direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C10, C15⋊4C23, C30⋊4C22, C6⋊(C2×C10), C3⋊(C22×C10), (C2×C30)⋊7C2, (C2×C6)⋊3C10, SmallGroup(120,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C10 |
Generators and relations for S3×C2×C10
G = < a,b,c,d | a2=b10=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C10, C10, D6, C2×C6, C15, C2×C10, C2×C10, C22×S3, C5×S3, C30, C22×C10, S3×C10, C2×C30, S3×C2×C10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C2×C10, C22×S3, C5×S3, C22×C10, S3×C10, S3×C2×C10
►On 60 points
Generators in S60
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)
(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 49 39)(2 50 40)(3 41 31)(4 42 32)(5 43 33)(6 44 34)(7 45 35)(8 46 36)(9 47 37)(10 48 38)(11 30 51)(12 21 52)(13 22 53)(14 23 54)(15 24 55)(16 25 56)(17 26 57)(18 27 58)(19 28 59)(20 29 60)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 39)(12 40)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 50)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)
G:=sub<Sym(60)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39), (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), (1,49,39)(2,50,40)(3,41,31)(4,42,32)(5,43,33)(6,44,34)(7,45,35)(8,46,36)(9,47,37)(10,48,38)(11,30,51)(12,21,52)(13,22,53)(14,23,54)(15,24,55)(16,25,56)(17,26,57)(18,27,58)(19,28,59)(20,29,60), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,39)(12,40)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39), (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), (1,49,39)(2,50,40)(3,41,31)(4,42,32)(5,43,33)(6,44,34)(7,45,35)(8,46,36)(9,47,37)(10,48,38)(11,30,51)(12,21,52)(13,22,53)(14,23,54)(15,24,55)(16,25,56)(17,26,57)(18,27,58)(19,28,59)(20,29,60), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,39)(12,40)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,49),(12,50),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,40),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39)], [(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)], [(1,49,39),(2,50,40),(3,41,31),(4,42,32),(5,43,33),(6,44,34),(7,45,35),(8,46,36),(9,47,37),(10,48,38),(11,30,51),(12,21,52),(13,22,53),(14,23,54),(15,24,55),(16,25,56),(17,26,57),(18,27,58),(19,28,59),(20,29,60)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,39),(12,40),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,50),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49)]])
S3×C2×C10 is a maximal subgroup of
D6⋊Dic5
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | S3×C10 |
| kernel | S3×C2×C10 | S3×C10 | C2×C30 | C22×S3 | D6 | C2×C6 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 6 | 1 | 4 | 24 | 4 | 1 | 3 | 4 | 12 |
Matrix representation of S3×C2×C10 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 1 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 30 |
| 0 | 0 | 0 | 30 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30] >;
S3×C2×C10 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{10} % in TeX
G:=Group("S3xC2xC10"); // GroupNames label
G:=SmallGroup(120,45);
// by ID
G=gap.SmallGroup(120,45);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-3,2004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations