direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D19, C19⋊3C6, C57⋊2C2, SmallGroup(114,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C3×D19 |
Generators and relations for C3×D19
G = < a,b,c | a3=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D19
►On 57 points
Generators in S57
(1 45 33)(2 46 34)(3 47 35)(4 48 36)(5 49 37)(6 50 38)(7 51 20)(8 52 21)(9 53 22)(10 54 23)(11 55 24)(12 56 25)(13 57 26)(14 39 27)(15 40 28)(16 41 29)(17 42 30)(18 43 31)(19 44 32)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 26)(21 25)(22 24)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(51 57)(52 56)(53 55)
G:=sub<Sym(57)| (1,45,33)(2,46,34)(3,47,35)(4,48,36)(5,49,37)(6,50,38)(7,51,20)(8,52,21)(9,53,22)(10,54,23)(11,55,24)(12,56,25)(13,57,26)(14,39,27)(15,40,28)(16,41,29)(17,42,30)(18,43,31)(19,44,32), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(51,57)(52,56)(53,55)>;
G:=Group( (1,45,33)(2,46,34)(3,47,35)(4,48,36)(5,49,37)(6,50,38)(7,51,20)(8,52,21)(9,53,22)(10,54,23)(11,55,24)(12,56,25)(13,57,26)(14,39,27)(15,40,28)(16,41,29)(17,42,30)(18,43,31)(19,44,32), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(51,57)(52,56)(53,55) );
G=PermutationGroup([[(1,45,33),(2,46,34),(3,47,35),(4,48,36),(5,49,37),(6,50,38),(7,51,20),(8,52,21),(9,53,22),(10,54,23),(11,55,24),(12,56,25),(13,57,26),(14,39,27),(15,40,28),(16,41,29),(17,42,30),(18,43,31),(19,44,32)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,26),(21,25),(22,24),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(51,57),(52,56),(53,55)]])
C3×D19 is a maximal subgroup of
C57.C6
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 19A | ··· | 19I | 57A | ··· | 57R |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 19 | ··· | 19 | 57 | ··· | 57 |
| size | 1 | 19 | 1 | 1 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D19 | C3×D19 |
| kernel | C3×D19 | C57 | D19 | C19 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 9 | 18 |
Matrix representation of C3×D19 ►in GL2(𝔽37) generated by
| 10 | 0 |
| 0 | 10 |
| 3 | 5 |
| 24 | 28 |
| 28 | 28 |
| 13 | 9 |
G:=sub<GL(2,GF(37))| [10,0,0,10],[3,24,5,28],[28,13,28,9] >;
C3×D19 in GAP, Magma, Sage, TeX
C_3\times D_{19} % in TeX
G:=Group("C3xD19"); // GroupNames label
G:=SmallGroup(114,4);
// by ID
G=gap.SmallGroup(114,4);
# by ID
G:=PCGroup([3,-2,-3,-19,974]);
// Polycyclic
G:=Group<a,b,c|a^3=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export