direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C21, C4⋊C42, C28⋊7C6, C84⋊7C2, C12⋊3C14, C22⋊2C42, C42.23C22, (C2×C6)⋊1C14, (C2×C42)⋊1C2, (C2×C14)⋊9C6, C2.1(C2×C42), C6.6(C2×C14), C14.14(C2×C6), SmallGroup(168,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C21
G = < a,b,c | a21=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C21
►On 84 points
Generators in S84
(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 80 22 48)(2 81 23 49)(3 82 24 50)(4 83 25 51)(5 84 26 52)(6 64 27 53)(7 65 28 54)(8 66 29 55)(9 67 30 56)(10 68 31 57)(11 69 32 58)(12 70 33 59)(13 71 34 60)(14 72 35 61)(15 73 36 62)(16 74 37 63)(17 75 38 43)(18 76 39 44)(19 77 40 45)(20 78 41 46)(21 79 42 47)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 43)(18 44)(19 45)(20 46)(21 47)(22 80)(23 81)(24 82)(25 83)(26 84)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)
G:=sub<Sym(84)| (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), (1,80,22,48)(2,81,23,49)(3,82,24,50)(4,83,25,51)(5,84,26,52)(6,64,27,53)(7,65,28,54)(8,66,29,55)(9,67,30,56)(10,68,31,57)(11,69,32,58)(12,70,33,59)(13,71,34,60)(14,72,35,61)(15,73,36,62)(16,74,37,63)(17,75,38,43)(18,76,39,44)(19,77,40,45)(20,78,41,46)(21,79,42,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,43)(18,44)(19,45)(20,46)(21,47)(22,80)(23,81)(24,82)(25,83)(26,84)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)>;
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)(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), (1,80,22,48)(2,81,23,49)(3,82,24,50)(4,83,25,51)(5,84,26,52)(6,64,27,53)(7,65,28,54)(8,66,29,55)(9,67,30,56)(10,68,31,57)(11,69,32,58)(12,70,33,59)(13,71,34,60)(14,72,35,61)(15,73,36,62)(16,74,37,63)(17,75,38,43)(18,76,39,44)(19,77,40,45)(20,78,41,46)(21,79,42,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,43)(18,44)(19,45)(20,46)(21,47)(22,80)(23,81)(24,82)(25,83)(26,84)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79) );
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),(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)], [(1,80,22,48),(2,81,23,49),(3,82,24,50),(4,83,25,51),(5,84,26,52),(6,64,27,53),(7,65,28,54),(8,66,29,55),(9,67,30,56),(10,68,31,57),(11,69,32,58),(12,70,33,59),(13,71,34,60),(14,72,35,61),(15,73,36,62),(16,74,37,63),(17,75,38,43),(18,76,39,44),(19,77,40,45),(20,78,41,46),(21,79,42,47)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,43),(18,44),(19,45),(20,46),(21,47),(22,80),(23,81),(24,82),(25,83),(26,84),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79)]])
D4×C21 is a maximal subgroup of
D4⋊D21 D4.D21 D4⋊2D21
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7A | ··· | 7F | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21L | 28A | ··· | 28F | 42A | ··· | 42L | 42M | ··· | 42AJ | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C7 | C14 | C14 | C21 | C42 | C42 | D4 | C3×D4 | C7×D4 | D4×C21 |
| kernel | D4×C21 | C84 | C2×C42 | C7×D4 | C28 | C2×C14 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 12 | 24 | 1 | 2 | 6 | 12 |
Matrix representation of D4×C21 ►in GL2(𝔽43) generated by
| 14 | 0 |
| 0 | 14 |
| 0 | 42 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(43))| [14,0,0,14],[0,1,42,0],[0,1,1,0] >;
D4×C21 in GAP, Magma, Sage, TeX
D_4\times C_{21} % in TeX
G:=Group("D4xC21"); // GroupNames label
G:=SmallGroup(168,40);
// by ID
G=gap.SmallGroup(168,40);
# by ID
G:=PCGroup([5,-2,-2,-3,-7,-2,861]);
// Polycyclic
G:=Group<a,b,c|a^21=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export