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G = D4×D21order 336 = 24·3·7

Direct product of D4 and D21

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D4×D21
 Chief series C1 — C7 — C21 — C42 — D42 — C22×D21 — D4×D21
 Lower central C21 — C42 — D4×D21
 Upper central C1 — C2 — D4

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, C217D4, 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

Smallest permutation representation of 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

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