direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×Dic21, C21⋊5C12, C42.9C6, C42.4S3, C6.4D21, C21⋊2Dic3, C32⋊2Dic7, C6.(C3×D7), (C3×C21)⋊4C4, C3⋊(C3×Dic7), C2.(C3×D21), (C3×C6).1D7, C7⋊3(C3×Dic3), C14.3(C3×S3), (C3×C42).2C2, SmallGroup(252,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C3×Dic21 |
Generators and relations for C3×Dic21
G = < a,b,c | a3=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic21
►On 84 points
Generators in S84
(1 15 29)(2 16 30)(3 17 31)(4 18 32)(5 19 33)(6 20 34)(7 21 35)(8 22 36)(9 23 37)(10 24 38)(11 25 39)(12 26 40)(13 27 41)(14 28 42)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)
(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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)
(1 57 22 78)(2 56 23 77)(3 55 24 76)(4 54 25 75)(5 53 26 74)(6 52 27 73)(7 51 28 72)(8 50 29 71)(9 49 30 70)(10 48 31 69)(11 47 32 68)(12 46 33 67)(13 45 34 66)(14 44 35 65)(15 43 36 64)(16 84 37 63)(17 83 38 62)(18 82 39 61)(19 81 40 60)(20 80 41 59)(21 79 42 58)
G:=sub<Sym(84)| (1,15,29)(2,16,30)(3,17,31)(4,18,32)(5,19,33)(6,20,34)(7,21,35)(8,22,36)(9,23,37)(10,24,38)(11,25,39)(12,26,40)(13,27,41)(14,28,42)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,57,22,78)(2,56,23,77)(3,55,24,76)(4,54,25,75)(5,53,26,74)(6,52,27,73)(7,51,28,72)(8,50,29,71)(9,49,30,70)(10,48,31,69)(11,47,32,68)(12,46,33,67)(13,45,34,66)(14,44,35,65)(15,43,36,64)(16,84,37,63)(17,83,38,62)(18,82,39,61)(19,81,40,60)(20,80,41,59)(21,79,42,58)>;
G:=Group( (1,15,29)(2,16,30)(3,17,31)(4,18,32)(5,19,33)(6,20,34)(7,21,35)(8,22,36)(9,23,37)(10,24,38)(11,25,39)(12,26,40)(13,27,41)(14,28,42)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,57,22,78)(2,56,23,77)(3,55,24,76)(4,54,25,75)(5,53,26,74)(6,52,27,73)(7,51,28,72)(8,50,29,71)(9,49,30,70)(10,48,31,69)(11,47,32,68)(12,46,33,67)(13,45,34,66)(14,44,35,65)(15,43,36,64)(16,84,37,63)(17,83,38,62)(18,82,39,61)(19,81,40,60)(20,80,41,59)(21,79,42,58) );
G=PermutationGroup([[(1,15,29),(2,16,30),(3,17,31),(4,18,32),(5,19,33),(6,20,34),(7,21,35),(8,22,36),(9,23,37),(10,24,38),(11,25,39),(12,26,40),(13,27,41),(14,28,42),(43,71,57),(44,72,58),(45,73,59),(46,74,60),(47,75,61),(48,76,62),(49,77,63),(50,78,64),(51,79,65),(52,80,66),(53,81,67),(54,82,68),(55,83,69),(56,84,70)], [(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),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)], [(1,57,22,78),(2,56,23,77),(3,55,24,76),(4,54,25,75),(5,53,26,74),(6,52,27,73),(7,51,28,72),(8,50,29,71),(9,49,30,70),(10,48,31,69),(11,47,32,68),(12,46,33,67),(13,45,34,66),(14,44,35,65),(15,43,36,64),(16,84,37,63),(17,83,38,62),(18,82,39,61),(19,81,40,60),(20,80,41,59),(21,79,42,58)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 21A | ··· | 21X | 42A | ··· | 42X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 21 | 21 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | D7 | C3×S3 | Dic7 | C3×Dic3 | C3×D7 | D21 | C3×Dic7 | Dic21 | C3×D21 | C3×Dic21 |
| kernel | C3×Dic21 | C3×C42 | Dic21 | C3×C21 | C42 | C21 | C42 | C21 | C3×C6 | C14 | C32 | C7 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 3 | 2 | 3 | 2 | 6 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C3×Dic21 ►in GL2(𝔽43) generated by
| 6 | 0 |
| 0 | 6 |
| 28 | 0 |
| 0 | 20 |
| 0 | 42 |
| 1 | 0 |
G:=sub<GL(2,GF(43))| [6,0,0,6],[28,0,0,20],[0,1,42,0] >;
C3×Dic21 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{21} % in TeX
G:=Group("C3xDic21"); // GroupNames label
G:=SmallGroup(252,22);
// by ID
G=gap.SmallGroup(252,22);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,5404]);
// Polycyclic
G:=Group<a,b,c|a^3=b^42=1,c^2=b^21,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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