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## G = S3×C27⋊C3order 486 = 2·35

### Direct product of S3 and C27⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×C27⋊C3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C27 — C3×C27⋊C3 — S3×C27⋊C3
 Lower central C3 — C32 — S3×C27⋊C3
 Upper central C1 — C9 — C27⋊C3

Generators and relations for S3×C27⋊C3
G = < a,b,c,d | a3=b2=c27=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c10 >

Subgroups: 128 in 66 conjugacy classes, 33 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, C18, C3×S3, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C3×C9, C33, C54, S3×C9, S3×C9, C3×C18, S3×C32, C3×C27, C27⋊C3, C27⋊C3, C32×C9, S3×C27, C2×C27⋊C3, S3×C3×C9, C3×C27⋊C3, S3×C27⋊C3
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, C27⋊C3, C2×C27⋊C3, S3×C3×C9, S3×C27⋊C3

Smallest permutation representation of S3×C27⋊C3
On 54 points
Generators in S54
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 28)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(28 46 37)(29 38 47)(31 49 40)(32 41 50)(34 52 43)(35 44 53)

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

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

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

99 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 9M 9N 9O 9P 9Q 9R 9S 9T 18A ··· 18F 18G 18H 18I 18J 27A ··· 27R 27S ··· 27AJ 54A ··· 54R order 1 2 3 3 3 3 3 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 9 9 9 9 9 9 9 9 18 ··· 18 18 18 18 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 3 1 1 2 2 2 3 3 6 6 3 3 9 9 1 ··· 1 2 ··· 2 3 3 3 3 6 6 6 6 3 ··· 3 9 9 9 9 3 ··· 3 6 ··· 6 9 ··· 9

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C6 C6 C9 C9 C18 C18 S3 C3×S3 C3×S3 S3×C9 S3×C9 C27⋊C3 C2×C27⋊C3 S3×C27⋊C3 kernel S3×C27⋊C3 C3×C27⋊C3 S3×C27 S3×C3×C9 C3×C27 C32×C9 S3×C9 S3×C32 C3×C9 C33 C27⋊C3 C27 C3×C9 C9 C32 S3 C3 C1 # reps 1 1 6 2 6 2 12 6 12 6 1 6 2 12 6 6 6 6

Matrix representation of S3×C27⋊C3 in GL5(𝔽109)

 108 1 0 0 0 108 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 108 0 0 0 108 0 0 0 0 0 0 108 0 0 0 0 0 108 0 0 0 0 0 108
,
 63 0 0 0 0 0 63 0 0 0 0 0 6 1 36 0 0 4 0 93 0 0 11 0 103
,
 45 0 0 0 0 0 45 0 0 0 0 0 1 0 24 0 0 0 63 43 0 0 0 0 45

G:=sub<GL(5,GF(109))| [108,108,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,108,0,0,0,108,0,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,108],[63,0,0,0,0,0,63,0,0,0,0,0,6,4,11,0,0,1,0,0,0,0,36,93,103],[45,0,0,0,0,0,45,0,0,0,0,0,1,0,0,0,0,0,63,0,0,0,24,43,45] >;

S3×C27⋊C3 in GAP, Magma, Sage, TeX

S_3\times C_{27}\rtimes C_3
% in TeX

G:=Group("S3xC27:C3");
// GroupNames label

G:=SmallGroup(486,114);
// by ID

G=gap.SmallGroup(486,114);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,68,93,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^27=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^10>;
// generators/relations

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