C2xS3xD12 - GroupNames

G = C₂xS₃xD₁₂ order 288 = 2⁵·3²

Direct product of C₂, S₃ and D₁₂

direct product, metabelian, supersoluble, monomial

Aliases: C₂xS₃xD₁₂, C₆².₁₃₅C₂³, C₆:₁(S₃xD₄), C₆:₁(C₂xD₁₂), (C₄xS₃):₁₄D₆, (S₃xC₆):₁₄D₄, (C₂xC₁₂):₂₁D₆, (C₆xD₁₂):₁₅C₂, (S₃xC₆):₁C₂³, (C₆xC₁₂):₃C₂², (C₃xC₁₂):₂C₂³, C₁₂:₄(C₂²xS₃), (C₂²xS₃):₉D₆, D₆:₁(C₂²xS₃), C₃:₁(C₂²xD₁₂), C₃²:₂(C₂²xD₄), (C₂xDic₃):₁₉D₆, C₆.₁₀(S₃xC₂³), (C₃xC₆).₁₀C₂⁴, (S₃xC₁₂):₁₈C₂², (C₃xD₁₂):₂₇C₂², C₃:D₁₂:₈C₂², (C₃xDic₃):₄C₂³, Dic₃:₄(C₂²xS₃), C₁₂:S₃:₂₁C₂², (C₆xDic₃):₂₆C₂², C₄:₂(C₂xS₃²), (C₂xC₄):₆S₃², C₃:₁(C₂xS₃xD₄), (S₃xC₂xC₄):₅S₃, (S₃xC₂xC₁₂):₉C₂, (C₃xC₆):₂(C₂xD₄), (C₂xS₃²):₈C₂², (C₂²xS₃²):₅C₂, (C₃xS₃):₁(C₂xD₄), (S₃xC₂xC₆):₆C₂², (C₂xC₃:S₃):₁C₂³, C₂².₆₅(C₂xS₃²), C₂.₁₂(C₂²xS₃²), (C₂xC₁₂:S₃):₁₈C₂, (C₂xC₃:D₁₂):₁₄C₂, (C₂²xC₃:S₃):₅C₂², (C₂xC₆).₁₅₂(C₂²xS₃), SmallGroup(288,951)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₂xS₃xD₁₂

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — C₂xS₃² — C₂²xS₃² — C₂xS₃xD₁₂

Lower central

C₃² — C₃xC₆ — C₂xS₃xD₁₂

Upper central

C₁ — C₂² — C₂xC₄

Generators and relations for C₂xS₃xD₁₂
G = < a,b,c,d,e | a²=b³=c²=d¹²=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: 2242 in 499 conjugacy classes, 132 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xD₄, C₂⁴, C₃xS₃, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xD₄, C₃xDic₃, C₃xC₁₂, S₃², S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₂xC₃:S₃, C₆², S₃xC₂xC₄, C₂xD₁₂, C₂xD₁₂, S₃xD₄, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, S₃xC₂³, C₃:D₁₂, S₃xC₁₂, C₃xD₁₂, C₆xDic₃, C₁₂:S₃, C₆xC₁₂, C₂xS₃², C₂xS₃², S₃xC₂xC₆, S₃xC₂xC₆, C₂²xC₃:S₃, C₂²xD₁₂, C₂xS₃xD₄, S₃xD₁₂, C₂xC₃:D₁₂, S₃xC₂xC₁₂, C₆xD₁₂, C₂xC₁₂:S₃, C₂²xS₃², C₂xS₃xD₁₂
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, D₁₂, C₂²xS₃, C₂²xD₄, S₃², C₂xD₁₂, S₃xD₄, S₃xC₂³, C₂xS₃², C₂²xD₁₂, C₂xS₃xD₄, S₃xD₁₂, C₂²xS₃², C₂xS₃xD₁₂

Smallest permutation representation of C₂xS₃xD₁₂
►On 48 points

Generators in S₄₈

(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)

(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)

(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)

(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)(37 38 39 40 41 42 43 44 45 46 47 48)

(1 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 25)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)

G:=sub<Sym(48)| (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,25)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)>;

G:=Group( (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,25)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26) );

G=PermutationGroup([[(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,25),(24,26)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,39),(2,38),(3,37),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,25),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26)]])

54 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	6H	6I	6J	6K	6L	6M	6N	6O	6P	6Q	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N
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	6	6	6	6	6	12	12	12	12	12	···	12	12	12	12	12
size	1	1	1	1	3	3	3	3	6	6	6	6	18	18	18	18	2	2	4	2	2	6	6	2	···	2	4	4	4	6	6	6	6	12	12	12	12	2	2	2	2	4	···	4	6	6	6	6

54 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

D₁₂

S₃²

S₃xD₄

C₂xS₃²

S₃xD₁₂

kernel

C₂xS₃xD₁₂

S₃xD₁₂

C₂xC₃:D₁₂

S₃xC₂xC₁₂

C₆xD₁₂

C₂xC₁₂:S₃

C₂²xS₃²

S₃xC₂xC₄

C₂xD₁₂

S₃xC₆

C₄xS₃

D₁₂

C₂xDic₃

C₂xC₁₂

C₂²xS₃

D₆

C₂xC₄

C₆

C₄

C₂²

C₂

# reps

Matrix representation of C₂xS₃xD₁₂ ►in GL₆(Z)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	-1	0
0	0	0	0	0	-1

0	1	0	0	0	0
-1	-1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
-1	-1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	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	1	2	0	0
0	0	-1	-1	0	0
0	0	0	0	0	1
0	0	0	0	-1	-1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	-1	-1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂xS₃xD₁₂ in GAP, Magma, Sage, TeX

C_2\times S_3\times D_{12}

% in TeX

G:=Group("C2xS3xD12");

// GroupNames label

G:=SmallGroup(288,951);

// by ID

G=gap.SmallGroup(288,951);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,346,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^12=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 = C2xS3xD12 order 288 = 25·32

Direct product of C2, S3 and D12

G = C₂xS₃xD₁₂ order 288 = 2⁵·3²

Direct product of C₂, S₃ and D₁₂