non-abelian, supersoluble, monomial
Aliases: He3⋊2D7, C32⋊2D21, (C3×C21)⋊2S3, C7⋊(He3⋊C2), (C7×He3)⋊2C2, C21.2(C3⋊S3), C3.2(C3⋊D21), SmallGroup(378,43)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×He3 — C32⋊D21 |
Generators and relations for C32⋊D21
G = < a,b,c,d | a3=b3=c21=d2=1, cac-1=ab=ba, dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >
63C2
3C3
3C3
3C3
3C3
21S3
21S3
21S3
21S3
63C6
9D7
3C21
3C21
3C21
3C21
21C3×S3
21C3×S3
21C3×S3
21C3×S3
3D21
3D21
3D21
3D21
Smallest permutation representation of C32⋊D21
►On 63 points
Generators in S63
(2 46 34)(3 35 47)(5 49 37)(6 38 50)(8 52 40)(9 41 53)(11 55 22)(12 23 56)(14 58 25)(15 26 59)(17 61 28)(18 29 62)(20 43 31)(21 32 44)
(1 45 33)(2 46 34)(3 47 35)(4 48 36)(5 49 37)(6 50 38)(7 51 39)(8 52 40)(9 53 41)(10 54 42)(11 55 22)(12 56 23)(13 57 24)(14 58 25)(15 59 26)(16 60 27)(17 61 28)(18 62 29)(19 63 30)(20 43 31)(21 44 32)
(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 61 62 63)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(43 46)(44 45)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)
G:=sub<Sym(63)| (2,46,34)(3,35,47)(5,49,37)(6,38,50)(8,52,40)(9,41,53)(11,55,22)(12,23,56)(14,58,25)(15,26,59)(17,61,28)(18,29,62)(20,43,31)(21,32,44), (1,45,33)(2,46,34)(3,47,35)(4,48,36)(5,49,37)(6,50,38)(7,51,39)(8,52,40)(9,53,41)(10,54,42)(11,55,22)(12,56,23)(13,57,24)(14,58,25)(15,59,26)(16,60,27)(17,61,28)(18,62,29)(19,63,30)(20,43,31)(21,44,32), (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(43,46)(44,45)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)>;
G:=Group( (2,46,34)(3,35,47)(5,49,37)(6,38,50)(8,52,40)(9,41,53)(11,55,22)(12,23,56)(14,58,25)(15,26,59)(17,61,28)(18,29,62)(20,43,31)(21,32,44), (1,45,33)(2,46,34)(3,47,35)(4,48,36)(5,49,37)(6,50,38)(7,51,39)(8,52,40)(9,53,41)(10,54,42)(11,55,22)(12,56,23)(13,57,24)(14,58,25)(15,59,26)(16,60,27)(17,61,28)(18,62,29)(19,63,30)(20,43,31)(21,44,32), (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(43,46)(44,45)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56) );
G=PermutationGroup([[(2,46,34),(3,35,47),(5,49,37),(6,38,50),(8,52,40),(9,41,53),(11,55,22),(12,23,56),(14,58,25),(15,26,59),(17,61,28),(18,29,62),(20,43,31),(21,32,44)], [(1,45,33),(2,46,34),(3,47,35),(4,48,36),(5,49,37),(6,50,38),(7,51,39),(8,52,40),(9,53,41),(10,54,42),(11,55,22),(12,56,23),(13,57,24),(14,58,25),(15,59,26),(16,60,27),(17,61,28),(18,62,29),(19,63,30),(20,43,31),(21,44,32)], [(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),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(43,46),(44,45),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56)]])
43 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 7A | 7B | 7C | 21A | ··· | 21F | 21G | ··· | 21AD |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 21 | ··· | 21 | 21 | ··· | 21 |
| size | 1 | 63 | 1 | 1 | 6 | 6 | 6 | 6 | 63 | 63 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
43 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 3 | 6 |
| type | + | + | + | + | + | ||
| image | C1 | C2 | S3 | D7 | D21 | He3⋊C2 | C32⋊D21 |
| kernel | C32⋊D21 | C7×He3 | C3×C21 | He3 | C32 | C7 | C1 |
| # reps | 1 | 1 | 4 | 3 | 24 | 4 | 6 |
Matrix representation of C32⋊D21 ►in GL5(𝔽43)
| 42 | 42 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 1 |
| 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 17 | 15 | 0 | 0 | 0 |
| 28 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 42 | 42 | 0 |
| 0 | 0 | 0 | 8 | 1 |
| 26 | 28 | 0 | 0 | 0 |
| 2 | 17 | 0 | 0 | 0 |
| 0 | 0 | 42 | 42 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 35 | 42 |
G:=sub<GL(5,GF(43))| [42,1,0,0,0,42,0,0,0,0,0,0,36,0,0,0,0,0,6,0,0,0,1,6,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[17,28,0,0,0,15,2,0,0,0,0,0,0,42,0,0,0,1,42,8,0,0,0,0,1],[26,2,0,0,0,28,17,0,0,0,0,0,42,0,0,0,0,42,1,35,0,0,0,0,42] >;
C32⋊D21 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_{21} % in TeX
G:=Group("C3^2:D21"); // GroupNames label
G:=SmallGroup(378,43);
// by ID
G=gap.SmallGroup(378,43);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,41,182,997,2163]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^21=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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