direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C20, C60⋊6C2, D6.C10, C12⋊2C10, Dic3○C20, C10.14D6, Dic3⋊2C10, C30.19C22, C3⋊1(C2×C20), C15⋊9(C2×C4), C4○(C5×Dic3), C2.1(S3×C10), C6.2(C2×C10), C20○(C5×Dic3), (S3×C10).2C2, (C5×Dic3)⋊5C2, SmallGroup(120,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C20 |
Generators and relations for S3×C20
G = < a,b,c | a20=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C20
►On 60 points
Generators in S60
(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)
(1 21 48)(2 22 49)(3 23 50)(4 24 51)(5 25 52)(6 26 53)(7 27 54)(8 28 55)(9 29 56)(10 30 57)(11 31 58)(12 32 59)(13 33 60)(14 34 41)(15 35 42)(16 36 43)(17 37 44)(18 38 45)(19 39 46)(20 40 47)
(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
G:=sub<Sym(60)| (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), (1,21,48)(2,22,49)(3,23,50)(4,24,51)(5,25,52)(6,26,53)(7,27,54)(8,28,55)(9,29,56)(10,30,57)(11,31,58)(12,32,59)(13,33,60)(14,34,41)(15,35,42)(16,36,43)(17,37,44)(18,38,45)(19,39,46)(20,40,47), (21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)>;
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), (1,21,48)(2,22,49)(3,23,50)(4,24,51)(5,25,52)(6,26,53)(7,27,54)(8,28,55)(9,29,56)(10,30,57)(11,31,58)(12,32,59)(13,33,60)(14,34,41)(15,35,42)(16,36,43)(17,37,44)(18,38,45)(19,39,46)(20,40,47), (21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47) );
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)], [(1,21,48),(2,22,49),(3,23,50),(4,24,51),(5,25,52),(6,26,53),(7,27,54),(8,28,55),(9,29,56),(10,30,57),(11,31,58),(12,32,59),(13,33,60),(14,34,41),(15,35,42),(16,36,43),(17,37,44),(18,38,45),(19,39,46),(20,40,47)], [(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)]])
S3×C20 is a maximal subgroup of
D6.Dic5 D20⋊5S3 D60⋊C2 D6.D10
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D6 | C4×S3 | C5×S3 | S3×C10 | S3×C20 |
| kernel | S3×C20 | C5×Dic3 | C60 | S3×C10 | C5×S3 | C4×S3 | Dic3 | C12 | D6 | S3 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of S3×C20 ►in GL2(𝔽41) generated by
| 21 | 0 |
| 0 | 21 |
| 0 | 2 |
| 20 | 40 |
| 1 | 2 |
| 0 | 40 |
G:=sub<GL(2,GF(41))| [21,0,0,21],[0,20,2,40],[1,0,2,40] >;
S3×C20 in GAP, Magma, Sage, TeX
S_3\times C_{20} % in TeX
G:=Group("S3xC20"); // GroupNames label
G:=SmallGroup(120,22);
// by ID
G=gap.SmallGroup(120,22);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-3,106,2004]);
// Polycyclic
G:=Group<a,b,c|a^20=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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