direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×D21, C42⋊5C6, C42⋊2S3, C21⋊7D6, C32⋊5D14, C6⋊(C3×D7), C7⋊5(S3×C6), (C3×C6)⋊1D7, C3⋊2(C6×D7), C14⋊3(C3×S3), C21⋊7(C2×C6), (C3×C42)⋊2C2, (C3×C21)⋊7C22, SmallGroup(252,43)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C6×D21 |
Generators and relations for C6×D21
G = < a,b,c | a6=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D21
►On 84 points
Generators in S84
(1 28 15 42 8 35)(2 29 16 22 9 36)(3 30 17 23 10 37)(4 31 18 24 11 38)(5 32 19 25 12 39)(6 33 20 26 13 40)(7 34 21 27 14 41)(43 78 50 64 57 71)(44 79 51 65 58 72)(45 80 52 66 59 73)(46 81 53 67 60 74)(47 82 54 68 61 75)(48 83 55 69 62 76)(49 84 56 70 63 77)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 63)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)
G:=sub<Sym(84)| (1,28,15,42,8,35)(2,29,16,22,9,36)(3,30,17,23,10,37)(4,31,18,24,11,38)(5,32,19,25,12,39)(6,33,20,26,13,40)(7,34,21,27,14,41)(43,78,50,64,57,71)(44,79,51,65,58,72)(45,80,52,66,59,73)(46,81,53,67,60,74)(47,82,54,68,61,75)(48,83,55,69,62,76)(49,84,56,70,63,77), (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,84)(17,83)(18,82)(19,81)(20,80)(21,79)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)>;
G:=Group( (1,28,15,42,8,35)(2,29,16,22,9,36)(3,30,17,23,10,37)(4,31,18,24,11,38)(5,32,19,25,12,39)(6,33,20,26,13,40)(7,34,21,27,14,41)(43,78,50,64,57,71)(44,79,51,65,58,72)(45,80,52,66,59,73)(46,81,53,67,60,74)(47,82,54,68,61,75)(48,83,55,69,62,76)(49,84,56,70,63,77), (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,84)(17,83)(18,82)(19,81)(20,80)(21,79)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57) );
G=PermutationGroup([[(1,28,15,42,8,35),(2,29,16,22,9,36),(3,30,17,23,10,37),(4,31,18,24,11,38),(5,32,19,25,12,39),(6,33,20,26,13,40),(7,34,21,27,14,41),(43,78,50,64,57,71),(44,79,51,65,58,72),(45,80,52,66,59,73),(46,81,53,67,60,74),(47,82,54,68,61,75),(48,83,55,69,62,76),(49,84,56,70,63,77)], [(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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,84),(17,83),(18,82),(19,81),(20,80),(21,79),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,63),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 7A | 7B | 7C | 14A | 14B | 14C | 21A | ··· | 21X | 42A | ··· | 42X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | 14 | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 21 | 21 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 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 | C2 | C3 | C6 | C6 | S3 | D6 | D7 | C3×S3 | D14 | S3×C6 | C3×D7 | D21 | C6×D7 | D42 | C3×D21 | C6×D21 |
| kernel | C6×D21 | C3×D21 | C3×C42 | D42 | D21 | C42 | C42 | C21 | C3×C6 | C14 | C32 | C7 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 3 | 2 | 3 | 2 | 6 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C6×D21 ►in GL2(𝔽43) generated by
| 37 | 0 |
| 0 | 37 |
| 14 | 0 |
| 0 | 40 |
| 0 | 18 |
| 12 | 0 |
G:=sub<GL(2,GF(43))| [37,0,0,37],[14,0,0,40],[0,12,18,0] >;
C6×D21 in GAP, Magma, Sage, TeX
C_6\times D_{21} % in TeX
G:=Group("C6xD21"); // GroupNames label
G:=SmallGroup(252,43);
// by ID
G=gap.SmallGroup(252,43);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,483,5404]);
// Polycyclic
G:=Group<a,b,c|a^6=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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