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## G = S32×C7order 252 = 22·32·7

### Direct product of C7, S3 and S3

Aliases: S32×C7, C215D6, C3⋊S3⋊C14, (C3×S3)⋊C14, C32⋊(C2×C14), C31(S3×C14), (S3×C21)⋊3C2, (C3×C21)⋊5C22, (C7×C3⋊S3)⋊3C2, SmallGroup(252,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C7
 Chief series C1 — C3 — C32 — C3×C21 — S3×C21 — S32×C7
 Lower central C32 — S32×C7
 Upper central C1 — C7

Generators and relations for S32×C7
G = < a,b,c,d,e | a7=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Smallest permutation representation of S32×C7
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 32 28)(2 33 22)(3 34 23)(4 35 24)(5 29 25)(6 30 26)(7 31 27)(8 16 42)(9 17 36)(10 18 37)(11 19 38)(12 20 39)(13 21 40)(14 15 41)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 36)(8 26)(9 27)(10 28)(11 22)(12 23)(13 24)(14 25)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)
(1 28 32)(2 22 33)(3 23 34)(4 24 35)(5 25 29)(6 26 30)(7 27 31)(8 16 42)(9 17 36)(10 18 37)(11 19 38)(12 20 39)(13 21 40)(14 15 41)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 36)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 29)(15 25)(16 26)(17 27)(18 28)(19 22)(20 23)(21 24)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 7A ··· 7F 14A ··· 14L 14M ··· 14R 21A ··· 21L 21M ··· 21R 42A ··· 42L order 1 2 2 2 3 3 3 6 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 21 ··· 21 42 ··· 42 size 1 3 3 9 2 2 4 6 6 1 ··· 1 3 ··· 3 9 ··· 9 2 ··· 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 S3×C7 S3×C14 S32 S32×C7 kernel S32×C7 S3×C21 C7×C3⋊S3 S32 C3×S3 C3⋊S3 S3×C7 C21 S3 C3 C7 C1 # reps 1 2 1 6 12 6 2 2 12 12 1 6

Matrix representation of S32×C7 in GL4(𝔽43) generated by

 1 0 0 0 0 1 0 0 0 0 21 0 0 0 0 21
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 42 42
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 42 42 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 42 42 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(43))| [1,0,0,0,0,1,0,0,0,0,21,0,0,0,0,21],[1,0,0,0,0,1,0,0,0,0,0,42,0,0,1,42],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,42,0,0,1,42,0,0,0,0,1,0,0,0,0,1],[1,42,0,0,0,42,0,0,0,0,1,0,0,0,0,1] >;

S32×C7 in GAP, Magma, Sage, TeX

S_3^2\times C_7
% in TeX

G:=Group("S3^2xC7");
// GroupNames label

G:=SmallGroup(252,35);
// by ID

G=gap.SmallGroup(252,35);
# by ID

G:=PCGroup([5,-2,-2,-7,-3,-3,568,4204]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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