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## G = C3×He3.S3order 486 = 2·35

### Direct product of C3 and He3.S3

Aliases: C3×He3.S3, C9⋊S36C32, (C32×C9)⋊8C6, He3.C38C6, (C3×He3).8S3, He3.1(C3×S3), C33.58(C3×S3), C32.14(S3×C32), C32.43(C32⋊C6), (C3×C9⋊S3)⋊2C3, (C3×C9)⋊10(C3×C6), (C3×He3.C3)⋊3C2, C3.5(C3×C32⋊C6), SmallGroup(486,119)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×He3.S3
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=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, ede-1=b-1cd, df=fd, fef=ce2 >

Subgroups: 492 in 78 conjugacy classes, 22 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, He3.C3, He3.C3, C32×C9, C3×He3, C3×3- 1+2, He3.S3, C3×C32⋊C6, C3×C9⋊S3, C3×He3.C3, C3×He3.S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, He3.S3, C3×C32⋊C6, C3×He3.S3

Smallest permutation representation of C3×He3.S3
On 54 points
Generators in S54
(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 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)
(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 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 14 20)(2 12 21)(3 10 22)(4 17 23)(5 15 24)(6 13 25)(7 11 26)(8 18 27)(9 16 19)(28 43 46)(29 41 47)(30 39 48)(31 37 49)(32 44 50)(33 42 51)(34 40 52)(35 38 53)(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 36)(9 35)(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)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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), (1,7,4)(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), (1,14,20)(2,12,21)(3,10,22)(4,17,23)(5,15,24)(6,13,25)(7,11,26)(8,18,27)(9,16,19)(28,43,46)(29,41,47)(30,39,48)(31,37,49)(32,44,50)(33,42,51)(34,40,52)(35,38,53)(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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,36)(9,35)(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( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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), (1,7,4)(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), (1,14,20)(2,12,21)(3,10,22)(4,17,23)(5,15,24)(6,13,25)(7,11,26)(8,18,27)(9,16,19)(28,43,46)(29,41,47)(30,39,48)(31,37,49)(32,44,50)(33,42,51)(34,40,52)(35,38,53)(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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,36)(9,35)(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([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(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)], [(1,7,4),(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)], [(1,14,20),(2,12,21),(3,10,22),(4,17,23),(5,15,24),(6,13,25),(7,11,26),(8,18,27),(9,16,19),(28,43,46),(29,41,47),(30,39,48),(31,37,49),(32,44,50),(33,42,51),(34,40,52),(35,38,53),(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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,36),(9,35),(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 3G 3H 3I ··· 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 6 6 6 9 ··· 9 27 ··· 27 6 ··· 6 18 ··· 18

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 6 6 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 C32⋊C6 He3.S3 C3×C32⋊C6 C3×He3.S3 kernel C3×He3.S3 C3×He3.C3 He3.S3 C3×C9⋊S3 He3.C3 C32×C9 C3×He3 He3 C33 C32 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 1 3 2 6

Matrix representation of C3×He3.S3 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 0 7 0 0 0 0 1 18 11 0 0 0 0 0 0 1 0 0 8 12 0 8 11 0 8 0 0 1 0 7
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 12 7 0 0 7 0 1 18 0 0 0 7
,
 12 8 6 0 0 0 1 0 0 0 0 0 0 0 7 0 0 0 8 11 0 1 0 6 0 0 18 0 0 18 0 0 11 0 1 18
,
 6 0 0 0 0 0 0 4 0 0 0 0 0 9 6 0 0 0 0 0 0 16 0 0 3 3 0 14 5 0 7 3 0 0 0 16
,
 0 0 0 1 0 0 1 18 0 1 6 0 0 0 0 0 11 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 8 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],[1,0,1,0,8,8,0,7,18,0,12,0,0,0,11,0,0,0,0,0,0,1,8,1,0,0,0,0,11,0,0,0,0,0,0,7],[11,0,0,0,12,1,0,11,0,0,7,18,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[12,1,0,8,0,0,8,0,0,11,0,0,6,0,7,0,18,11,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,6,18,18],[6,0,0,0,3,7,0,4,9,0,3,3,0,0,6,0,0,0,0,0,0,16,14,0,0,0,0,0,5,0,0,0,0,0,0,16],[0,1,0,1,0,0,0,18,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,6,11,0,1,8,0,0,1,0,0,0] >;

C3×He3.S3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(486,119);
// by ID

G=gap.SmallGroup(486,119);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,8643,873,237,3244,3250,11669]);
// 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=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*c*d,d*f=f*d,f*e*f=c*e^2>;
// generators/relations

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