S3^2xDic3 - GroupNames

G = S₃²×Dic₃ order 432 = 2⁴·3³

Direct product of S₃, S₃ and Dic₃

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

Aliases: S₃²×Dic₃, D₆.₇S₃², C₃⋊Dic₃⋊₁₁D₆, (C₃×Dic₃)⋊₇D₆, (S₃×C₆).₁₇D₆, C₃³⋊₂(C₂²×C₄), C₃³⋊₅C₄⋊₁C₂², (C₃²×C₆).₁C₂³, C₃²⋊₄(C₂²×Dic₃), (C₃²×Dic₃)⋊₉C₂², C₂.₁S₃³, C₃⋊₅(C₄×S₃²), (C₃×S₃²)⋊₃C₄, C₆.₁(C₂×S₃²), (S₃²×C₆).₃C₂, (C₂×S₃²).₂S₃, C₃⋊₁(C₂×S₃×Dic₃), (C₃×S₃)⋊₃(C₄×S₃), C₃²⋊₁₁(S₃×C₂×C₄), (C₃×S₃×Dic₃)⋊₉C₂, C₃³⋊₉(C₂×C₄)⋊₉C₂, (Dic₃×C₃⋊S₃)⋊₆C₂, C₃⋊S₃⋊₂(C₂×Dic₃), (S₃×C₃⋊Dic₃)⋊₇C₂, (C₂×C₃⋊S₃).₂₇D₆, (S₃×C₃×C₆).₁C₂², (S₃×C₃²)⋊₂(C₂×C₄), (C₃×S₃)⋊₁(C₂×Dic₃), (C₆×C₃⋊S₃).₁₄C₂², (C₃×C₆).₅₀(C₂²×S₃), (C₃×C₃⋊Dic₃)⋊₈C₂², (C₃×C₃⋊S₃)⋊₁(C₂×C₄), SmallGroup(432,594)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃²×Dic₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — S₃×C₃×C₆ — S₃²×C₆ — S₃²×Dic₃

Lower central

C₃³ — S₃²×Dic₃

Upper central

C₁ — C₂

Generators and relations for S₃²×Dic₃
G = < a,b,c,d,e,f | a³=b²=c³=d²=e⁶=1, f²=e³, 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, df=fd, fef^-1=e^-1 >

Subgroups: 1244 in 270 conjugacy classes, 74 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, S₃, C₆, C₆, C₆, C₂×C₄, C₂³, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂²×C₄, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂²×C₆, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃², S₃×C₆, S₃×C₆, C₂×C₃⋊S₃, C₆², S₃×C₂×C₄, C₂²×Dic₃, S₃×C₃², C₃×C₃⋊S₃, C₃²×C₆, S₃×Dic₃, S₃×Dic₃, C₆.D₆, S₃×C₁₂, C₆×Dic₃, C₄×C₃⋊S₃, C₂×C₃⋊Dic₃, C₂×S₃², S₃×C₂×C₆, C₃²×Dic₃, C₃×C₃⋊Dic₃, C₃³⋊₅C₄, C₃×S₃², S₃×C₃×C₆, C₆×C₃⋊S₃, C₄×S₃², C₂×S₃×Dic₃, C₃×S₃×Dic₃, S₃×C₃⋊Dic₃, Dic₃×C₃⋊S₃, C₃³⋊₉(C₂×C₄), S₃²×C₆, S₃²×Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, Dic₃, D₆, C₂²×C₄, C₄×S₃, C₂×Dic₃, C₂²×S₃, S₃², S₃×C₂×C₄, C₂²×Dic₃, S₃×Dic₃, C₂×S₃², C₄×S₃², C₂×S₃×Dic₃, S₃³, S₃²×Dic₃

Smallest permutation representation of S₃²×Dic₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	3F	3G	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	6F	6G	···	6N	6O	6P	6Q	6R	6S	6T	6U	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J
order	1	2	2	2	2	2	2	2	3	3	3	3	3	3	3	4	4	4	4	4	4	4	4	6	6	6	6	6	6	6	···	6	6	6	6	6	6	6	6	12	12	12	12	12	12	12	12	12	12
size	1	1	3	3	3	3	9	9	2	2	2	4	4	4	8	3	3	9	9	9	9	27	27	2	2	2	4	4	4	6	···	6	8	12	12	12	12	18	18	6	6	6	6	12	12	18	18	18	18

54 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	8	8
type	+	+	+	+	+	+		+	+	+	+	-	+	+		+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	S₃	S₃	D₆	D₆	Dic₃	D₆	D₆	C₄×S₃	S₃²	S₃²	S₃×Dic₃	C₂×S₃²	C₄×S₃²	S₃³	S₃²×Dic₃
kernel	S₃²×Dic₃	C₃×S₃×Dic₃	S₃×C₃⋊Dic₃	Dic₃×C₃⋊S₃	C₃³⋊₉(C₂×C₄)	S₃²×C₆	C₃×S₃²	S₃×Dic₃	C₂×S₃²	C₃×Dic₃	C₃⋊Dic₃	S₃²	S₃×C₆	C₂×C₃⋊S₃	C₃×S₃	Dic₃	D₆	S₃	C₆	C₃	C₂	C₁
# reps	1	2	2	1	1	1	8	2	1	2	2	4	4	1	8	1	2	4	3	2	1	1

Matrix representation of S₃²×Dic₃ ►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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	1
0	0	0	0	12	0

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

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

5	0	0	0	0	0
0	5	0	0	0	0
0	0	0	1	0	0
0	0	1	0	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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃²×Dic₃ in GAP, Magma, Sage, TeX

S_3^2\times {\rm Dic}_3

% in TeX

G:=Group("S3^2xDic3");

// GroupNames label

G:=SmallGroup(432,594);

// by ID

G=gap.SmallGroup(432,594);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,298,2028,14118]);

// Polycyclic

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

// generators/relations

G = S32×Dic3 order 432 = 24·33

Direct product of S3, S3 and Dic3

G = S₃²×Dic₃ order 432 = 2⁴·3³

Direct product of S₃, S₃ and Dic₃