C3:S3xC3^3 - GroupNames

G = C₃⋊S₃×C₃³ order 486 = 2·3⁵

Direct product of C₃³ and C₃⋊S₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃⋊S₃×C₃³, C₃⁵⋊₂C₂, C₃⁴⋊₁₄S₃, C₃⁴⋊₁₅C₆, C₃⋊(S₃×C₃³), C₃³⋊₁₄(C₃×C₆), C₃³⋊₁₈(C₃×S₃), C₃²⋊₄(C₃²×C₆), C₃²⋊₅(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

G = C3⋊S3×C33 order 486 = 2·35

Direct product of C33 and C3⋊S3

G = C₃⋊S₃×C₃³ order 486 = 2·3⁵

Direct product of C₃³ and C₃⋊S₃