direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D21, C4⋊1D42, C28⋊3D6, D84⋊3C2, C12⋊3D14, C84⋊1C22, C22⋊2D42, D42⋊6C22, C42.32C23, Dic21⋊3C22, C3⋊4(D4×D7), C7⋊4(S3×D4), (C3×D4)⋊2D7, (C7×D4)⋊2S3, (C2×C6)⋊3D14, (C2×C14)⋊6D6, C21⋊13(C2×D4), (D4×C21)⋊2C2, (C4×D21)⋊1C2, C21⋊7D4⋊1C2, (C2×C42)⋊1C22, (C22×D21)⋊2C2, C6.32(C22×D7), C2.6(C22×D21), C14.32(C22×S3), SmallGroup(336,198)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D21
G = < a,b,c,d | a4=b2=c21=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 840 in 108 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, D4, C23, Dic3, C12, D6, C2×C6, D7, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, Dic7, C28, D14, C2×C14, D21, D21, C42, C42, S3×D4, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, Dic21, C84, D42, D42, D42, C2×C42, D4×D7, C4×D21, D84, C21⋊7D4, D4×C21, C22×D21, D4×D21
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C22×S3, D14, D21, S3×D4, C22×D7, D42, D4×D7, C22×D21, D4×D21
►On 84 points
Generators in S84
(1 66 24 43)(2 67 25 44)(3 68 26 45)(4 69 27 46)(5 70 28 47)(6 71 29 48)(7 72 30 49)(8 73 31 50)(9 74 32 51)(10 75 33 52)(11 76 34 53)(12 77 35 54)(13 78 36 55)(14 79 37 56)(15 80 38 57)(16 81 39 58)(17 82 40 59)(18 83 41 60)(19 84 42 61)(20 64 22 62)(21 65 23 63)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 25)(23 24)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(64 67)(65 66)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(84)| (1,66,24,43)(2,67,25,44)(3,68,26,45)(4,69,27,46)(5,70,28,47)(6,71,29,48)(7,72,30,49)(8,73,31,50)(9,74,32,51)(10,75,33,52)(11,76,34,53)(12,77,35,54)(13,78,36,55)(14,79,37,56)(15,80,38,57)(16,81,39,58)(17,82,40,59)(18,83,41,60)(19,84,42,61)(20,64,22,62)(21,65,23,63), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,25)(23,24)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(64,67)(65,66)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,66,24,43)(2,67,25,44)(3,68,26,45)(4,69,27,46)(5,70,28,47)(6,71,29,48)(7,72,30,49)(8,73,31,50)(9,74,32,51)(10,75,33,52)(11,76,34,53)(12,77,35,54)(13,78,36,55)(14,79,37,56)(15,80,38,57)(16,81,39,58)(17,82,40,59)(18,83,41,60)(19,84,42,61)(20,64,22,62)(21,65,23,63), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,25)(23,24)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(64,67)(65,66)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([[(1,66,24,43),(2,67,25,44),(3,68,26,45),(4,69,27,46),(5,70,28,47),(6,71,29,48),(7,72,30,49),(8,73,31,50),(9,74,32,51),(10,75,33,52),(11,76,34,53),(12,77,35,54),(13,78,36,55),(14,79,37,56),(15,80,38,57),(16,81,39,58),(17,82,40,59),(18,83,41,60),(19,84,42,61),(20,64,22,62),(21,65,23,63)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,25),(23,24),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(64,67),(65,66),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 7A | 7B | 7C | 12 | 14A | 14B | 14C | 14D | ··· | 14I | 21A | ··· | 21F | 28A | 28B | 28C | 42A | ··· | 42F | 42G | ··· | 42R | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | 28 | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 21 | 21 | 42 | 42 | 2 | 2 | 42 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D7 | D14 | D14 | D21 | D42 | D42 | S3×D4 | D4×D7 | D4×D21 |
| kernel | D4×D21 | C4×D21 | D84 | C21⋊7D4 | D4×C21 | C22×D21 | C7×D4 | D21 | C28 | C2×C14 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C7 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 12 | 1 | 3 | 6 |
Matrix representation of D4×D21 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 336 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 70 | 323 | 0 | 0 |
| 14 | 262 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 70 | 323 | 0 | 0 |
| 37 | 267 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,0,336,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[70,14,0,0,323,262,0,0,0,0,1,0,0,0,0,1],[70,37,0,0,323,267,0,0,0,0,1,0,0,0,0,1] >;
D4×D21 in GAP, Magma, Sage, TeX
D_4\times D_{21} % in TeX
G:=Group("D4xD21"); // GroupNames label
G:=SmallGroup(336,198);
// by ID
G=gap.SmallGroup(336,198);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,116,964,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^21=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations