C3xC2^3.6D6 - GroupNames

G = C₃xC₂³.₆D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₆D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₆D₆, C₆².₃₃D₄, (C₂²xS₃):C₁₂, (C₂xDic₃):C₁₂, (C₆xDic₃):₂C₄, (C₂xC₆).₄₄D₁₂, C₂³.₆(S₃xC₆), C₆.₄₃(D₆:C₄), C₆.D₄:₁C₆, C₃²:₆(C₂³:C₄), C₂².₃(S₃xC₁₂), C₆².₃₆(C₂xC₄), C₂².₂(C₃xD₁₂), (C₂²xC₆).₂₄D₆, (C₂xC₆²).₉C₂², (S₃xC₂xC₆):₂C₄, C₃:₁(C₃xC₂³:C₄), C₂.₄(C₃xD₆:C₄), (C₂xC₆).₁(C₃xD₄), (C₃xC₂²:C₄):₁S₃, C₂²:C₄:₁(C₃xS₃), (C₃xC₂²:C₄):₁C₆, (C₂xC₆).₁(C₂xC₁₂), (C₂xC₆).₅₈(C₄xS₃), (C₆xC₃:D₄).₅C₂, (C₂xC₃:D₄).₁C₆, C₆.₂(C₃xC₂²:C₄), C₂².₈(C₃xC₃:D₄), (C₃xC₆.D₄):₃C₂, (C₂xC₆).₆₁(C₃:D₄), (C₃²xC₂²:C₄):₁C₂, (C₂²xC₆).₁₆(C₂xC₆), (C₃xC₆).₄₂(C₂²:C₄), SmallGroup(288,240)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂³.₆D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xC₆ — C₂xC₆² — C₆xC₃:D₄ — C₃xC₂³.₆D₆

Lower central

C₃ — C₆ — C₂xC₆ — C₃xC₂³.₆D₆

Upper central

C₁ — C₆ — C₂²xC₆ — C₃xC₂²:C₄

Generators and relations for C₃xC₂³.₆D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁶=b, f²=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf^-1=bd=db, be=eb, ece^-1=fcf^-1=cd=dc, de=ed, df=fd, fef^-1=cde⁵ >

Subgroups: 354 in 121 conjugacy classes, 38 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂²:C₄, C₂xD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂³:C₄, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², C₆², C₆.D₄, C₃xC₂²:C₄, C₃xC₂²:C₄, C₂xC₃:D₄, C₆xD₄, C₆xDic₃, C₆xDic₃, C₃xC₃:D₄, C₆xC₁₂, S₃xC₂xC₆, C₂xC₆², C₂³.₆D₆, C₃xC₂³:C₄, C₃xC₆.D₄, C₃²xC₂²:C₄, C₆xC₃:D₄, C₃xC₂³.₆D₆
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₄, C₁₂, D₆, C₂xC₆, C₂²:C₄, C₃xS₃, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂³:C₄, S₃xC₆, D₆:C₄, C₃xC₂²:C₄, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₂³.₆D₆, C₃xC₂³:C₄, C₃xD₆:C₄, C₃xC₂³.₆D₆

Permutation representations of C₃xC₂³.₆D₆
►On 24 points - transitive group 24T587

Generators in S₂₄

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

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(2 12)(4 8)(6 10)(13 19)(15 21)(17 23)

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

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

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

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

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

G:=TransitiveGroup(24,587);

►On 24 points - transitive group 24T629

Generators in S₂₄

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

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

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

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

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

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

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

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

G:=TransitiveGroup(24,629);

63 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6Q	6R	···	6W	6X	6Y	12A	···	12P	12Q	···	12V
order	1	2	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
size	1	1	2	2	2	12	1	1	2	2	2	4	4	12	12	12	1	1	2	···	2	4	···	4	12	12	4	···	4	12	···	12

63 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+									+	+	+			+							+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	S₃	D₄	D₆	C₃xS₃	C₄xS₃	D₁₂	C₃:D₄	C₃xD₄	S₃xC₆	S₃xC₁₂	C₃xD₁₂	C₃xC₃:D₄	C₂³:C₄	C₂³.₆D₆	C₃xC₂³:C₄	C₃xC₂³.₆D₆
kernel	C₃xC₂³.₆D₆	C₃xC₆.D₄	C₃²xC₂²:C₄	C₆xC₃:D₄	C₂³.₆D₆	C₆xDic₃	S₃xC₂xC₆	C₆.D₄	C₃xC₂²:C₄	C₂xC₃:D₄	C₂xDic₃	C₂²xS₃	C₃xC₂²:C₄	C₆²	C₂²xC₆	C₂²:C₄	C₂xC₆	C₂xC₆	C₂xC₆	C₂xC₆	C₂³	C₂²	C₂²	C₂²	C₃²	C₃	C₃	C₁
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	2	1	2	2	2	2	4	2	4	4	4	1	2	2	4

Matrix representation of C₃xC₂³.₆D₆ ►in GL₄(F₇) generated by

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

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

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

6	0	0	0
0	6	0	0
0	0	6	0
0	0	0	6

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

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

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

C₃xC₂³.₆D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._6D_6

% in TeX

G:=Group("C3xC2^3.6D6");

// GroupNames label

G:=SmallGroup(288,240);

// by ID

G=gap.SmallGroup(288,240);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,1271,9414]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;

// generators/relations

G = C3xC23.6D6 order 288 = 25·32

Direct product of C3 and C23.6D6

G = C₃xC₂³.₆D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₆D₆