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## G = C3×He3.3S3order 486 = 2·35

### Direct product of C3 and He3.3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3.C3 — C3×He3.3S3
 Chief series C1 — C3 — C32 — C3×C9 — He3.C3 — C3×He3.C3 — C3×He3.3S3
 Lower central He3.C3 — C3×He3.3S3
 Upper central C1 — C3

Generators and relations for C3×He3.3S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=fcf=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=fbf=bc-1, be=eb, cd=dc, ce=ec, ede-1=b-1cd, fdf=d-1, fef=ce2 >

Subgroups: 524 in 96 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, S3×C32, C3×C3⋊S3, He3.C3, He3.C3, C32×C9, C3×He3, C3×3- 1+2, He3.3S3, C32×D9, C3×C32⋊C6, C3×C9⋊C6, C3×He3.C3, C3×He3.3S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, He3.3S3, C3×He3⋊C2, C3×He3.3S3

Smallest permutation representation of C3×He3.3S3
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)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(46 49 52)(47 50 53)(48 51 54)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(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 25 10)(2 23 11)(3 21 12)(4 19 13)(5 26 14)(6 24 15)(7 22 16)(8 20 17)(9 27 18)(28 37 52)(29 38 50)(30 39 48)(31 40 46)(32 41 53)(33 42 51)(34 43 49)(35 44 47)(36 45 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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 36)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 45)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(46,49,52)(47,50,53)(48,51,54), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(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), (1,25,10)(2,23,11)(3,21,12)(4,19,13)(5,26,14)(6,24,15)(7,22,16)(8,20,17)(9,27,18)(28,37,52)(29,38,50)(30,39,48)(31,40,46)(32,41,53)(33,42,51)(34,43,49)(35,44,47)(36,45,54), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,36)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,54)>;

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 6 6 type + + + + + + image C1 C2 C3 C6 S3 S3 S3 C3×S3 C3×S3 C3×S3 He3⋊C2 He3.3S3 C3×He3.3S3 kernel C3×He3.3S3 C3×He3.C3 He3.3S3 He3.C3 C32×C9 C3×He3 C3×3- 1+2 C3×C9 He3 3- 1+2 C32 C3 C1 # reps 1 1 2 2 1 1 2 2 2 4 12 3 6

Matrix representation of C3×He3.3S3 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 1 0 0 0 0 0 6 7 0 0 0 0 3 0 11 0 0 0 0 0 0 11 0 0 0 0 0 9 1 0 5 0 0 11 0 7
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 16 0 0 0 0 11
,
 6 6 0 0 0 0 0 13 1 0 0 0 18 2 0 0 0 0 5 0 0 12 0 6 14 0 0 4 0 13 4 14 0 15 1 7
,
 16 0 0 0 0 0 0 16 0 0 0 0 10 0 5 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 9 0 0 15 0 4
,
 0 0 0 1 0 0 0 0 0 0 1 0 4 0 0 15 0 1 1 0 0 0 0 0 0 1 0 0 0 0 4 0 1 15 0 0

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,6,3,0,0,5,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,9,11,0,0,0,0,1,0,0,0,0,0,0,7],[7,0,0,0,0,16,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[6,0,18,5,14,4,6,13,2,0,0,14,0,1,0,0,0,0,0,0,0,12,4,15,0,0,0,0,0,1,0,0,0,6,13,7],[16,0,10,0,0,9,0,16,0,0,0,0,0,0,5,0,0,0,0,0,0,6,0,15,0,0,0,0,6,0,0,0,0,0,0,4],[0,0,4,1,0,4,0,0,0,0,1,0,0,0,0,0,0,1,1,0,15,0,0,15,0,1,0,0,0,0,0,0,1,0,0,0] >;

C3×He3.3S3 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._3S_3
% in TeX

G:=Group("C3xHe3.3S3");
// GroupNames label

G:=SmallGroup(486,168);
// by ID

G=gap.SmallGroup(486,168);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,8643,303,237,11344,1096,11669]);
// Polycyclic

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

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