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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 — C3×He3 — 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=f3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf-1=cd-1, de=ed, df=fd, fef-1=cde >

Subgroups: 488 in 90 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, C33, C33, S3×C9, C3×C18, C2×He3, S3×C32, S3×C32, He3⋊C3, He3⋊C3, C32×C9, C3×He3, C2×He3⋊C3, S3×C3×C9, S3×He3, 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 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 29)(2 28)(3 30)(4 32)(5 31)(6 33)(7 35)(8 34)(9 36)(10 38)(11 37)(12 39)(13 41)(14 40)(15 42)(16 44)(17 43)(18 45)(19 47)(20 46)(21 48)(22 50)(23 49)(24 51)(25 53)(26 52)(27 54)
(1 3 2)(7 8 9)(13 14 15)(16 18 17)(19 20 21)(22 24 23)(28 29 30)(34 36 35)(40 42 41)(43 44 45)(46 48 47)(49 50 51)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 15 14)(16 18 17)(19 21 20)(22 24 23)(25 27 26)(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 26 15)(2 27 13)(3 25 14)(4 20 17)(5 21 18)(6 19 16)(7 23 10)(8 24 11)(9 22 12)(28 54 41)(29 52 42)(30 53 40)(31 48 45)(32 46 43)(33 47 44)(34 51 37)(35 49 38)(36 50 39)
(1 10 20)(2 11 21)(3 12 19)(4 14 22)(5 15 23)(6 13 24)(7 18 27)(8 16 25)(9 17 26)(28 37 48)(29 38 46)(30 39 47)(31 42 49)(32 40 50)(33 41 51)(34 44 53)(35 45 54)(36 43 52)

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

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

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

51 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J ··· 3O 3P ··· 3U 6A 6B 6C 6D 6E ··· 6J 9A ··· 9F 9G ··· 9L 18A ··· 18F 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 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

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 6 6 type + + + image C1 C2 C3 C3 C6 C6 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 C32×C9 C3×He3 He3⋊C3 C3×C9 He3 C3×S3 C32 S3 C3 C3 C1 # reps 1 1 2 6 2 6 1 2 6 2 2 6 6 2 6

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

 0 1 0 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 0 0 0 0 1 1 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 7 0 0 0 0 0 1 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 17 0 0 17 0 0 0 0 0 5 0
,
 11 0 0 0 0 0 11 0 0 0 0 0 0 0 9 0 0 9 0 0 0 0 0 4 0

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,1,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11],[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],[7,0,0,0,0,0,7,0,0,0,0,0,0,17,0,0,0,0,0,5,0,0,17,0,0],[11,0,0,0,0,0,11,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,9,0,0] >;

S3×He3⋊C3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(486,123);
// by ID

G=gap.SmallGroup(486,123);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,873,303,11669]);
// Polycyclic

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

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