Dic3:6S3^2 - GroupNames

G = Dic₃⋊₆S₃² order 432 = 2⁴·3³

2^nd semidirect product of Dic₃ and S₃² acting through Inn(Dic₃)

metabelian, supersoluble, monomial, A-group

Aliases: Dic₃⋊₆S₃², (C₃×Dic₃)⋊₉D₆, C₆.D₆⋊₅S₃, C₃³⋊₄(C₂²×C₄), C₃²⋊₄D₆⋊₃C₄, C₃³⋊₅C₄⋊₂C₂², (C₃²×C₆).₃C₂³, (C₃²×Dic₃)⋊₁₁C₂², C₂.₃S₃³, C₃⋊₂(C₄×S₃²), C₆.₃(C₂×S₃²), C₃⋊S₃⋊₂(C₄×S₃), C₃²⋊₇(S₃×C₂×C₄), (Dic₃×C₃⋊S₃)⋊₇C₂, (C₂×C₃⋊S₃).₂₉D₆, (C₃×C₆.D₆)⋊₁₀C₂, (C₆×C₃⋊S₃).₁₆C₂², (C₃×C₆).₅₂(C₂²×S₃), (C₂×C₃²⋊₄D₆).₃C₂, (C₃×C₃⋊S₃)⋊₃(C₂×C₄), SmallGroup(432,596)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — Dic₃⋊₆S₃²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃²×Dic₃ — C₃×C₆.D₆ — Dic₃⋊₆S₃²

Lower central

C₃³ — Dic₃⋊₆S₃²

Upper central

C₁ — C₂

Generators and relations for Dic₃⋊₆S₃²
G = < a,b,c,d,e,f | a⁶=c³=d²=e³=f²=1, b²=a³, bab^-1=dad=faf=a^-1, ac=ca, ae=ea, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1452 in 270 conjugacy classes, 62 normal (7 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₂×C₄, C₂³, C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂²×C₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃², S₃×C₆, C₂×C₃⋊S₃, S₃×C₂×C₄, C₃×C₃⋊S₃, C₃²×C₆, S₃×Dic₃, C₆.D₆, S₃×C₁₂, C₄×C₃⋊S₃, C₂×S₃², C₃²×Dic₃, C₃³⋊₅C₄, C₃²⋊₄D₆, C₆×C₃⋊S₃, C₄×S₃², C₃×C₆.D₆, Dic₃×C₃⋊S₃, C₂×C₃²⋊₄D₆, Dic₃⋊₆S₃²
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₆, C₂²×C₄, C₄×S₃, C₂²×S₃, S₃², S₃×C₂×C₄, C₂×S₃², C₄×S₃², S₃³, Dic₃⋊₆S₃²

Smallest permutation representation of Dic₃⋊₆S₃²
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	···	2G	3A	3B	3C	3D	3E	3F	3G	4A	···	4F	4G	4H	6A	6B	6C	6D	6E	6F	6G	6H	···	6M	12A	···	12L	12M	···	12R
order	1	2	2	···	2	3	3	3	3	3	3	3	4	···	4	4	4	6	6	6	6	6	6	6	6	···	6	12	···	12	12	···	12
size	1	1	9	···	9	2	2	2	4	4	4	8	3	···	3	27	27	2	2	2	4	4	4	8	18	···	18	6	···	6	12	···	12

54 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4	4	4	8	8
type	+	+	+	+		+	+	+		+	+		+	-
image	C₁	C₂	C₂	C₂	C₄	S₃	D₆	D₆	C₄×S₃	S₃²	C₂×S₃²	C₄×S₃²	S₃³	Dic₃⋊₆S₃²
kernel	Dic₃⋊₆S₃²	C₃×C₆.D₆	Dic₃×C₃⋊S₃	C₂×C₃²⋊₄D₆	C₃²⋊₄D₆	C₆.D₆	C₃×Dic₃	C₂×C₃⋊S₃	C₃⋊S₃	Dic₃	C₆	C₃	C₂	C₁
# reps	1	3	3	1	8	3	6	3	12	3	3	6	1	1

Matrix representation of Dic₃⋊₆S₃² ►in GL₆(𝔽₁₃)

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	12	12
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	0	0
0	0	0	8	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	1	0	0
0	0	12	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

0	1	0	0	0	0
12	12	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	12	12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

Dic₃⋊₆S₃² in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_6S_3^2

% in TeX

G:=Group("Dic3:6S3^2");

// GroupNames label

G:=SmallGroup(432,596);

// by ID

G=gap.SmallGroup(432,596);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,58,298,2028,14118]);

// Polycyclic

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

// generators/relations

G = Dic3⋊6S32 order 432 = 24·33

2nd semidirect product of Dic3 and S32 acting through Inn(Dic3)

G = Dic₃⋊₆S₃² order 432 = 2⁴·3³

2^nd semidirect product of Dic₃ and S₃² acting through Inn(Dic₃)