direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D7×C21, C7⋊3C42, C21⋊2C14, C72⋊11C6, (C7×C21)⋊3C2, SmallGroup(294,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C21 |
Generators and relations for D7×C21
G = < a,b,c | a21=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×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 10 19 7 16 4 13)(2 11 20 8 17 5 14)(3 12 21 9 18 6 15)(22 34 25 37 28 40 31)(23 35 26 38 29 41 32)(24 36 27 39 30 42 33)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)
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), (1,10,19,7,16,4,13)(2,11,20,8,17,5,14)(3,12,21,9,18,6,15)(22,34,25,37,28,40,31)(23,35,26,38,29,41,32)(24,36,27,39,30,42,33), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)>;
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), (1,10,19,7,16,4,13)(2,11,20,8,17,5,14)(3,12,21,9,18,6,15)(22,34,25,37,28,40,31)(23,35,26,38,29,41,32)(24,36,27,39,30,42,33), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34) );
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)], [(1,10,19,7,16,4,13),(2,11,20,8,17,5,14),(3,12,21,9,18,6,15),(22,34,25,37,28,40,31),(23,35,26,38,29,41,32),(24,36,27,39,30,42,33)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F | 21A | ··· | 21L | 21M | ··· | 21BB | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
105 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 | D7 | C3×D7 | C7×D7 | D7×C21 |
| kernel | D7×C21 | C7×C21 | C7×D7 | C72 | C3×D7 | C21 | D7 | C7 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 3 | 6 | 18 | 36 |
Matrix representation of D7×C21 ►in GL2(𝔽43) generated by
| 24 | 0 |
| 0 | 24 |
| 16 | 0 |
| 36 | 35 |
| 8 | 9 |
| 36 | 35 |
G:=sub<GL(2,GF(43))| [24,0,0,24],[16,36,0,35],[8,36,9,35] >;
D7×C21 in GAP, Magma, Sage, TeX
D_7\times C_{21} % in TeX
G:=Group("D7xC21"); // GroupNames label
G:=SmallGroup(294,18);
// by ID
G=gap.SmallGroup(294,18);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,4035]);
// Polycyclic
G:=Group<a,b,c|a^21=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export