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## G = D7×C21order 294 = 2·3·72

### Direct product of C21 and D7

Aliases: D7×C21, C73C42, C212C14, C7211C6, (C7×C21)⋊3C2, SmallGroup(294,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C21
 Chief series C1 — C7 — C72 — C7×C21 — D7×C21
 Lower central C7 — D7×C21
 Upper central C1 — C21

Generators and relations for D7×C21
G = < a,b,c | a21=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D7×C21
On 42 points
Generators in S42
(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)
(1 10 19 7 16 4 13)(2 11 20 8 17 5 14)(3 12 21 9 18 6 15)(22 34 25 37 28 40 31)(23 35 26 38 29 41 32)(24 36 27 39 30 42 33)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)

G:=sub<Sym(42)| (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), (1,10,19,7,16,4,13)(2,11,20,8,17,5,14)(3,12,21,9,18,6,15)(22,34,25,37,28,40,31)(23,35,26,38,29,41,32)(24,36,27,39,30,42,33), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)>;

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), (1,10,19,7,16,4,13)(2,11,20,8,17,5,14)(3,12,21,9,18,6,15)(22,34,25,37,28,40,31)(23,35,26,38,29,41,32)(24,36,27,39,30,42,33), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31) );

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)], [(1,10,19,7,16,4,13),(2,11,20,8,17,5,14),(3,12,21,9,18,6,15),(22,34,25,37,28,40,31),(23,35,26,38,29,41,32),(24,36,27,39,30,42,33)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31)])

105 conjugacy classes

 class 1 2 3A 3B 6A 6B 7A ··· 7F 7G ··· 7AA 14A ··· 14F 21A ··· 21L 21M ··· 21BB 42A ··· 42L order 1 2 3 3 6 6 7 ··· 7 7 ··· 7 14 ··· 14 21 ··· 21 21 ··· 21 42 ··· 42 size 1 7 1 1 7 7 1 ··· 1 2 ··· 2 7 ··· 7 1 ··· 1 2 ··· 2 7 ··· 7

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C7 C14 C21 C42 D7 C3×D7 C7×D7 D7×C21 kernel D7×C21 C7×C21 C7×D7 C72 C3×D7 C21 D7 C7 C21 C7 C3 C1 # reps 1 1 2 2 6 6 12 12 3 6 18 36

Matrix representation of D7×C21 in GL2(𝔽43) generated by

 24 0 0 24
,
 16 0 36 35
,
 8 9 36 35
G:=sub<GL(2,GF(43))| [24,0,0,24],[16,36,0,35],[8,36,9,35] >;

D7×C21 in GAP, Magma, Sage, TeX

D_7\times C_{21}
% in TeX

G:=Group("D7xC21");
// GroupNames label

G:=SmallGroup(294,18);
// by ID

G=gap.SmallGroup(294,18);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,4035]);
// Polycyclic

G:=Group<a,b,c|a^21=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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