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## G = C33.(C3×S3)  order 486 = 2·35

### 8th non-split extension by C33 of C3×S3 acting faithfully

Aliases: C32⋊C9.3S3, C32⋊C9.3C6, C33.8(C3×S3), C322D9.2C3, C32.29He32C2, C32.32(C32⋊C6), C3.2(He3.C6), C3.7(He3.2S3), SmallGroup(486,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C33.(C3×S3)
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.29He3 — C33.(C3×S3)
 Lower central C32⋊C9 — C33.(C3×S3)
 Upper central C1 — C3

Generators and relations for C33.(C3×S3)
G = < a,b,c,d,e,f | a3=b3=c3=f2=1, d3=c-1, e3=fbf=b-1, dad-1=ab=ba, eae-1=ac=ca, faf=a-1c-1, bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, ede-1=a-1c-1d, df=fd, fef=be2 >

Smallest permutation representation of C33.(C3×S3)
On 54 points
Generators in S54
(2 5 8)(3 9 6)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 32 35)(30 36 33)(37 43 40)(39 42 45)(47 53 50)(48 51 54)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(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)
(1 33 45 7 30 42 4 36 39)(2 31 37 8 28 43 5 34 40)(3 32 44 9 29 41 6 35 38)(10 52 19 13 46 22 16 49 25)(11 53 26 14 47 20 17 50 23)(12 51 27 15 54 21 18 48 24)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 46)(9 47)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)

G:=sub<Sym(54)| (2,5,8)(3,9,6)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,32,35)(30,36,33)(37,43,40)(39,42,45)(47,53,50)(48,51,54), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (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), (1,33,45,7,30,42,4,36,39)(2,31,37,8,28,43,5,34,40)(3,32,44,9,29,41,6,35,38)(10,52,19,13,46,22,16,49,25)(11,53,26,14,47,20,17,50,23)(12,51,27,15,54,21,18,48,24), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)>;

G:=Group( (2,5,8)(3,9,6)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,32,35)(30,36,33)(37,43,40)(39,42,45)(47,53,50)(48,51,54), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (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), (1,33,45,7,30,42,4,36,39)(2,31,37,8,28,43,5,34,40)(3,32,44,9,29,41,6,35,38)(10,52,19,13,46,22,16,49,25)(11,53,26,14,47,20,17,50,23)(12,51,27,15,54,21,18,48,24), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42) );

G=PermutationGroup([[(2,5,8),(3,9,6),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(29,32,35),(30,36,33),(37,43,40),(39,42,45),(47,53,50),(48,51,54)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51)], [(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)], [(1,33,45,7,30,42,4,36,39),(2,31,37,8,28,43,5,34,40),(3,32,44,9,29,41,6,35,38),(10,52,19,13,46,22,16,49,25),(11,53,26,14,47,20,17,50,23),(12,51,27,15,54,21,18,48,24)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,46),(9,47),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42)]])

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A ··· 9F 9G ··· 9O 18A ··· 18F order 1 2 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 18 27 27 9 ··· 9 18 ··· 18 27 ··· 27

31 irreducible representations

 dim 1 1 1 1 2 2 3 6 6 6 type + + + + + image C1 C2 C3 C6 S3 C3×S3 He3.C6 C32⋊C6 He3.2S3 C33.(C3×S3) kernel C33.(C3×S3) C32.29He3 C32⋊2D9 C32⋊C9 C32⋊C9 C33 C3 C32 C3 C1 # reps 1 1 2 2 1 2 12 1 3 6

Matrix representation of C33.(C3×S3) in GL6(𝔽19)

 1 0 0 0 0 0 8 11 0 0 0 0 1 0 7 0 0 0 11 0 0 11 0 0 0 0 0 0 1 0 18 0 0 0 0 7
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 7 0 0 7 0 0 7 0 0 0 7 0 7 0 0 0 0 7
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 9 0 16 0 0 0 15 0 10 0 0 0 15 4 10 0 0 0 0 0 9 0 0 9 4 0 9 4 0 0 4 0 9 0 4 0
,
 11 9 0 0 0 0 0 8 11 0 0 0 11 8 0 0 0 0 8 11 0 0 0 1 7 11 0 7 0 0 7 11 0 0 7 0
,
 18 0 0 6 0 0 0 0 0 18 1 0 0 0 0 18 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0

G:=sub<GL(6,GF(19))| [1,8,1,11,0,18,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[11,0,0,7,7,7,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[9,15,15,0,4,4,0,0,4,0,0,0,16,10,10,9,9,9,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,9,0,0],[11,0,11,8,7,7,9,8,8,11,11,11,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,1,0,0],[18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,6,18,18,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0] >;

C33.(C3×S3) in GAP, Magma, Sage, TeX

C_3^3.(C_3\times S_3)
% in TeX

G:=Group("C3^3.(C3xS3)");
// GroupNames label

G:=SmallGroup(486,11);
// by ID

G=gap.SmallGroup(486,11);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,8643,873,1383,3244]);
// Polycyclic

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

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