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## G = C3×He3.2S3order 486 = 2·35

### Direct product of C3 and He3.2S3

Aliases: C3×He3.2S3, C9⋊S37C32, (C32×C9)⋊9C6, He3.2(C3×S3), (C3×He3).9S3, C33.59(C3×S3), He3⋊C311C6, C32.15(S3×C32), C32.44(C32⋊C6), (C3×C9⋊S3)⋊3C3, (C3×C9)⋊11(C3×C6), C3.6(C3×C32⋊C6), (C3×He3⋊C3)⋊4C2, SmallGroup(486,122)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C3×He3.2S3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C3×He3⋊C3 — C3×He3.2S3
 Lower central C3×C9 — C3×He3.2S3
 Upper central C1 — C3

Generators and relations for C3×He3.2S3
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, ede-1=b-1cd, df=fd, fef=c-1e2 >

Subgroups: 600 in 84 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, C33, C33, C3×D9, C32⋊C6, C9⋊S3, S3×C32, C3×C3⋊S3, He3⋊C3, He3⋊C3, C32×C9, C3×He3, C3×He3, He3.2S3, C3×C32⋊C6, C3×C9⋊S3, C3×He3⋊C3, C3×He3.2S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, He3.2S3, C3×C32⋊C6, C3×He3.2S3

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

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 He3.2S3 C3×C32⋊C6 C3×He3.2S3 kernel C3×He3.2S3 C3×He3⋊C3 He3.2S3 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×He3.2S3 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
,
 7 0 0 0 0 0 8 1 0 0 0 0 11 0 11 0 0 0 0 0 0 11 0 0 0 0 0 7 1 0 0 0 0 8 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 0 0 0 0 0 11
,
 7 0 6 0 0 0 0 0 8 0 0 0 0 1 12 0 0 0 0 0 0 7 0 6 0 0 0 0 0 8 0 0 0 0 1 12
,
 6 0 0 0 0 0 0 6 0 0 0 0 13 0 9 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 17 0 17
,
 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],[7,8,11,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,7,8,0,0,0,0,1,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,1,0,0,0,6,8,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,6,8,12],[6,0,13,0,0,0,0,6,0,0,0,0,0,0,9,0,0,0,0,0,0,16,0,17,0,0,0,0,16,0,0,0,0,0,0,17],[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×He3.2S3 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._2S_3
% in TeX

G:=Group("C3xHe3.2S3");
// GroupNames label

G:=SmallGroup(486,122);
// by ID

G=gap.SmallGroup(486,122);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,873,453,3244,3250,11669]);
// 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,e*d*e^-1=b^-1*c*d,d*f=f*d,f*e*f=c^-1*e^2>;
// generators/relations

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