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## G = S3×C33order 162 = 2·34

### Direct product of C33 and S3

Aliases: S3×C33, C341C2, C337C6, C3⋊(C32×C6), C323(C3×C6), SmallGroup(162,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C33
 Chief series C1 — C3 — C32 — C33 — C34 — S3×C33
 Lower central C3 — S3×C33
 Upper central C1 — C33

Generators and relations for S3×C33
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 324 in 190 conjugacy classes, 84 normal (6 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C3×S3, C3×C6, C33, C33, C33, S3×C32, C32×C6, C34, S3×C33
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C33, S3×C32, C32×C6, S3×C33

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

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

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

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

81 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3BA 6A ··· 6Z order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3

81 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 S3 C3×S3 kernel S3×C33 C34 S3×C32 C33 C33 C32 # reps 1 1 26 26 1 26

Matrix representation of S3×C33 in GL4(𝔽7) generated by

 4 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 0 0 1 0 0 0 0 2 0 0 0 4 4
,
 6 0 0 0 0 6 0 0 0 0 6 3 0 0 0 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,2,4,0,0,0,4],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,3,1] >;

S3×C33 in GAP, Magma, Sage, TeX

S_3\times C_3^3
% in TeX

G:=Group("S3xC3^3");
// GroupNames label

G:=SmallGroup(162,51);
// by ID

G=gap.SmallGroup(162,51);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,2704]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,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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