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 54 38)(2 55 39)(3 56 40)(4 57 41)(5 58 42)(6 59 43)(7 60 44)(8 61 45)(9 62 46)(10 63 24)(11 64 25)(12 65 26)(13 66 27)(14 67 28)(15 68 29)(16 69 30)(17 47 31)(18 48 32)(19 49 33)(20 50 34)(21 51 35)(22 52 36)(23 53 37)
(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 28)(25 27)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 69)(62 68)(63 67)(64 66)
G:=sub<Sym(69)| (1,54,38)(2,55,39)(3,56,40)(4,57,41)(5,58,42)(6,59,43)(7,60,44)(8,61,45)(9,62,46)(10,63,24)(11,64,25)(12,65,26)(13,66,27)(14,67,28)(15,68,29)(16,69,30)(17,47,31)(18,48,32)(19,49,33)(20,50,34)(21,51,35)(22,52,36)(23,53,37), (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)(24,28)(25,27)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,69)(62,68)(63,67)(64,66)>;
G:=Group( (1,54,38)(2,55,39)(3,56,40)(4,57,41)(5,58,42)(6,59,43)(7,60,44)(8,61,45)(9,62,46)(10,63,24)(11,64,25)(12,65,26)(13,66,27)(14,67,28)(15,68,29)(16,69,30)(17,47,31)(18,48,32)(19,49,33)(20,50,34)(21,51,35)(22,52,36)(23,53,37), (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)(24,28)(25,27)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,69)(62,68)(63,67)(64,66) );
G=PermutationGroup([[(1,54,38),(2,55,39),(3,56,40),(4,57,41),(5,58,42),(6,59,43),(7,60,44),(8,61,45),(9,62,46),(10,63,24),(11,64,25),(12,65,26),(13,66,27),(14,67,28),(15,68,29),(16,69,30),(17,47,31),(18,48,32),(19,49,33),(20,50,34),(21,51,35),(22,52,36),(23,53,37)], [(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),(24,28),(25,27),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(37,38),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,69),(62,68),(63,67),(64,66)]])
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