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## G = C3×S3×D7order 252 = 22·32·7

### Direct product of C3, S3 and D7

Aliases: C3×S3×D7, C214D6, D213C6, C323D14, C74(S3×C6), C31(C6×D7), (S3×C7)⋊3C6, C215(C2×C6), (C3×D7)⋊3C6, (S3×C21)⋊2C2, (C3×D21)⋊1C2, (C3×C21)⋊1C22, (C32×D7)⋊1C2, SmallGroup(252,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C3×S3×D7
 Chief series C1 — C7 — C21 — C3×C21 — C32×D7 — C3×S3×D7
 Lower central C21 — C3×S3×D7
 Upper central C1 — C3

Generators and relations for C3×S3×D7
G = < a,b,c,d,e | a3=b3=c2=d7=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 C3×S3×D7
On 42 points
Generators in S42
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 7A 7B 7C 14A 14B 14C 21A ··· 21F 21G ··· 21O 42A ··· 42F order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 7 7 7 14 14 14 21 ··· 21 21 ··· 21 42 ··· 42 size 1 3 7 21 1 1 2 2 2 3 3 7 7 14 14 14 21 21 2 2 2 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 D7 C3×S3 D14 S3×C6 C3×D7 C6×D7 S3×D7 C3×S3×D7 kernel C3×S3×D7 C32×D7 S3×C21 C3×D21 S3×D7 S3×C7 C3×D7 D21 C3×D7 C21 C3×S3 D7 C32 C7 S3 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 3 2 3 2 6 6 3 6

Matrix representation of C3×S3×D7 in GL4(𝔽43) generated by

 1 0 0 0 0 1 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 0 6 0 0 0 38 36
,
 1 0 0 0 0 1 0 0 0 0 1 37 0 0 0 42
,
 16 1 0 0 26 42 0 0 0 0 1 0 0 0 0 1
,
 19 34 0 0 40 24 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(43))| [1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,6,38,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,37,42],[16,26,0,0,1,42,0,0,0,0,1,0,0,0,0,1],[19,40,0,0,34,24,0,0,0,0,1,0,0,0,0,1] >;

C3×S3×D7 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_7
% in TeX

G:=Group("C3xS3xD7");
// GroupNames label

G:=SmallGroup(252,33);
// by ID

G=gap.SmallGroup(252,33);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,248,5404]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^7=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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