direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D23, C23⋊C6, C69⋊2C2, SmallGroup(138,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C3×D23 |
Generators and relations for C3×D23
G = < a,b,c | a3=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D23
►On 69 points
Generators in S69
(1 68 29)(2 69 30)(3 47 31)(4 48 32)(5 49 33)(6 50 34)(7 51 35)(8 52 36)(9 53 37)(10 54 38)(11 55 39)(12 56 40)(13 57 41)(14 58 42)(15 59 43)(16 60 44)(17 61 45)(18 62 46)(19 63 24)(20 64 25)(21 65 26)(22 66 27)(23 67 28)
(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 33)(25 32)(26 31)(27 30)(28 29)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(66 69)(67 68)
G:=sub<Sym(69)| (1,68,29)(2,69,30)(3,47,31)(4,48,32)(5,49,33)(6,50,34)(7,51,35)(8,52,36)(9,53,37)(10,54,38)(11,55,39)(12,56,40)(13,57,41)(14,58,42)(15,59,43)(16,60,44)(17,61,45)(18,62,46)(19,63,24)(20,64,25)(21,65,26)(22,66,27)(23,67,28), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,33)(25,32)(26,31)(27,30)(28,29)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(66,69)(67,68)>;
G:=Group( (1,68,29)(2,69,30)(3,47,31)(4,48,32)(5,49,33)(6,50,34)(7,51,35)(8,52,36)(9,53,37)(10,54,38)(11,55,39)(12,56,40)(13,57,41)(14,58,42)(15,59,43)(16,60,44)(17,61,45)(18,62,46)(19,63,24)(20,64,25)(21,65,26)(22,66,27)(23,67,28), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,33)(25,32)(26,31)(27,30)(28,29)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(66,69)(67,68) );
G=PermutationGroup([(1,68,29),(2,69,30),(3,47,31),(4,48,32),(5,49,33),(6,50,34),(7,51,35),(8,52,36),(9,53,37),(10,54,38),(11,55,39),(12,56,40),(13,57,41),(14,58,42),(15,59,43),(16,60,44),(17,61,45),(18,62,46),(19,63,24),(20,64,25),(21,65,26),(22,66,27),(23,67,28)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,33),(25,32),(26,31),(27,30),(28,29),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(66,69),(67,68)])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 23A | ··· | 23K | 69A | ··· | 69V |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 23 | 1 | 1 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D23 | C3×D23 |
| kernel | C3×D23 | C69 | D23 | C23 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 11 | 22 |
Matrix representation of C3×D23 ►in GL2(𝔽139) generated by
| 42 | 0 |
| 0 | 42 |
| 0 | 1 |
| 138 | 123 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(139))| [42,0,0,42],[0,138,1,123],[0,1,1,0] >;
C3×D23 in GAP, Magma, Sage, TeX
C_3\times D_{23} % in TeX
G:=Group("C3xD23"); // GroupNames label
G:=SmallGroup(138,2);
// by ID
G=gap.SmallGroup(138,2);
# by ID
G:=PCGroup([3,-2,-3,-23,1190]);
// Polycyclic
G:=Group<a,b,c|a^3=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export