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G = D4xD21order 336 = 24·3·7

Direct product of D4 and D21

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4xD21, C4:1D42, C28:3D6, D84:3C2, C12:3D14, C84:1C22, C22:2D42, D42:6C22, C42.32C23, Dic21:3C22, C3:4(D4xD7), C7:4(S3xD4), (C3xD4):2D7, (C7xD4):2S3, (C2xC6):3D14, (C2xC14):6D6, C21:13(C2xD4), (D4xC21):2C2, (C4xD21):1C2, C21:7D4:1C2, (C2xC42):1C22, (C22xD21):2C2, C6.32(C22xD7), C2.6(C22xD21), C14.32(C22xS3), SmallGroup(336,198)

Series: Derived Chief Lower central Upper central

C1C42 — D4xD21
C1C7C21C42D42C22xD21 — D4xD21
C21C42 — D4xD21
C1C2D4

Generators and relations for D4xD21
 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, C2xC4, D4, D4, C23, Dic3, C12, D6, C2xC6, D7, C14, C14, C2xD4, C21, C4xS3, D12, C3:D4, C3xD4, C22xS3, Dic7, C28, D14, C2xC14, D21, D21, C42, C42, S3xD4, C4xD7, D28, C7:D4, C7xD4, C22xD7, Dic21, C84, D42, D42, D42, C2xC42, D4xD7, C4xD21, D84, C21:7D4, D4xC21, C22xD21, D4xD21
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2xD4, C22xS3, D14, D21, S3xD4, C22xD7, D42, D4xD7, C22xD21, D4xD21

Smallest permutation representation of D4xD21
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 2A2B2C2D2E2F2G 3 4A4B6A6B6C7A7B7C 12 14A14B14C14D···14I21A···21F28A28B28C42A···42F42G···42R84A···84F
order122222223446667771214141414···1421···2128282842···4242···4284···84
size112221214242224224422242224···42···24442···24···44···4

60 irreducible representations

dim1111112222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D6D6D7D14D14D21D42D42S3xD4D4xD7D4xD21
kernelD4xD21C4xD21D84C21:7D4D4xC21C22xD21C7xD4D21C28C2xC14C3xD4C12C2xC6D4C4C22C7C3C1
# reps11121212123366612136

Matrix representation of D4xD21 in GL4(F337) generated by

1000
0100
0001
003360
,
1000
0100
0001
0010
,
7032300
1426200
0010
0001
,
7032300
3726700
0010
0001
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] >;

D4xD21 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

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