 21 71 57 83)(2 22 72 39 84)(3 23 73 40 85)(4 24 74 41 86)(5 25 75 42 87)(6 26 76 43 88)(7 27 58 44 89)(8 28 59 45 90)(9 29 60 46 91)(10 30 61 47 92)(11 31 62 48 93)(12 32 63 49 94)(13 33 64 50 95)(14 34 65 51 77)(15 35 66 52 78)(16 36 67 53 79)(17 37 68 54 80)(18 38 69 55 81)(19 20 70 56 82)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 56)(21 57)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)
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), (1,21,71,57,83)(2,22,72,39,84)(3,23,73,40,85)(4,24,74,41,86)(5,25,75,42,87)(6,26,76,43,88)(7,27,58,44,89)(8,28,59,45,90)(9,29,60,46,91)(10,30,61,47,92)(11,31,62,48,93)(12,32,63,49,94)(13,33,64,50,95)(14,34,65,51,77)(15,35,66,52,78)(16,36,67,53,79)(17,37,68,54,80)(18,38,69,55,81)(19,20,70,56,82), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,56)(21,57)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)>;
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), (1,21,71,57,83)(2,22,72,39,84)(3,23,73,40,85)(4,24,74,41,86)(5,25,75,42,87)(6,26,76,43,88)(7,27,58,44,89)(8,28,59,45,90)(9,29,60,46,91)(10,30,61,47,92)(11,31,62,48,93)(12,32,63,49,94)(13,33,64,50,95)(14,34,65,51,77)(15,35,66,52,78)(16,36,67,53,79)(17,37,68,54,80)(18,38,69,55,81)(19,20,70,56,82), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,56)(21,57)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55) );
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)], [(1,21,71,57,83),(2,22,72,39,84),(3,23,73,40,85),(4,24,74,41,86),(5,25,75,42,87),(6,26,76,43,88),(7,27,58,44,89),(8,28,59,45,90),(9,29,60,46,91),(10,30,61,47,92),(11,31,62,48,93),(12,32,63,49,94),(13,33,64,50,95),(14,34,65,51,77),(15,35,66,52,78),(16,36,67,53,79),(17,37,68,54,80),(18,38,69,55,81),(19,20,70,56,82)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,56),(21,57),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55)]])
D5×C19 is a maximal subgroup of
C19⋊F5
76 conjugacy classes
| class | 1 | 2 | 5A | 5B | 19A | ··· | 19R | 38A | ··· | 38R | 95A | ··· | 95AJ |
| order | 1 | 2 | 5 | 5 | 19 | ··· | 19 | 38 | ··· | 38 | 95 | ··· | 95 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C19 | C38 | D5 | D5×C19 |
| kernel | D5×C19 | C95 | D5 | C5 | C19 | C1 |
| # reps | 1 | 1 | 18 | 18 | 2 | 36 |
Matrix representation of D5×C19 ►in GL2(𝔽191) generated by
| 121 | 0 |
| 0 | 121 |
| 88 | 1 |
| 190 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(191))| [121,0,0,121],[88,190,1,0],[0,1,1,0] >;
D5×C19 in GAP, Magma, Sage, TeX
D_5\times C_{19} % in TeX
G:=Group("D5xC19"); // GroupNames label
G:=SmallGroup(190,1);
// by ID
G=gap.SmallGroup(190,1);
# by ID
G:=PCGroup([3,-2,-19,-5,1370]);
// Polycyclic
G:=Group<a,b,c|a^19=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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