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## G = S3×He3.C3order 486 = 2·35

### Direct product of S3 and He3.C3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×He3.C3
G = < a,b,c,d,e,f | a3=b2=c3=d3=e3=1, f3=d, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, cf=fc, de=ed, df=fd, fef-1=cd-1e >

Subgroups: 308 in 78 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, S3×C9, C3×C18, C2×He3, C2×3- 1+2, S3×C32, S3×C32, He3.C3, He3.C3, C32×C9, C3×He3, C3×3- 1+2, C2×He3.C3, S3×C3×C9, S3×He3, S3×3- 1+2, C3×He3.C3, S3×He3.C3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, He3, C2×He3, S3×C32, He3.C3, C2×He3.C3, S3×He3, S3×He3.C3

Smallest permutation representation of S3×He3.C3
On 54 points
Generators in S54
(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 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 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(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 26 14)(2 24 15)(3 22 16)(4 20 17)(5 27 18)(6 25 10)(7 23 11)(8 21 12)(9 19 13)(28 52 37)(29 50 38)(30 48 39)(31 46 40)(32 53 41)(33 51 42)(34 49 43)(35 47 44)(36 54 45)
(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)

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 type + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×S3 C3×S3 C3×S3 He3 C2×He3 He3.C3 C2×He3.C3 S3×He3 S3×He3.C3 kernel S3×He3.C3 C3×He3.C3 S3×C3×C9 S3×He3 S3×3- 1+2 C32×C9 C3×He3 C3×3- 1+2 He3.C3 C3×C9 He3 3- 1+2 C3×S3 C32 S3 C3 C3 C1 # reps 1 1 2 2 4 2 2 4 1 2 2 4 2 2 6 6 2 6

Matrix representation of S3×He3.C3 in GL5(𝔽19)

 18 1 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 18 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 11
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 7 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 11 0 0 0 0 0 11 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 4

G:=sub<GL(5,GF(19))| [18,18,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[7,0,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[11,0,0,0,0,0,11,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,4] >;

S3×He3.C3 in GAP, Magma, Sage, TeX

S_3\times {\rm He}_3.C_3
% in TeX

G:=Group("S3xHe3.C3");
// GroupNames label

G:=SmallGroup(486,120);
// by ID

G=gap.SmallGroup(486,120);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,176,873,11669]);
// Polycyclic

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

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