direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D21, C21⋊5C6, C21⋊2S3, C32⋊1D7, C3⋊(C3×D7), C7⋊3(C3×S3), (C3×C21)⋊2C2, SmallGroup(126,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C3×D21 |
Generators and relations for C3×D21
G = < a,b,c | a3=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D21
►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)]])
C3×D21 is a maximal subgroup of
C3×S3×D7 D21⋊S3 D21⋊C9 He3⋊D7 D63⋊C3 C32⋊D21
C3×D21 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 | C3×S3 | C3×D7 | D21 | C3×D21 |
| kernel | C3×D21 | C3×C21 | D21 | C21 | C21 | C32 | C7 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 6 | 12 |
Matrix representation of C3×D21 ►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] >;
C3×D21 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