C3xS3xD4 - GroupNames

G = C₃xS₃xD₄ order 144 = 2⁴·3²

Direct product of C₃, S₃ and D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xS₃xD₄, C₁₂:₅D₆, D₁₂:₃C₆, C₆²:₃C₂², C₁₂:(C₂xC₆), C₄:₁(S₃xC₆), (C₂xC₆):₇D₆, C₃:₂(C₆xD₄), (C₄xS₃):₁C₆, (C₃xD₄):₂C₆, D₆:₂(C₂xC₆), C₃:D₄:₁C₆, (S₃xC₁₂):₅C₂, (C₃xD₁₂):₈C₂, C₂²:₃(S₃xC₆), C₃²:₁₀(C₂xD₄), (S₃xC₆):₈C₂², (C₂²xS₃):₃C₆, (C₃xC₁₂):₃C₂², Dic₃:₁(C₂xC₆), (D₄xC₃²):₃C₂, C₆.₅(C₂²xC₆), C₆.₄₄(C₂²xS₃), (C₃xC₆).₂₃C₂³, (C₃xDic₃):₈C₂², (S₃xC₂xC₆):₅C₂, C₂.₆(S₃xC₂xC₆), (C₂xC₆):₂(C₂xC₆), (C₃xC₃:D₄):₅C₂, SmallGroup(144,162)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₃xS₃xD₄

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — S₃xC₆ — S₃xC₂xC₆ — C₃xS₃xD₄

Lower central

C₃ — C₆ — C₃xS₃xD₄

Upper central

C₁ — C₆ — C₃xD₄

Generators and relations for C₃xS₃xD₄
G = < a,b,c,d,e | a³=b³=c²=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 260 in 116 conjugacy classes, 50 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₂xC₄, D₄, D₄, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xS₃, C₃xC₆, C₃xC₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₃xDic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, S₃xC₆, C₆², S₃xD₄, C₆xD₄, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, D₄xC₃², S₃xC₂xC₆, C₃xS₃xD₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xD₄, C₂²xS₃, C₂²xC₆, S₃xC₆, S₃xD₄, C₆xD₄, S₃xC₂xC₆, C₃xS₃xD₄

Permutation representations of C₃xS₃xD₄
►On 24 points - transitive group 24T208

Generators in S₂₄

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

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

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

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

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

G:=sub<Sym(24)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)>;

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

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

G:=TransitiveGroup(24,208);

C₃xS₃xD₄ is a maximal subgroup of D₁₂:₉D₆ D₁₂.₇D₆ D₁₂:₁₂D₆ D₁₂:₁₃D₆

45 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	···	6I	6J	6K	6L	6M	6N	···	6S	6T	6U	6V	6W	12A	12B	12C	12D	12E	12F	12G
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	6	6	6	12	12	12	12	12	12	12
size	1	1	2	2	3	3	6	6	1	1	2	2	2	2	6	1	1	2	···	2	3	3	3	3	4	···	4	6	6	6	6	2	2	4	4	4	6	6

45 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₃xS₃	C₃xD₄	S₃xC₆	S₃xC₆	S₃xD₄	C₃xS₃xD₄
kernel	C₃xS₃xD₄	S₃xC₁₂	C₃xD₁₂	C₃xC₃:D₄	D₄xC₃²	S₃xC₂xC₆	S₃xD₄	C₄xS₃	D₁₂	C₃:D₄	C₃xD₄	C₂²xS₃	C₃xD₄	C₃xS₃	C₁₂	C₂xC₆	D₄	S₃	C₄	C₂²	C₃	C₁
# reps	1	1	1	2	1	2	2	2	2	4	2	4	1	2	1	2	2	4	2	4	1	2

Matrix representation of C₃xS₃xD₄ ►in GL₄(F₇) generated by

2	0	0	0
0	2	0	0
0	0	2	0
0	0	0	2

6	0	3	1
5	3	6	0
1	6	4	5
3	3	2	6

6	6	1	1
0	6	0	1
0	5	1	1
0	0	0	1

3	1	0	0
4	4	0	0
4	5	6	3
0	3	4	1

3	2	6	3
6	0	4	5
6	6	5	2
0	0	0	6

G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[6,0,0,0,6,6,5,0,1,0,1,0,1,1,1,1],[3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[3,6,6,0,2,0,6,0,6,4,5,0,3,5,2,6] >;

C₃xS₃xD₄ in GAP, Magma, Sage, TeX

C_3\times S_3\times D_4

% in TeX

G:=Group("C3xS3xD4");

// GroupNames label

G:=SmallGroup(144,162);

// by ID

G=gap.SmallGroup(144,162);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,260,3461]);

// Polycyclic

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

// generators/relations

G = C3xS3xD4 order 144 = 24·32

Direct product of C3, S3 and D4

G = C₃xS₃xD₄ order 144 = 2⁴·3²

Direct product of C₃, S₃ and D₄