direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×D21, C21⋊3D7, C72⋊2S3, C21⋊1C14, C3⋊(C7×D7), C7⋊(S3×C7), (C7×C21)⋊2C2, SmallGroup(294,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C7×D21 |
Generators and relations for C7×D21
G = < a,b,c | a7=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D21
►On 42 points
Generators in S42
(1 13 4 16 7 19 10)(2 14 5 17 8 20 11)(3 15 6 18 9 21 12)(22 31 40 28 37 25 34)(23 32 41 29 38 26 35)(24 33 42 30 39 27 36)
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)
G:=sub<Sym(42)| (1,13,4,16,7,19,10)(2,14,5,17,8,20,11)(3,15,6,18,9,21,12)(22,31,40,28,37,25,34)(23,32,41,29,38,26,35)(24,33,42,30,39,27,36), (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)>;
G:=Group( (1,13,4,16,7,19,10)(2,14,5,17,8,20,11)(3,15,6,18,9,21,12)(22,31,40,28,37,25,34)(23,32,41,29,38,26,35)(24,33,42,30,39,27,36), (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35) );
G=PermutationGroup([[(1,13,4,16,7,19,10),(2,14,5,17,8,20,11),(3,15,6,18,9,21,12),(22,31,40,28,37,25,34),(23,32,41,29,38,26,35),(24,33,42,30,39,27,36)], [(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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35)]])
84 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F | 21A | ··· | 21AV |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 |
| size | 1 | 21 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 21 | ··· | 21 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C7 | C14 | S3 | D7 | S3×C7 | D21 | C7×D7 | C7×D21 |
| kernel | C7×D21 | C7×C21 | D21 | C21 | C72 | C21 | C7 | C7 | C3 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 3 | 6 | 6 | 18 | 36 |
Matrix representation of C7×D21 ►in GL2(𝔽43) generated by
| 4 | 0 |
| 0 | 4 |
| 25 | 0 |
| 6 | 31 |
| 35 | 35 |
| 24 | 8 |
G:=sub<GL(2,GF(43))| [4,0,0,4],[25,6,0,31],[35,24,35,8] >;
C7×D21 in GAP, Magma, Sage, TeX
C_7\times D_{21} % in TeX
G:=Group("C7xD21"); // GroupNames label
G:=SmallGroup(294,21);
// by ID
G=gap.SmallGroup(294,21);
# by ID
G:=PCGroup([4,-2,-7,-3,-7,338,4035]);
// Polycyclic
G:=Group<a,b,c|a^7=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export