direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3xD21, C21:5C6, C21:2S3, C32:1D7, C3:(C3xD7), C7:3(C3xS3), (C3xC21):2C2, SmallGroup(126,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C3xD21 |
Generators and relations for C3xD21
G = < a,b,c | a3=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, S3, C6, D7, C3xS3, C3xD7, D21, C3xD21
Smallest permutation representation of C3xD21
►On 42 points
Generators in S42
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)
G:=sub<Sym(42)| (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)>;
G:=Group( (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33) );
G=PermutationGroup([[(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33)]])
C3xD21 is a maximal subgroup of
C3xS3xD7 D21:S3 D21:C9 He3:D7 D63:C3 C32:D21
C3xD21 is a maximal quotient of He3:D7 D63:C3
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | 7B | 7C | 21A | ··· | 21X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 21 | ··· | 21 |
| size | 1 | 21 | 1 | 1 | 2 | 2 | 2 | 21 | 21 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | D7 | C3xS3 | C3xD7 | D21 | C3xD21 |
| kernel | C3xD21 | C3xC21 | D21 | C21 | C21 | C32 | C7 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 6 | 12 |
Matrix representation of C3xD21 ►in GL2(F43) generated by
| 36 | 0 |
| 0 | 36 |
| 23 | 0 |
| 0 | 15 |
| 0 | 15 |
| 23 | 0 |
G:=sub<GL(2,GF(43))| [36,0,0,36],[23,0,0,15],[0,23,15,0] >;
C3xD21 in GAP, Magma, Sage, TeX
C_3\times D_{21} % in TeX
G:=Group("C3xD21"); // GroupNames label
G:=SmallGroup(126,13);
// by ID
G=gap.SmallGroup(126,13);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,146,1731]);
// Polycyclic
G:=Group<a,b,c|a^3=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export