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G = C3×He3⋊S3order 486 = 2·35

Direct product of C3 and He3⋊S3

Aliases: C3×He3⋊S3, He33(C3×S3), (C3×He3)⋊11S3, (C32×C9)⋊16S3, He3⋊C312C6, C33.34(C3⋊S3), C32.6(He3⋊C2), (C3×C9)⋊14(C3×S3), C32.5(C3×C3⋊S3), (C3×He3⋊C3)⋊5C2, C3.7(C3×He3⋊C2), SmallGroup(486,171)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3⋊C3 — C3×He3⋊S3
 Chief series C1 — C3 — C32 — He3 — He3⋊C3 — C3×He3⋊C3 — C3×He3⋊S3
 Lower central He3⋊C3 — 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=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, dbd-1=bc-1, bf=fb, cd=dc, ce=ec, fcf=c-1, ede-1=b-1c-1d, fdf=bc-1d-1, fef=e-1 >

Subgroups: 740 in 108 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, He3, He3, C33, C33, C3×D9, C32⋊C6, S3×C32, C3×C3⋊S3, He3⋊C3, He3⋊C3, C32×C9, C3×He3, He3⋊S3, C32×D9, C3×C32⋊C6, C3×He3⋊C3, C3×He3⋊S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, He3⋊S3, C3×He3⋊C2, C3×He3⋊S3

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

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

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

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 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
,
 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
,
 0 4 0 0 0 0 0 0 9 0 0 0 9 0 0 0 0 0 0 0 0 0 0 5 0 0 0 17 0 0 0 0 0 0 17 0
,
 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
,
 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))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[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],[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],[0,0,9,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,5,0,0],[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],[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⋊S3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(486,171);
// by ID

G=gap.SmallGroup(486,171);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,867,303,1096,652,11669]);
// Polycyclic

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

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