direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19×F5, C5⋊C76, C95⋊2C4, D5.C38, (D5×C19).2C2, SmallGroup(380,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C19×F5 |
Generators and relations for C19×F5
G = < a,b,c | a19=b5=c4=1, ab=ba, ac=ca, 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 33 87 76 56)(2 34 88 58 57)(3 35 89 59 39)(4 36 90 60 40)(5 37 91 61 41)(6 38 92 62 42)(7 20 93 63 43)(8 21 94 64 44)(9 22 95 65 45)(10 23 77 66 46)(11 24 78 67 47)(12 25 79 68 48)(13 26 80 69 49)(14 27 81 70 50)(15 28 82 71 51)(16 29 83 72 52)(17 30 84 73 53)(18 31 85 74 54)(19 32 86 75 55)
(20 93 43 63)(21 94 44 64)(22 95 45 65)(23 77 46 66)(24 78 47 67)(25 79 48 68)(26 80 49 69)(27 81 50 70)(28 82 51 71)(29 83 52 72)(30 84 53 73)(31 85 54 74)(32 86 55 75)(33 87 56 76)(34 88 57 58)(35 89 39 59)(36 90 40 60)(37 91 41 61)(38 92 42 62)
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), (1,33,87,76,56)(2,34,88,58,57)(3,35,89,59,39)(4,36,90,60,40)(5,37,91,61,41)(6,38,92,62,42)(7,20,93,63,43)(8,21,94,64,44)(9,22,95,65,45)(10,23,77,66,46)(11,24,78,67,47)(12,25,79,68,48)(13,26,80,69,49)(14,27,81,70,50)(15,28,82,71,51)(16,29,83,72,52)(17,30,84,73,53)(18,31,85,74,54)(19,32,86,75,55), (20,93,43,63)(21,94,44,64)(22,95,45,65)(23,77,46,66)(24,78,47,67)(25,79,48,68)(26,80,49,69)(27,81,50,70)(28,82,51,71)(29,83,52,72)(30,84,53,73)(31,85,54,74)(32,86,55,75)(33,87,56,76)(34,88,57,58)(35,89,39,59)(36,90,40,60)(37,91,41,61)(38,92,42,62)>;
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), (1,33,87,76,56)(2,34,88,58,57)(3,35,89,59,39)(4,36,90,60,40)(5,37,91,61,41)(6,38,92,62,42)(7,20,93,63,43)(8,21,94,64,44)(9,22,95,65,45)(10,23,77,66,46)(11,24,78,67,47)(12,25,79,68,48)(13,26,80,69,49)(14,27,81,70,50)(15,28,82,71,51)(16,29,83,72,52)(17,30,84,73,53)(18,31,85,74,54)(19,32,86,75,55), (20,93,43,63)(21,94,44,64)(22,95,45,65)(23,77,46,66)(24,78,47,67)(25,79,48,68)(26,80,49,69)(27,81,50,70)(28,82,51,71)(29,83,52,72)(30,84,53,73)(31,85,54,74)(32,86,55,75)(33,87,56,76)(34,88,57,58)(35,89,39,59)(36,90,40,60)(37,91,41,61)(38,92,42,62) );
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)], [(1,33,87,76,56),(2,34,88,58,57),(3,35,89,59,39),(4,36,90,60,40),(5,37,91,61,41),(6,38,92,62,42),(7,20,93,63,43),(8,21,94,64,44),(9,22,95,65,45),(10,23,77,66,46),(11,24,78,67,47),(12,25,79,68,48),(13,26,80,69,49),(14,27,81,70,50),(15,28,82,71,51),(16,29,83,72,52),(17,30,84,73,53),(18,31,85,74,54),(19,32,86,75,55)], [(20,93,43,63),(21,94,44,64),(22,95,45,65),(23,77,46,66),(24,78,47,67),(25,79,48,68),(26,80,49,69),(27,81,50,70),(28,82,51,71),(29,83,52,72),(30,84,53,73),(31,85,54,74),(32,86,55,75),(33,87,56,76),(34,88,57,58),(35,89,39,59),(36,90,40,60),(37,91,41,61),(38,92,42,62)]])
95 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 19A | ··· | 19R | 38A | ··· | 38R | 76A | ··· | 76AJ | 95A | ··· | 95R |
| order | 1 | 2 | 4 | 4 | 5 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 95 | ··· | 95 |
| size | 1 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C19 | C38 | C76 | F5 | C19×F5 |
| kernel | C19×F5 | D5×C19 | C95 | F5 | D5 | C5 | C19 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 | 1 | 18 |
Matrix representation of C19×F5 ►in GL4(𝔽761) generated by
| 680 | 0 | 0 | 0 |
| 0 | 680 | 0 | 0 |
| 0 | 0 | 680 | 0 |
| 0 | 0 | 0 | 680 |
| 760 | 760 | 760 | 760 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 760 | 760 | 760 | 760 |
G:=sub<GL(4,GF(761))| [680,0,0,0,0,680,0,0,0,0,680,0,0,0,0,680],[760,1,0,0,760,0,1,0,760,0,0,1,760,0,0,0],[1,0,0,760,0,0,1,760,0,0,0,760,0,1,0,760] >;
C19×F5 in GAP, Magma, Sage, TeX
C_{19}\times F_5 % in TeX
G:=Group("C19xF5"); // GroupNames label
G:=SmallGroup(380,5);
// by ID
G=gap.SmallGroup(380,5);
# by ID
G:=PCGroup([4,-2,-19,-2,-5,152,2435,139]);
// Polycyclic
G:=Group<a,b,c|a^19=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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