S3xD6:S3 - GroupNames

G = S₃×D₆⋊S₃ order 432 = 2⁴·3³

Direct product of S₃ and D₆⋊S₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×D₆⋊S₃, D₆⋊₁S₃², (S₃×C₆)⋊₇D₆, C₃³⋊₅(C₂×D₄), C₃⋊Dic₃⋊₁₃D₆, (S₃×C₃²)⋊₂D₄, C₃²⋊₁₃(S₃×D₄), C₃³⋊₆D₄⋊₁C₂, C₃³⋊₉D₄⋊₇C₂, C₃³⋊₅C₄⋊₃C₂², (C₃²×C₆).₄C₂³, C₂.₄S₃³, (C₂×S₃²)⋊₁S₃, (S₃²×C₆)⋊₁C₂, C₆.₄(C₂×S₃²), (C₂×C₃⋊S₃)⋊₈D₆, C₃⋊₆(S₃×C₃⋊D₄), (S₃×C₃×C₆)⋊₁C₂², C₃⋊₁(C₂×D₆⋊S₃), (C₆×C₃⋊S₃)⋊₁C₂², (S₃×C₃⋊Dic₃)⋊₁C₂, C₃²⋊₉(C₂×C₃⋊D₄), (C₃×D₆⋊S₃)⋊₂C₂, (C₃×S₃)⋊₂(C₃⋊D₄), (C₃×C₆).₅₃(C₂²×S₃), (C₃×C₃⋊Dic₃)⋊₂C₂², SmallGroup(432,597)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₆ — S₃×D₆⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — S₃×C₃×C₆ — S₃²×C₆ — S₃×D₆⋊S₃

Lower central

C₃³ — C₃²×C₆ — S₃×D₆⋊S₃

Upper central

C₁ — C₂

Generators and relations for S₃×D₆⋊S₃
G = < a,b,c,d,e,f | a³=b²=c⁶=d²=e³=f²=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, fdf=c³d, fef=e^-1 >

Subgroups: 1508 in 270 conjugacy classes, 54 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, S₃, C₆, C₆, C₆, C₂×C₄, D₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, D₆, D₆, D₆, C₂×C₆, C₂×D₄, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₃×D₄, C₂²×S₃, C₂²×C₆, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, S₃², S₃×C₆, S₃×C₆, C₂×C₃⋊S₃, C₆², S₃×D₄, C₂×C₃⋊D₄, S₃×C₃², S₃×C₃², C₃×C₃⋊S₃, C₃²×C₆, S₃×Dic₃, D₆⋊S₃, D₆⋊S₃, C₃⋊D₁₂, C₃×C₃⋊D₄, C₂×C₃⋊Dic₃, C₃²⋊₇D₄, C₂×S₃², S₃×C₂×C₆, C₃×C₃⋊Dic₃, C₃³⋊₅C₄, C₃×S₃², S₃×C₃×C₆, S₃×C₃×C₆, C₆×C₃⋊S₃, C₂×D₆⋊S₃, S₃×C₃⋊D₄, C₃×D₆⋊S₃, S₃×C₃⋊Dic₃, C₃³⋊₆D₄, C₃³⋊₉D₄, S₃²×C₆, S₃×D₆⋊S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, S₃², S₃×D₄, C₂×C₃⋊D₄, D₆⋊S₃, C₂×S₃², C₂×D₆⋊S₃, S₃×C₃⋊D₄, S₃³, S₃×D₆⋊S₃

Smallest permutation representation of S₃×D₆⋊S₃
►On 48 points

Generators in S₄₈

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

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

(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)

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

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

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

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

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

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

45 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	3F	3G	4A	4B	6A	6B	6C	6D	6E	6F	6G	···	6N	6O	6P	···	6W	6X	6Y	6Z	6AA	12
order	1	2	2	2	2	2	2	2	3	3	3	3	3	3	3	4	4	6	6	6	6	6	6	6	···	6	6	6	···	6	6	6	6	6	12
size	1	1	3	3	6	6	18	18	2	2	2	4	4	4	8	18	54	2	2	2	4	4	4	6	···	6	8	12	···	12	18	18	18	18	36

45 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	8	8
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₄	D₆	D₆	D₆	C₃⋊D₄	S₃²	S₃×D₄	D₆⋊S₃	C₂×S₃²	S₃×C₃⋊D₄	S₃³	S₃×D₆⋊S₃
kernel	S₃×D₆⋊S₃	C₃×D₆⋊S₃	S₃×C₃⋊Dic₃	C₃³⋊₆D₄	C₃³⋊₉D₄	S₃²×C₆	D₆⋊S₃	C₂×S₃²	S₃×C₃²	C₃⋊Dic₃	S₃×C₆	C₂×C₃⋊S₃	C₃×S₃	D₆	C₃²	S₃	C₆	C₃	C₂	C₁
# reps	1	1	1	2	1	2	1	2	2	1	6	2	8	3	1	2	3	4	1	1

Matrix representation of S₃×D₆⋊S₃ ►in GL₆(𝔽₁₃)

12	1	0	0	0	0
12	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	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	12	0	0	0
0	0	0	12	0	0
0	0	0	0	0	12
0	0	0	0	1	12

12	0	0	0	0	0
0	12	0	0	0	0
0	0	11	4	0	0
0	0	9	2	0	0
0	0	0	0	1	12
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	1	0	0
0	0	12	0	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	12	0	0	0
0	0	12	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃×D₆⋊S₃ in GAP, Magma, Sage, TeX

S_3\times D_6\rtimes S_3

% in TeX

G:=Group("S3xD6:S3");

// GroupNames label

G:=SmallGroup(432,597);

// by ID

G=gap.SmallGroup(432,597);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,298,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^6=d^2=e^3=f^2=1,b*a*b=a^-1,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,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=c^3*d,f*e*f=e^-1>;

// generators/relations

G = S3×D6⋊S3 order 432 = 24·33

Direct product of S3 and D6⋊S3

G = S₃×D₆⋊S₃ order 432 = 2⁴·3³

Direct product of S₃ and D₆⋊S₃