C3xD4:2S3 - GroupNames

G = C₃xD₄:₂S₃ order 144 = 2⁴·3²

Direct product of C₃ and D₄:₂S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃xD₄:₂S₃, Dic₆:₃C₆, C₁₂.₃₇D₆, C₆².₁₃C₂², (C₄xS₃):₂C₆, D₄:₂(C₃xS₃), (C₃xD₄):₅S₃, (C₃xD₄):₃C₆, C₃:D₄:₂C₆, C₄.₅(S₃xC₆), (C₂xC₆).₇D₆, (S₃xC₁₂):₆C₂, C₁₂.₅(C₂xC₆), D₆.₂(C₂xC₆), (C₂xDic₃):₃C₆, (C₃xDic₆):₈C₂, (C₆xDic₃):₉C₂, (D₄xC₃²):₄C₂, C₃²:₉(C₄oD₄), C₆.₆(C₂²xC₆), C₂².₁(S₃xC₆), (C₃xC₆).₂₄C₂³, C₆.₄₅(C₂²xS₃), Dic₃.₃(C₂xC₆), (S₃xC₆).₁₁C₂², (C₃xC₁₂).₂₁C₂², (C₃xDic₃).₁₃C₂², (C₂xC₆).(C₂xC₆), C₂.₇(S₃xC₂xC₆), C₃:₂(C₃xC₄oD₄), (C₃xC₃:D₄):₆C₂, SmallGroup(144,163)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₃xD₄:₂S₃

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — S₃xC₆ — S₃xC₁₂ — C₃xD₄:₂S₃

Lower central

C₃ — C₆ — C₃xD₄:₂S₃

Upper central

C₁ — C₆ — C₃xD₄

Generators and relations for C₃xD₄:₂S₃
G = < a,b,c,d,e | a³=b⁴=c²=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece=b²c, ede=d^-1 >

Subgroups: 164 in 88 conjugacy classes, 46 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, D₄, Q₈, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₃xDic₃, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², D₄:₂S₃, C₃xC₄oD₄, C₃xDic₆, S₃xC₁₂, C₆xDic₃, C₃xC₃:D₄, D₄xC₃², C₃xD₄:₂S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₂²xS₃, C₂²xC₆, S₃xC₆, D₄:₂S₃, C₃xC₄oD₄, S₃xC₂xC₆, C₃xD₄:₂S₃

Permutation representations of C₃xD₄:₂S₃
►On 24 points - transitive group 24T209

Generators in S₂₄

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

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

(2 4)(6 8)(9 11)(13 15)(18 20)(22 24)

(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)

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

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

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

G:=TransitiveGroup(24,209);

C₃xD₄:₂S₃ is a maximal subgroup of
Dic₆:₃D₆ Dic₆.₁₉D₆ D₁₂.₂₂D₆ Dic₆.₂₀D₆ Dic₆.₂₄D₆ Dic₆:₁₂D₆ D₁₂:₁₃D₆ C₃xS₃xC₄oD₄ C₆².₁₃D₆ Dic₁₈:₂C₆ C₆².₁₆D₆
C₃xD₄:₂S₃ is a maximal quotient of
C₃xD₄xDic₃ C₆².₁₃D₆ Dic₁₈:₂C₆

45 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6I	6J	···	6O	6P	6Q	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J	12K	12L	12M
order	1	2	2	2	2	3	3	3	3	3	4	4	4	4	4	6	6	6	···	6	6	···	6	6	6	12	12	12	12	12	12	12	12	12	12	12	12	12
size	1	1	2	2	6	1	1	2	2	2	2	3	3	6	6	1	1	2	···	2	4	···	4	6	6	2	2	3	3	3	3	4	4	4	6	6	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	D₆	D₆	C₄oD₄	C₃xS₃	S₃xC₆	S₃xC₆	C₃xC₄oD₄	D₄:₂S₃	C₃xD₄:₂S₃
kernel	C₃xD₄:₂S₃	C₃xDic₆	S₃xC₁₂	C₆xDic₃	C₃xC₃:D₄	D₄xC₃²	D₄:₂S₃	Dic₆	C₄xS₃	C₂xDic₃	C₃:D₄	C₃xD₄	C₃xD₄	C₁₂	C₂xC₆	C₃²	D₄	C₄	C₂²	C₃	C₃	C₁
# reps	1	1	1	2	2	1	2	2	2	4	4	2	1	1	2	2	2	2	4	4	1	2

Matrix representation of C₃xD₄:₂S₃ ►in GL₄(F₇) generated by

4	0	0	0
0	4	0	0
0	0	4	0
0	0	0	4

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

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

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

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

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

C₃xD₄:₂S₃ in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_2S_3

% in TeX

G:=Group("C3xD4:2S3");

// GroupNames label

G:=SmallGroup(144,163);

// by ID

G=gap.SmallGroup(144,163);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,260,3461]);

// Polycyclic

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

// generators/relations

G = C3xD4:2S3 order 144 = 24·32

Direct product of C3 and D4:2S3

G = C₃xD₄:₂S₃ order 144 = 2⁴·3²

Direct product of C₃ and D₄:₂S₃