A4xD12 - GroupNames

G = A₄xD₁₂ order 288 = 2⁵·3²

Direct product of A₄ and D₁₂

direct product, metabelian, soluble, monomial

Aliases: A₄xD₁₂, C₄:(S₃xA₄), C₃:₁(D₄xA₄), (C₄xA₄):₃S₃, C₁₂:₁(C₂xA₄), D₆:₁(C₂xA₄), (C₃xA₄):₆D₄, (C₁₂xA₄):₃C₂, (C₂²xD₁₂):C₃, (S₃xC₂³):₁C₆, (C₂²xC₁₂):₁C₆, (C₂xA₄).₁₅D₆, C₂²:₂(C₃xD₁₂), C₆.₃(C₂²xA₄), C₂³.₁₈(S₃xC₆), (C₆xA₄).₂₀C₂², (C₂xS₃xA₄):₃C₂, C₂.₄(C₂xS₃xA₄), (C₂xC₆):₁(C₃xD₄), (C₂²xC₄):₃(C₃xS₃), (C₂²xC₆).₃(C₂xC₆), SmallGroup(288,920)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²xC₆ — A₄xD₁₂

Chief series

C₁ — C₃ — C₂xC₆ — C₂²xC₆ — C₆xA₄ — C₂xS₃xA₄ — A₄xD₁₂

Lower central

C₂xC₆ — C₂²xC₆ — A₄xD₁₂

Upper central

C₁ — C₂ — C₄

Generators and relations for A₄xD₁₂
G = < a,b,c,d,e | a²=b²=c³=d¹²=e²=1, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 778 in 138 conjugacy classes, 27 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₂³, C₃², C₁₂, C₁₂, A₄, A₄, D₆, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xD₄, C₂⁴, C₃xS₃, C₃xC₆, D₁₂, D₁₂, C₂xC₁₂, C₃xD₄, C₂xA₄, C₂xA₄, C₂²xS₃, C₂²xC₆, C₂²xD₄, C₃xC₁₂, C₃xA₄, S₃xC₆, C₄xA₄, C₄xA₄, C₂xD₁₂, C₂²xC₁₂, C₂²xA₄, S₃xC₂³, C₃xD₁₂, S₃xA₄, C₆xA₄, D₄xA₄, C₂²xD₁₂, C₁₂xA₄, C₂xS₃xA₄, A₄xD₁₂
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, A₄, D₆, C₂xC₆, C₃xS₃, D₁₂, C₃xD₄, C₂xA₄, S₃xC₆, C₂²xA₄, C₃xD₁₂, S₃xA₄, D₄xA₄, C₂xS₃xA₄, A₄xD₁₂

Smallest permutation representation of A₄xD₁₂
►On 36 points

Generators in S₃₆

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	12A	12B	12C	12D	12E	···	12J
order	1	2	2	2	2	2	2	2	3	3	3	3	3	4	4	6	6	6	6	6	6	6	6	6	6	6	12	12	12	12	12	···	12
size	1	1	3	3	6	6	18	18	2	4	4	8	8	2	6	2	4	4	6	6	8	8	24	24	24	24	2	2	6	6	8	···	8

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	6	6	6	6
type	+	+	+				+	+	+		+				+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₆	C₃xS₃	D₁₂	C₃xD₄	S₃xC₆	C₃xD₁₂	A₄	C₂xA₄	C₂xA₄	S₃xA₄	D₄xA₄	C₂xS₃xA₄	A₄xD₁₂
kernel	A₄xD₁₂	C₁₂xA₄	C₂xS₃xA₄	C₂²xD₁₂	C₂²xC₁₂	S₃xC₂³	C₄xA₄	C₃xA₄	C₂xA₄	C₂²xC₄	A₄	C₂xC₆	C₂³	C₂²	D₁₂	C₁₂	D₆	C₄	C₃	C₂	C₁
# reps	1	1	2	2	2	4	1	1	1	2	2	2	2	4	1	1	2	1	1	1	2

Matrix representation of A₄xD₁₂ ►in GL₅(F₁₃)

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

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

3	0	0	0	0
0	3	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

2	6	0	0	0
7	2	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

11	7	0	0	0
7	2	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[2,7,0,0,0,6,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,7,0,0,0,7,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A₄xD₁₂ in GAP, Magma, Sage, TeX

A_4\times D_{12}

% in TeX

G:=Group("A4xD12");

// GroupNames label

G:=SmallGroup(288,920);

// by ID

G=gap.SmallGroup(288,920);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,197,92,648,271,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = A4xD12 order 288 = 25·32

Direct product of A4 and D12

G = A₄xD₁₂ order 288 = 2⁵·3²

Direct product of A₄ and D₁₂