Dic3xC3^2:C4 - GroupNames

G = Dic₃×C₃²⋊C₄ order 432 = 2⁴·3³

Direct product of Dic₃ and C₃²⋊C₄

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

Aliases: Dic₃×C₃²⋊C₄, C₃³⋊C₄², C₃³⋊₅C₄⋊₁C₄, C₃³⋊C₄⋊₁C₄, C₃²⋊₄(C₄×Dic₃), (C₃²×Dic₃)⋊₁C₄, C₃⋊₂(C₄×C₃²⋊C₄), C₃⋊S₃.₆(C₄×S₃), (C₃×C₃²⋊C₄)⋊₃C₄, C₂.₁(S₃×C₃²⋊C₄), C₆.₁(C₂×C₃²⋊C₄), (C₃×C₆).₂₆(C₄×S₃), (C₂×C₃⋊S₃).₂₄D₆, (C₂×C₃²⋊C₄).₇S₃, (C₆×C₃²⋊C₄).₈C₂, (C₆×C₃⋊S₃).₁C₂², C₃⋊S₃.₅(C₂×Dic₃), (Dic₃×C₃⋊S₃).₃C₂, (C₃²×C₆).₁(C₂×C₄), (C₂×C₃³⋊C₄).₁C₂, (C₃×C₃⋊S₃).₇(C₂×C₄), SmallGroup(432,567)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — Dic₃×C₃²⋊C₄

Chief series

C₁ — C₃ — C₃³ — C₃×C₃⋊S₃ — C₆×C₃⋊S₃ — Dic₃×C₃⋊S₃ — Dic₃×C₃²⋊C₄

Lower central

C₃³ — Dic₃×C₃²⋊C₄

Upper central

C₁ — C₂

Generators and relations for Dic₃×C₃²⋊C₄
G = < a,b,c,d,e | a⁶=c³=d³=e⁴=1, b²=a³, bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede^-1=cd=dc, ece^-1=c^-1d >

Subgroups: 704 in 104 conjugacy classes, 28 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₂×C₄, C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₄², C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃²⋊C₄, C₃²⋊C₄, S₃×C₆, C₂×C₃⋊S₃, C₄×Dic₃, C₃×C₃⋊S₃, C₃²×C₆, S₃×Dic₃, C₄×C₃⋊S₃, C₂×C₃²⋊C₄, C₂×C₃²⋊C₄, C₃²×Dic₃, C₃³⋊₅C₄, C₃×C₃²⋊C₄, C₃³⋊C₄, C₆×C₃⋊S₃, C₄×C₃²⋊C₄, Dic₃×C₃⋊S₃, C₆×C₃²⋊C₄, C₂×C₃³⋊C₄, Dic₃×C₃²⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Dic₃, D₆, C₄², C₄×S₃, C₂×Dic₃, C₃²⋊C₄, C₄×Dic₃, C₂×C₃²⋊C₄, C₄×C₃²⋊C₄, S₃×C₃²⋊C₄, Dic₃×C₃²⋊C₄

Smallest permutation representation of Dic₃×C₃²⋊C₄
►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 3 5)(2 4 6)(7 11 9)(8 12 10)(25 29 27)(26 30 28)(31 33 35)(32 34 36)

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

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4	8	8
type	+	+	+	+					+	-	+			+	+		+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	S₃	Dic₃	D₆	C₄×S₃	C₄×S₃	C₃²⋊C₄	C₂×C₃²⋊C₄	C₄×C₃²⋊C₄	S₃×C₃²⋊C₄	Dic₃×C₃²⋊C₄
kernel	Dic₃×C₃²⋊C₄	Dic₃×C₃⋊S₃	C₆×C₃²⋊C₄	C₂×C₃³⋊C₄	C₃²×Dic₃	C₃³⋊₅C₄	C₃×C₃²⋊C₄	C₃³⋊C₄	C₂×C₃²⋊C₄	C₃²⋊C₄	C₂×C₃⋊S₃	C₃⋊S₃	C₃×C₆	Dic₃	C₆	C₃	C₂	C₁
# reps	1	1	1	1	2	2	4	4	1	2	1	2	2	2	2	4	2	2

Matrix representation of Dic₃×C₃²⋊C₄ ►in GL₆(𝔽₁₃)

12	12	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	12	0
0	0	0	0	0	12

12	0	0	0	0	0
1	1	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	5	0
0	0	0	0	0	5

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	1	0	0
0	0	12	11	12	12
0	0	0	1	1	0
0	0	0	1	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	1
0	0	12	11	12	12
0	0	1	1	0	0
0	0	0	1	1	0

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

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,11,1,1,0,0,0,12,1,0,0,0,0,12,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,1,11,1,1,0,0,0,12,0,1,0,0,1,12,0,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0,0,1,12,0,0,0,0,0,1,12,12] >;

Dic₃×C₃²⋊C₄ in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3^2\rtimes C_4

% in TeX

G:=Group("Dic3xC3^2:C4");

// GroupNames label

G:=SmallGroup(432,567);

// by ID

G=gap.SmallGroup(432,567);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,36,1411,298,1356,1027,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^3=d^3=e^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;

// generators/relations

G = Dic3×C32⋊C4 order 432 = 24·33

Direct product of Dic3 and C32⋊C4

G = Dic₃×C₃²⋊C₄ order 432 = 2⁴·3³

Direct product of Dic₃ and C₃²⋊C₄