direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D23, D69⋊C2, C23⋊1D6, C3⋊1D46, C69⋊C22, (S3×C23)⋊C2, (C3×D23)⋊C2, SmallGroup(276,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C69 — S3×D23 |
Generators and relations for S3×D23
G = < a,b,c,d | a3=b2=c23=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of S3×D23
►On 69 points
Generators in S69
(1 38 50)(2 39 51)(3 40 52)(4 41 53)(5 42 54)(6 43 55)(7 44 56)(8 45 57)(9 46 58)(10 24 59)(11 25 60)(12 26 61)(13 27 62)(14 28 63)(15 29 64)(16 30 65)(17 31 66)(18 32 67)(19 33 68)(20 34 69)(21 35 47)(22 36 48)(23 37 49)
(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)
(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 52)(48 51)(49 50)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
G:=sub<Sym(69)| (1,38,50)(2,39,51)(3,40,52)(4,41,53)(5,42,54)(6,43,55)(7,44,56)(8,45,57)(9,46,58)(10,24,59)(11,25,60)(12,26,61)(13,27,62)(14,28,63)(15,29,64)(16,30,65)(17,31,66)(18,32,67)(19,33,68)(20,34,69)(21,35,47)(22,36,48)(23,37,49), (24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58), (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)(47,52)(48,51)(49,50)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)>;
G:=Group( (1,38,50)(2,39,51)(3,40,52)(4,41,53)(5,42,54)(6,43,55)(7,44,56)(8,45,57)(9,46,58)(10,24,59)(11,25,60)(12,26,61)(13,27,62)(14,28,63)(15,29,64)(16,30,65)(17,31,66)(18,32,67)(19,33,68)(20,34,69)(21,35,47)(22,36,48)(23,37,49), (24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58), (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)(47,52)(48,51)(49,50)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62) );
G=PermutationGroup([[(1,38,50),(2,39,51),(3,40,52),(4,41,53),(5,42,54),(6,43,55),(7,44,56),(8,45,57),(9,46,58),(10,24,59),(11,25,60),(12,26,61),(13,27,62),(14,28,63),(15,29,64),(16,30,65),(17,31,66),(18,32,67),(19,33,68),(20,34,69),(21,35,47),(22,36,48),(23,37,49)], [(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58)], [(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),(47,52),(48,51),(49,50),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 23A | ··· | 23K | 46A | ··· | 46K | 69A | ··· | 69K |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 |
| size | 1 | 3 | 23 | 69 | 2 | 46 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D23 | D46 | S3×D23 |
| kernel | S3×D23 | S3×C23 | C3×D23 | D69 | D23 | C23 | S3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 11 | 11 |
Matrix representation of S3×D23 ►in GL4(𝔽139) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 123 |
| 0 | 0 | 61 | 137 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 138 | 16 |
| 0 | 0 | 0 | 1 |
| 29 | 1 | 0 | 0 |
| 97 | 128 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 104 | 134 | 0 | 0 |
| 78 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(139))| [1,0,0,0,0,1,0,0,0,0,1,61,0,0,123,137],[1,0,0,0,0,1,0,0,0,0,138,0,0,0,16,1],[29,97,0,0,1,128,0,0,0,0,1,0,0,0,0,1],[104,78,0,0,134,35,0,0,0,0,1,0,0,0,0,1] >;
S3×D23 in GAP, Magma, Sage, TeX
S_3\times D_{23} % in TeX
G:=Group("S3xD23"); // GroupNames label
G:=SmallGroup(276,5);
// by ID
G=gap.SmallGroup(276,5);
# by ID
G:=PCGroup([4,-2,-2,-3,-23,54,4227]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^23=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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