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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 — C33 — C3×He3⋊4S3
 Chief series C1 — C3 — C32 — C33 — C34 — C32×He3 — C3×He3⋊4S3
 Lower central C33 — C3×He3⋊4S3
 Upper central C1 — C3

Generators and relations for C3×He34S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=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, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 1554 in 333 conjugacy classes, 50 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, C33, C33, C32⋊C6, S3×C32, C3×C3⋊S3, C33⋊C2, C3×He3, C3×He3, C34, C34, C3×C32⋊C6, He34S3, C32×C3⋊S3, C3×C33⋊C2, C32×He3, C3×He34S3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, C32⋊C6, S3×C32, C3×C3⋊S3, C3×C32⋊C6, He34S3, C32×C3⋊S3, C3×He34S3

Smallest permutation representation of C3×He34S3
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 31 20)(2 32 21)(3 33 19)(4 41 44)(5 42 45)(6 40 43)(7 36 46)(8 34 47)(9 35 48)(10 27 24)(11 25 22)(12 26 23)(13 29 16)(14 30 17)(15 28 18)(37 51 54)(38 49 52)(39 50 53)
(1 10 14)(2 11 15)(3 12 13)(4 54 8)(5 52 9)(6 53 7)(16 19 23)(17 20 24)(18 21 22)(25 28 32)(26 29 33)(27 30 31)(34 41 37)(35 42 38)(36 40 39)(43 50 46)(44 51 47)(45 49 48)
(1 28 16)(2 29 17)(3 30 18)(4 45 36)(5 43 34)(6 44 35)(7 47 38)(8 48 39)(9 46 37)(10 32 19)(11 33 20)(12 31 21)(13 27 22)(14 25 23)(15 26 24)(40 54 49)(41 52 50)(42 53 51)
(1 15 12)(2 13 10)(3 14 11)(4 7 52)(5 8 53)(6 9 54)(16 24 21)(17 22 19)(18 23 20)(25 33 30)(26 31 28)(27 32 29)(34 39 42)(35 37 40)(36 38 41)(43 48 51)(44 46 49)(45 47 50)
(1 34)(2 35)(3 36)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(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), (1,31,20)(2,32,21)(3,33,19)(4,41,44)(5,42,45)(6,40,43)(7,36,46)(8,34,47)(9,35,48)(10,27,24)(11,25,22)(12,26,23)(13,29,16)(14,30,17)(15,28,18)(37,51,54)(38,49,52)(39,50,53), (1,10,14)(2,11,15)(3,12,13)(4,54,8)(5,52,9)(6,53,7)(16,19,23)(17,20,24)(18,21,22)(25,28,32)(26,29,33)(27,30,31)(34,41,37)(35,42,38)(36,40,39)(43,50,46)(44,51,47)(45,49,48), (1,28,16)(2,29,17)(3,30,18)(4,45,36)(5,43,34)(6,44,35)(7,47,38)(8,48,39)(9,46,37)(10,32,19)(11,33,20)(12,31,21)(13,27,22)(14,25,23)(15,26,24)(40,54,49)(41,52,50)(42,53,51), (1,15,12)(2,13,10)(3,14,11)(4,7,52)(5,8,53)(6,9,54)(16,24,21)(17,22,19)(18,23,20)(25,33,30)(26,31,28)(27,32,29)(34,39,42)(35,37,40)(36,38,41)(43,48,51)(44,46,49)(45,47,50), (1,34)(2,35)(3,36)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(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), (1,31,20)(2,32,21)(3,33,19)(4,41,44)(5,42,45)(6,40,43)(7,36,46)(8,34,47)(9,35,48)(10,27,24)(11,25,22)(12,26,23)(13,29,16)(14,30,17)(15,28,18)(37,51,54)(38,49,52)(39,50,53), (1,10,14)(2,11,15)(3,12,13)(4,54,8)(5,52,9)(6,53,7)(16,19,23)(17,20,24)(18,21,22)(25,28,32)(26,29,33)(27,30,31)(34,41,37)(35,42,38)(36,40,39)(43,50,46)(44,51,47)(45,49,48), (1,28,16)(2,29,17)(3,30,18)(4,45,36)(5,43,34)(6,44,35)(7,47,38)(8,48,39)(9,46,37)(10,32,19)(11,33,20)(12,31,21)(13,27,22)(14,25,23)(15,26,24)(40,54,49)(41,52,50)(42,53,51), (1,15,12)(2,13,10)(3,14,11)(4,7,52)(5,8,53)(6,9,54)(16,24,21)(17,22,19)(18,23,20)(25,33,30)(26,31,28)(27,32,29)(34,39,42)(35,37,40)(36,38,41)(43,48,51)(44,46,49)(45,47,50), (1,34)(2,35)(3,36)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(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)], [(1,31,20),(2,32,21),(3,33,19),(4,41,44),(5,42,45),(6,40,43),(7,36,46),(8,34,47),(9,35,48),(10,27,24),(11,25,22),(12,26,23),(13,29,16),(14,30,17),(15,28,18),(37,51,54),(38,49,52),(39,50,53)], [(1,10,14),(2,11,15),(3,12,13),(4,54,8),(5,52,9),(6,53,7),(16,19,23),(17,20,24),(18,21,22),(25,28,32),(26,29,33),(27,30,31),(34,41,37),(35,42,38),(36,40,39),(43,50,46),(44,51,47),(45,49,48)], [(1,28,16),(2,29,17),(3,30,18),(4,45,36),(5,43,34),(6,44,35),(7,47,38),(8,48,39),(9,46,37),(10,32,19),(11,33,20),(12,31,21),(13,27,22),(14,25,23),(15,26,24),(40,54,49),(41,52,50),(42,53,51)], [(1,15,12),(2,13,10),(3,14,11),(4,7,52),(5,8,53),(6,9,54),(16,24,21),(17,22,19),(18,23,20),(25,33,30),(26,31,28),(27,32,29),(34,39,42),(35,37,40),(36,38,41),(43,48,51),(44,46,49),(45,47,50)], [(1,34),(2,35),(3,36),(4,30),(5,28),(6,29),(7,33),(8,31),(9,32),(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)]])

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 3O ··· 3T 3U ··· 3BA 6A ··· 6H order 1 2 3 3 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 size 1 27 1 1 2 ··· 2 3 ··· 3 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + + image C1 C2 C3 C3 C6 C6 S3 S3 C3×S3 C3×S3 C32⋊C6 C3×C32⋊C6 kernel C3×He3⋊4S3 C32×He3 He3⋊4S3 C3×C33⋊C2 C3×He3 C34 C3×He3 C34 He3 C33 C32 C3 # reps 1 1 6 2 6 2 3 1 18 14 3 6

Matrix representation of C3×He34S3 in GL8(𝔽7)

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

G:=sub<GL(8,GF(7))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,4,5,5,0,0,0,0,4,5,2,0,6,4,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,3,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,3,2,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,4,6,2,0,3,0,0,1,3,4,0,5,1,0,0,0,2,0,0,0,0,0,0,0,0,0,2,4,3,0,0,0,0,0,1,5,3,0,0,0,0,0,0,2,0],[4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,2,6,1,2,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,5,1,0,5,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;

C3×He34S3 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3\rtimes_4S_3
% in TeX

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

G:=SmallGroup(486,229);
// by ID

G=gap.SmallGroup(486,229);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,2169,3244,11669]);
// 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,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=e^-1>;
// generators/relations

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