C2xDic3:D6 - GroupNames

G = C₂xDic₃:D₆ order 288 = 2⁵·3²

Direct product of C₂ and Dic₃:D₆

direct product, metabelian, supersoluble, monomial, rational

Aliases: C₂xDic₃:D₆, C₆²:₃C₂³, C₂³:₅S₃², C₆:₃(S₃xD₄), C₃:D₄:₉D₆, (C₂²xC₆):₈D₆, (S₃xC₆):₅C₂³, D₆:₄(C₂²xS₃), C₃²:₈(C₂²xD₄), (C₂xDic₃):₁₆D₆, (C₂²xS₃):₁₃D₆, C₆.₃₆(S₃xC₂³), (C₃xC₆).₃₆C₂⁴, (C₂xC₆²):₇C₂², (C₆xDic₃):₉C₂², (C₃xDic₃):₃C₂³, Dic₃:₃(C₂²xS₃), C₃:D₁₂:₁₈C₂², C₆.D₆:₁₂C₂², C₃:₄(C₂xS₃xD₄), C₃:S₃:₄(C₂xD₄), C₂²:₄(C₂xS₃²), (C₃xC₆):₇(C₂xD₄), (C₂xC₃:S₃):₁₈D₄, (C₂²xS₃²):₉C₂, (C₆xC₃:D₄):₇C₂, (C₂xS₃²):₁₂C₂², (C₂xC₃:D₄):₁₁S₃, (C₂³xC₃:S₃):₅C₂, (C₂xC₃:S₃):₅C₂³, (S₃xC₂xC₆):₁₂C₂², (C₂xC₆):₅(C₂²xS₃), C₂.₃₆(C₂²xS₃²), (C₂xC₆.D₆):₆C₂, (C₂xC₃:D₁₂):₂₂C₂, (C₃xC₃:D₄):₁₃C₂², (C₂²xC₃:S₃):₁₄C₂², SmallGroup(288,977)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₂xDic₃:D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — C₂xS₃² — C₂²xS₃² — C₂xDic₃:D₆

Lower central

C₃² — C₃xC₆ — C₂xDic₃:D₆

Upper central

C₁ — C₂² — C₂³

Generators and relations for C₂xDic₃:D₆
G = < a,b,c,d,e | a²=b⁶=d⁶=e²=1, c²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe=b^-1, bd=db, dcd^-1=b³c, ce=ec, ede=d^-1 >

Subgroups: 2370 in 539 conjugacy classes, 124 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xD₄, C₂⁴, C₃xS₃, C₃:S₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²xD₄, C₃xDic₃, S₃², S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₆², C₆², S₃xC₂xC₄, C₂xD₁₂, S₃xD₄, C₂xC₃:D₄, C₂xC₃:D₄, C₆xD₄, S₃xC₂³, C₆.D₆, C₃:D₁₂, C₆xDic₃, C₃xC₃:D₄, C₂xS₃², C₂xS₃², S₃xC₂xC₆, C₂²xC₃:S₃, C₂²xC₃:S₃, C₂²xC₃:S₃, C₂xC₆², C₂xS₃xD₄, C₂xC₆.D₆, C₂xC₃:D₁₂, Dic₃:D₆, C₆xC₃:D₄, C₂²xS₃², C₂³xC₃:S₃, C₂xDic₃:D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₂²xS₃, C₂²xD₄, S₃², S₃xD₄, S₃xC₂³, C₂xS₃², C₂xS₃xD₄, Dic₃:D₆, C₂²xS₃², C₂xDic₃:D₆

Permutation representations of C₂xDic₃:D₆
►On 24 points - transitive group 24T671

Generators in S₂₄

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

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

(1 5)(2 4)(8 12)(9 11)(13 15)(16 18)(19 21)(22 24)

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

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

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

G:=TransitiveGroup(24,671);

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	2N	2O	3A	3B	3C	4A	4B	4C	4D	6A	···	6F	6G	···	6Q	6R	6S	6T	6U	12A	12B	12C	12D
order	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	3	3	3	4	4	4	4	6	···	6	6	···	6	6	6	6	6	12	12	12	12
size	1	1	1	1	2	2	6	6	6	6	9	9	9	9	18	18	2	2	4	6	6	6	6	2	···	2	4	···	4	12	12	12	12	12	12	12	12

48 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

S₃²

S₃xD₄

C₂xS₃²

Dic₃:D₆

kernel

C₂xDic₃:D₆

C₂xC₆.D₆

C₂xC₃:D₁₂

Dic₃:D₆

C₆xC₃:D₄

C₂²xS₃²

C₂³xC₃:S₃

C₂xC₃:D₄

C₂xC₃:S₃

C₂xDic₃

C₃:D₄

C₂²xS₃

C₂²xC₆

C₂³

C₆

C₂²

C₂

# reps

Matrix representation of C₂xDic₃:D₆ ►in GL₆(F₁₃)

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

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	5	0	0
0	0	10	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	3	1	0	0
0	0	0	0	0	1
0	0	0	0	12	12

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

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,3,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂xDic₃:D₆ in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\rtimes D_6

% in TeX

G:=Group("C2xDic3:D6");

// GroupNames label

G:=SmallGroup(288,977);

// by ID

G=gap.SmallGroup(288,977);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,346,1356,9414]);

// Polycyclic

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

// generators/relations

G = C2xDic3:D6 order 288 = 25·32

Direct product of C2 and Dic3:D6

G = C₂xDic₃:D₆ order 288 = 2⁵·3²

Direct product of C₂ and Dic₃:D₆