C3xDic3xA4 - GroupNames

G = C₃xDic₃xA₄ order 432 = 2⁴·3³

Direct product of C₃, Dic₃ and A₄

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

Aliases: C₃xDic₃xA₄, C₆²:₆C₁₂, C₃:(C₁₂xA₄), C₆.₃(C₆xA₄), (C₃xA₄):₅C₁₂, (C₆xA₄).₉C₆, C₆.₂₂(S₃xA₄), C₃²:₄(C₄xA₄), (C₆xA₄).₁₂S₃, (C₃²xA₄):₆C₄, (C₂xC₆²).₉C₆, C₂³.₂(S₃xC₃²), (C₂²xDic₃):C₃², C₂²:₂(C₃²xDic₃), (C₂xC₆):(C₃xC₁₂), C₂.₁(C₃xS₃xA₄), (Dic₃xC₂xC₆):C₃, (A₄xC₃xC₆).₄C₂, (C₂²xC₆).(C₃xC₆), (C₂xA₄).₂(C₃xS₃), (C₃xC₆).₁₇(C₂xA₄), (C₂xC₆):₄(C₃xDic₃), (C₂²xC₆).₂₀(C₃xS₃), SmallGroup(432,624)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xDic₃xA₄

Chief series

C₁ — C₃ — C₂xC₆ — C₂²xC₆ — C₂xC₆² — A₄xC₃xC₆ — C₃xDic₃xA₄

Lower central

C₂xC₆ — C₃xDic₃xA₄

Upper central

C₁ — C₆

Generators and relations for C₃xDic₃xA₄
G = < a,b,c,d,e,f | a³=b⁶=d²=e²=f³=1, c²=b³, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc^-1=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 484 in 136 conjugacy classes, 40 normal (25 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₆, C₆, C₂xC₄, C₂³, C₃², C₃², Dic₃, Dic₃, C₁₂, A₄, A₄, C₂xC₆, C₂xC₆, C₂²xC₄, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂xA₄, C₂xA₄, C₂²xC₆, C₂²xC₆, C₃³, C₃xDic₃, C₃xDic₃, C₃xC₁₂, C₃xA₄, C₃xA₄, C₃xA₄, C₆², C₆², C₄xA₄, C₂²xDic₃, C₂²xC₁₂, C₃²xC₆, C₆xDic₃, C₆xA₄, C₆xA₄, C₆xA₄, C₂xC₆², C₃²xDic₃, C₃²xA₄, Dic₃xA₄, C₁₂xA₄, Dic₃xC₂xC₆, A₄xC₃xC₆, C₃xDic₃xA₄
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₃², Dic₃, C₁₂, A₄, C₃xS₃, C₃xC₆, C₂xA₄, C₃xDic₃, C₃xC₁₂, C₃xA₄, C₄xA₄, S₃xC₃², S₃xA₄, C₆xA₄, C₃²xDic₃, Dic₃xA₄, C₁₂xA₄, C₃xS₃xA₄, C₃xDic₃xA₄

Smallest permutation representation of C₃xDic₃xA₄
►On 36 points

Generators in S₃₆

(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)

(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)

(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)

(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)

(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)

(1 15 7)(2 16 8)(3 17 9)(4 18 10)(5 13 11)(6 14 12)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)

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

G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28) );

G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(1,15,7),(2,16,8),(3,17,9),(4,18,10),(5,13,11),(6,14,12),(19,33,29),(20,34,30),(21,35,25),(22,36,26),(23,31,27),(24,32,28)]])

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	···	3K	3L	···	3Q	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	···	6O	6P	···	6U	6V	···	6AA	12A	12B	12C	12D	12E	12F	12G	12H	12I	···	12T
order	1	2	2	2	3	3	3	3	3	3	···	3	3	···	3	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	1	1	2	2	2	4	···	4	8	···	8	3	3	9	9	1	1	2	2	2	3	3	3	3	4	···	4	6	···	6	8	···	8	3	3	3	3	9	9	9	9	12	···	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	3	3	6	6	6	6
type	+	+								+	-					+	+					+	-
image	C₁	C₂	C₃	C₃	C₄	C₆	C₆	C₁₂	C₁₂	S₃	Dic₃	C₃xS₃	C₃xS₃	C₃xDic₃	C₃xDic₃	A₄	C₂xA₄	C₃xA₄	C₄xA₄	C₆xA₄	C₁₂xA₄	S₃xA₄	Dic₃xA₄	C₃xS₃xA₄	C₃xDic₃xA₄
kernel	C₃xDic₃xA₄	A₄xC₃xC₆	Dic₃xA₄	Dic₃xC₂xC₆	C₃²xA₄	C₆xA₄	C₂xC₆²	C₃xA₄	C₆²	C₆xA₄	C₃xA₄	C₂xA₄	C₂²xC₆	A₄	C₂xC₆	C₃xDic₃	C₃xC₆	Dic₃	C₃²	C₆	C₃	C₆	C₃	C₂	C₁
# reps	1	1	6	2	2	6	2	12	4	1	1	6	2	6	2	1	1	2	2	2	4	1	1	2	2

Matrix representation of C₃xDic₃xA₄ ►in GL₅(F₁₃)

9	0	0	0	0
0	9	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

10	0	0	0	0
3	4	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

5	3	0	0	0
0	8	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

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

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

1	0	0	0	0
0	1	0	0	0
0	0	0	0	9
0	0	4	0	0
0	0	0	4	0

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[10,3,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,3,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,9,0,0] >;

C₃xDic₃xA₄ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\times A_4

% in TeX

G:=Group("C3xDic3xA4");

// GroupNames label

G:=SmallGroup(432,624);

// by ID

G=gap.SmallGroup(432,624);

# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,2,-3,126,1901,768,14118]);

// Polycyclic

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

// generators/relations

G = C3xDic3xA4 order 432 = 24·33

Direct product of C3, Dic3 and A4

G = C₃xDic₃xA₄ order 432 = 2⁴·3³

Direct product of C₃, Dic₃ and A₄