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## G = C3×He3.4S3order 486 = 2·35

### Direct product of C3 and He3.4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C3×He3.4S3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C3×C9○He3 — C3×He3.4S3
 Lower central C3×C9 — C3×He3.4S3
 Upper central C1 — C3

Generators and relations for C3×He3.4S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=c, 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, fcf=c-1, de=ed, df=fd, fef=c-1e2 >

Subgroups: 636 in 186 conjugacy classes, 44 normal (17 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, C32×C9, C32×C9, C3×He3, C3×3- 1+2, C3×3- 1+2, C9○He3, C9○He3, C32×D9, C3×C32⋊C6, C3×C9⋊C6, C3×C9⋊S3, He3.4S3, C3×C9○He3, C3×He3.4S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, He3.4S3, C32×C3⋊S3, C3×He3.4S3

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

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3T 6A ··· 6H 9A ··· 9I 9J ··· 9AG order 1 2 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 6 ··· 6 27 ··· 27 2 ··· 2 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 6 6 type + + + + + + image C1 C2 C3 C3 C6 C6 S3 S3 S3 C3×S3 C3×S3 C3×S3 C3×S3 He3.4S3 C3×He3.4S3 kernel C3×He3.4S3 C3×C9○He3 C3×C9⋊S3 He3.4S3 C32×C9 C9○He3 C32×C9 C3×He3 C3×3- 1+2 C3×C9 He3 3- 1+2 C33 C3 C1 # reps 1 1 2 6 2 6 1 1 2 12 6 12 2 3 6

Matrix representation of C3×He3.4S3 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
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 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],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C3×He3.4S3 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._4S_3
% in TeX

G:=Group("C3xHe3.4S3");
// GroupNames label

G:=SmallGroup(486,234);
// by ID

G=gap.SmallGroup(486,234);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,453,3244,3250,11669]);
// Polycyclic

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

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