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## G = D4×C21order 168 = 23·3·7

### Direct product of C21 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C21, C4⋊C42, C287C6, C847C2, C123C14, C222C42, 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

 Derived series C1 — C2 — D4×C21
 Chief series C1 — C2 — C14 — C42 — C2×C42 — D4×C21
 Lower central C1 — C2 — D4×C21
 Upper central C1 — C42 — D4×C21

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 45 83 25)(2 46 84 26)(3 47 64 27)(4 48 65 28)(5 49 66 29)(6 50 67 30)(7 51 68 31)(8 52 69 32)(9 53 70 33)(10 54 71 34)(11 55 72 35)(12 56 73 36)(13 57 74 37)(14 58 75 38)(15 59 76 39)(16 60 77 40)(17 61 78 41)(18 62 79 42)(19 63 80 22)(20 43 81 23)(21 44 82 24)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 22)(20 23)(21 24)(43 81)(44 82)(45 83)(46 84)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)

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), (1,45,83,25)(2,46,84,26)(3,47,64,27)(4,48,65,28)(5,49,66,29)(6,50,67,30)(7,51,68,31)(8,52,69,32)(9,53,70,33)(10,54,71,34)(11,55,72,35)(12,56,73,36)(13,57,74,37)(14,58,75,38)(15,59,76,39)(16,60,77,40)(17,61,78,41)(18,62,79,42)(19,63,80,22)(20,43,81,23)(21,44,82,24), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,22)(20,23)(21,24)(43,81)(44,82)(45,83)(46,84)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)>;

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), (1,45,83,25)(2,46,84,26)(3,47,64,27)(4,48,65,28)(5,49,66,29)(6,50,67,30)(7,51,68,31)(8,52,69,32)(9,53,70,33)(10,54,71,34)(11,55,72,35)(12,56,73,36)(13,57,74,37)(14,58,75,38)(15,59,76,39)(16,60,77,40)(17,61,78,41)(18,62,79,42)(19,63,80,22)(20,43,81,23)(21,44,82,24), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,22)(20,23)(21,24)(43,81)(44,82)(45,83)(46,84)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80) );

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)], [(1,45,83,25),(2,46,84,26),(3,47,64,27),(4,48,65,28),(5,49,66,29),(6,50,67,30),(7,51,68,31),(8,52,69,32),(9,53,70,33),(10,54,71,34),(11,55,72,35),(12,56,73,36),(13,57,74,37),(14,58,75,38),(15,59,76,39),(16,60,77,40),(17,61,78,41),(18,62,79,42),(19,63,80,22),(20,43,81,23),(21,44,82,24)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,22),(20,23),(21,24),(43,81),(44,82),(45,83),(46,84),(47,64),(48,65),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80)])

D4×C21 is a maximal subgroup of   D4⋊D21  D4.D21  D42D21

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

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