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## G = C3×C33.S3order 486 = 2·35

### Direct product of C3 and C33.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C3×C33.S3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C32×3- 1+2 — C3×C33.S3
 Lower central C3×C9 — C3×C33.S3
 Upper central C1 — C3

Generators and relations for C3×C33.S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bd-1, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=d-1e2 >

Subgroups: 852 in 234 conjugacy classes, 50 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C33, C33, C3×D9, C9⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, C32×C9, C32×C9, C3×3- 1+2, C3×3- 1+2, C34, C3×C9⋊C6, C3×C9⋊S3, C33.S3, C32×C3⋊S3, C32×3- 1+2, C3×C33.S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, C9⋊C6, S3×C32, C3×C3⋊S3, C3×C9⋊C6, C33.S3, C32×C3⋊S3, C3×C33.S3

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

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 3O ··· 3T 3U ··· 3Z 6A ··· 6H 9A ··· 9AA order 1 2 3 3 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 27 1 1 2 ··· 2 3 ··· 3 6 ··· 6 27 ··· 27 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 S3 C3×S3 C3×S3 C3×S3 C9⋊C6 C3×C9⋊C6 kernel C3×C33.S3 C32×3- 1+2 C3×C9⋊S3 C33.S3 C32×C9 C3×3- 1+2 C3×3- 1+2 C34 C3×C9 3- 1+2 C33 C32 C3 # reps 1 1 2 6 2 6 3 1 6 18 8 3 6

Matrix representation of C3×C33.S3 in GL8(𝔽19)

 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11
,
 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7
,
 11 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 7 0 0
,
 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,GF(19))| [7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;

C3×C33.S3 in GAP, Magma, Sage, TeX

C_3\times C_3^3.S_3
% in TeX

G:=Group("C3xC3^3.S3");
// GroupNames label

G:=SmallGroup(486,232);
// by ID

G=gap.SmallGroup(486,232);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,2169,453,3244,11669]);
// Polycyclic

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

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