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G = C3⋊S3×C33order 486 = 2·35

Direct product of C33 and C3⋊S3

Aliases: C3⋊S3×C33, C352C2, C3414S3, C3415C6, C3⋊(S3×C33), C3314(C3×C6), C3318(C3×S3), C324(C32×C6), C325(S3×C32), SmallGroup(486,257)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C33
 Chief series C1 — C3 — C32 — C33 — C34 — C35 — C3⋊S3×C33
 Lower central C32 — C3⋊S3×C33
 Upper central C1 — C33

Generators and relations for C3⋊S3×C33
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, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 3280 in 1584 conjugacy classes, 196 normal (6 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, C33, C33, C33, S3×C32, C3×C3⋊S3, C32×C6, C34, C34, S3×C33, C32×C3⋊S3, C35, C3⋊S3×C33
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, C33, S3×C32, C3×C3⋊S3, C32×C6, S3×C33, C32×C3⋊S3, C3⋊S3×C33

Smallest permutation representation of C3⋊S3×C33
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 10 25)(2 11 26)(3 12 27)(4 8 34)(5 9 35)(6 7 36)(13 42 21)(14 40 19)(15 41 20)(16 33 28)(17 31 29)(18 32 30)(22 47 43)(23 48 44)(24 46 45)(37 49 54)(38 50 52)(39 51 53)
(1 6 20)(2 4 21)(3 5 19)(7 15 10)(8 13 11)(9 14 12)(16 24 52)(17 22 53)(18 23 54)(25 36 41)(26 34 42)(27 35 40)(28 45 50)(29 43 51)(30 44 49)(31 47 39)(32 48 37)(33 46 38)
(1 40 8)(2 41 9)(3 42 7)(4 25 14)(5 26 15)(6 27 13)(10 19 34)(11 20 35)(12 21 36)(16 47 49)(17 48 50)(18 46 51)(22 37 28)(23 38 29)(24 39 30)(31 44 52)(32 45 53)(33 43 54)
(1 19 4)(2 20 5)(3 21 6)(7 12 13)(8 10 14)(9 11 15)(16 22 54)(17 23 52)(18 24 53)(25 40 34)(26 41 35)(27 42 36)(28 43 49)(29 44 50)(30 45 51)(31 48 38)(32 46 39)(33 47 37)
(1 38)(2 39)(3 37)(4 31)(5 32)(6 33)(7 28)(8 29)(9 30)(10 50)(11 51)(12 49)(13 43)(14 44)(15 45)(16 36)(17 34)(18 35)(19 48)(20 46)(21 47)(22 42)(23 40)(24 41)(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), (1,10,25)(2,11,26)(3,12,27)(4,8,34)(5,9,35)(6,7,36)(13,42,21)(14,40,19)(15,41,20)(16,33,28)(17,31,29)(18,32,30)(22,47,43)(23,48,44)(24,46,45)(37,49,54)(38,50,52)(39,51,53), (1,6,20)(2,4,21)(3,5,19)(7,15,10)(8,13,11)(9,14,12)(16,24,52)(17,22,53)(18,23,54)(25,36,41)(26,34,42)(27,35,40)(28,45,50)(29,43,51)(30,44,49)(31,47,39)(32,48,37)(33,46,38), (1,40,8)(2,41,9)(3,42,7)(4,25,14)(5,26,15)(6,27,13)(10,19,34)(11,20,35)(12,21,36)(16,47,49)(17,48,50)(18,46,51)(22,37,28)(23,38,29)(24,39,30)(31,44,52)(32,45,53)(33,43,54), (1,19,4)(2,20,5)(3,21,6)(7,12,13)(8,10,14)(9,11,15)(16,22,54)(17,23,52)(18,24,53)(25,40,34)(26,41,35)(27,42,36)(28,43,49)(29,44,50)(30,45,51)(31,48,38)(32,46,39)(33,47,37), (1,38)(2,39)(3,37)(4,31)(5,32)(6,33)(7,28)(8,29)(9,30)(10,50)(11,51)(12,49)(13,43)(14,44)(15,45)(16,36)(17,34)(18,35)(19,48)(20,46)(21,47)(22,42)(23,40)(24,41)(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), (1,10,25)(2,11,26)(3,12,27)(4,8,34)(5,9,35)(6,7,36)(13,42,21)(14,40,19)(15,41,20)(16,33,28)(17,31,29)(18,32,30)(22,47,43)(23,48,44)(24,46,45)(37,49,54)(38,50,52)(39,51,53), (1,6,20)(2,4,21)(3,5,19)(7,15,10)(8,13,11)(9,14,12)(16,24,52)(17,22,53)(18,23,54)(25,36,41)(26,34,42)(27,35,40)(28,45,50)(29,43,51)(30,44,49)(31,47,39)(32,48,37)(33,46,38), (1,40,8)(2,41,9)(3,42,7)(4,25,14)(5,26,15)(6,27,13)(10,19,34)(11,20,35)(12,21,36)(16,47,49)(17,48,50)(18,46,51)(22,37,28)(23,38,29)(24,39,30)(31,44,52)(32,45,53)(33,43,54), (1,19,4)(2,20,5)(3,21,6)(7,12,13)(8,10,14)(9,11,15)(16,22,54)(17,23,52)(18,24,53)(25,40,34)(26,41,35)(27,42,36)(28,43,49)(29,44,50)(30,45,51)(31,48,38)(32,46,39)(33,47,37), (1,38)(2,39)(3,37)(4,31)(5,32)(6,33)(7,28)(8,29)(9,30)(10,50)(11,51)(12,49)(13,43)(14,44)(15,45)(16,36)(17,34)(18,35)(19,48)(20,46)(21,47)(22,42)(23,40)(24,41)(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)], [(1,10,25),(2,11,26),(3,12,27),(4,8,34),(5,9,35),(6,7,36),(13,42,21),(14,40,19),(15,41,20),(16,33,28),(17,31,29),(18,32,30),(22,47,43),(23,48,44),(24,46,45),(37,49,54),(38,50,52),(39,51,53)], [(1,6,20),(2,4,21),(3,5,19),(7,15,10),(8,13,11),(9,14,12),(16,24,52),(17,22,53),(18,23,54),(25,36,41),(26,34,42),(27,35,40),(28,45,50),(29,43,51),(30,44,49),(31,47,39),(32,48,37),(33,46,38)], [(1,40,8),(2,41,9),(3,42,7),(4,25,14),(5,26,15),(6,27,13),(10,19,34),(11,20,35),(12,21,36),(16,47,49),(17,48,50),(18,46,51),(22,37,28),(23,38,29),(24,39,30),(31,44,52),(32,45,53),(33,43,54)], [(1,19,4),(2,20,5),(3,21,6),(7,12,13),(8,10,14),(9,11,15),(16,22,54),(17,23,52),(18,24,53),(25,40,34),(26,41,35),(27,42,36),(28,43,49),(29,44,50),(30,45,51),(31,48,38),(32,46,39),(33,47,37)], [(1,38),(2,39),(3,37),(4,31),(5,32),(6,33),(7,28),(8,29),(9,30),(10,50),(11,51),(12,49),(13,43),(14,44),(15,45),(16,36),(17,34),(18,35),(19,48),(20,46),(21,47),(22,42),(23,40),(24,41),(25,52),(26,53),(27,54)]])

162 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3ED 6A ··· 6Z order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 9 1 ··· 1 2 ··· 2 9 ··· 9

162 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 S3 C3×S3 kernel C3⋊S3×C33 C35 C32×C3⋊S3 C34 C34 C33 # reps 1 1 26 26 4 104

Matrix representation of C3⋊S3×C33 in GL5(𝔽7)

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

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

C3⋊S3×C33 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_3^3
% in TeX

G:=Group("C3:S3xC3^3");
// GroupNames label

G:=SmallGroup(486,257);
// by ID

G=gap.SmallGroup(486,257);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3244,11669]);
// 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,b*d=d*b,b*e=e*b,b*f=f*b,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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