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## G = C3×He3⋊5S3order 486 = 2·35

### Direct product of C3 and He3⋊5S3

Aliases: C3×He35S3, C3412S3, C339(C3⋊S3), He310(C3×S3), (C3×He3)⋊24C6, (C3×He3)⋊23S3, C3311(C3×S3), (C32×He3)⋊7C2, C323(He3⋊C2), C32.14(C33⋊C2), C3⋊(C3×He3⋊C2), C322(C3×C3⋊S3), C3.6(C3×C33⋊C2), SmallGroup(486,243)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — C3×He3⋊5S3
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C32×He3 — C3×He3⋊5S3
 Lower central C3×He3 — C3×He3⋊5S3
 Upper central C1 — C32

Generators and relations for C3×He35S3
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, fbf=b-1, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1874 in 417 conjugacy classes, 66 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, C33, C33, He3⋊C2, S3×C32, C3×C3⋊S3, C3×He3, C3×He3, C3×He3, C34, C3×He3⋊C2, He35S3, C32×C3⋊S3, C32×He3, C3×He35S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, C33⋊C2, C3×He3⋊C2, He35S3, C3×C33⋊C2, C3×He35S3

Smallest permutation representation of C3×He35S3
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)
(1 25 17)(2 26 18)(3 27 16)(4 37 24)(5 38 22)(6 39 23)(7 14 29)(8 15 30)(9 13 28)(10 46 35)(11 47 36)(12 48 34)(19 40 50)(20 41 51)(21 42 49)(31 52 44)(32 53 45)(33 54 43)
(1 14 23)(2 15 24)(3 13 22)(4 26 30)(5 27 28)(6 25 29)(7 39 17)(8 37 18)(9 38 16)(10 41 32)(11 42 33)(12 40 31)(19 44 34)(20 45 35)(21 43 36)(46 51 53)(47 49 54)(48 50 52)
(4 26 30)(5 27 28)(6 25 29)(7 17 39)(8 18 37)(9 16 38)(10 32 41)(11 33 42)(12 31 40)(19 44 34)(20 45 35)(21 43 36)
(1 15 22)(2 13 23)(3 14 24)(4 5 6)(7 18 38)(8 16 39)(9 17 37)(10 12 11)(19 43 35)(20 44 36)(21 45 34)(25 26 27)(28 29 30)(31 33 32)(40 42 41)(46 52 49)(47 53 50)(48 54 51)
(1 52)(2 53)(3 54)(4 41)(5 42)(6 40)(7 34)(8 35)(9 36)(10 30)(11 28)(12 29)(13 47)(14 48)(15 46)(16 43)(17 44)(18 45)(19 39)(20 37)(21 38)(22 49)(23 50)(24 51)(25 31)(26 32)(27 33)

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

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

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

63 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 3R ··· 3BA 6A ··· 6H order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 size 1 27 1 ··· 1 2 ··· 2 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 6 type + + + + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 He3⋊C2 He3⋊5S3 kernel C3×He3⋊5S3 C32×He3 He3⋊5S3 C3×He3 C3×He3 C34 He3 C33 C32 C3 # reps 1 1 2 2 9 4 18 8 12 6

Matrix representation of C3×He35S3 in GL5(𝔽7)

 4 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 2 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 0 6 0 0 0 1 6 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 2
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 4
,
 0 6 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 5 0 0 0 3 0

G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[0,1,0,0,0,6,6,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4],[0,6,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,3,0,0,0,5,0] >;

C3×He35S3 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3\rtimes_5S_3
% in TeX

G:=Group("C3xHe3:5S3");
// GroupNames label

G:=SmallGroup(486,243);
// by ID

G=gap.SmallGroup(486,243);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,867,3244,382]);
// 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,f*b*f=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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