direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C21, C3⋊C42, C21⋊7C6, C32⋊1C14, (C3×C21)⋊4C2, SmallGroup(126,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C21 |
Generators and relations for S3×C21
G = < a,b,c | a21=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C21
►On 42 points
Generators in S42
(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)
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
G:=sub<Sym(42)| (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), (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)>;
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), (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42) );
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)], [(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,19),(6,13,20),(7,14,21),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42)]])
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 21M | ··· | 21AD | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | S3 | C3×S3 | S3×C7 | S3×C21 |
| kernel | S3×C21 | C3×C21 | S3×C7 | C21 | C3×S3 | C32 | S3 | C3 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of S3×C21 ►in GL2(𝔽43) generated by
| 40 | 0 |
| 0 | 40 |
| 36 | 0 |
| 0 | 6 |
| 0 | 11 |
| 4 | 0 |
G:=sub<GL(2,GF(43))| [40,0,0,40],[36,0,0,6],[0,4,11,0] >;
S3×C21 in GAP, Magma, Sage, TeX
S_3\times C_{21} % in TeX
G:=Group("S3xC21"); // GroupNames label
G:=SmallGroup(126,12);
// by ID
G=gap.SmallGroup(126,12);
# by ID
G:=PCGroup([4,-2,-3,-7,-3,1347]);
// Polycyclic
G:=Group<a,b,c|a^21=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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