C3xS3xD12 - GroupNames

G = C₃xS₃xD₁₂ order 432 = 2⁴·3³

Direct product of C₃, S₃ and D₁₂

direct product, metabelian, supersoluble, monomial

Aliases: C₃xS₃xD₁₂, C₁₂:₆S₃², C₁₂:₃(S₃xC₆), D₆:₁(S₃xC₆), C₃:₁(C₆xD₁₂), (S₃xC₁₂):₄C₆, (S₃xC₁₂):₄S₃, (S₃xC₆):₁₁D₆, (C₃xD₁₂):₆C₆, (C₃xC₁₂):₁₉D₆, C₃²:₅(C₆xD₄), C₃³:₁₁(C₂xD₄), C₃:D₁₂:₁C₆, Dic₃:₃(S₃xC₆), (S₃xC₃²):₄D₄, C₁₂:S₃:₁₁C₆, C₃²:₂₀(S₃xD₄), (C₃xDic₃):₁₇D₆, (C₃²xD₁₂):₈C₂, C₃²:₁₂(C₂xD₁₂), (C₃²xC₁₂):₁C₂², (C₃²xC₆).₂₇C₂³, (C₃²xDic₃):₁₄C₂², C₄:₁(C₃xS₃²), C₃:₁(C₃xS₃xD₄), (C₂xS₃²):₁C₆, (S₃²xC₆):₅C₂, C₆.₈(S₃xC₂xC₆), (S₃xC₃xC₁₂):₅C₂, C₂.₁₀(S₃²xC₆), (C₄xS₃):₃(C₃xS₃), (S₃xC₆):₁(C₂xC₆), C₆.₁₁₁(C₂xS₃²), (C₃xC₁₂):₅(C₂xC₆), (C₃xS₃):₁(C₃xD₄), (S₃xC₃xC₆):₇C₂², (C₆xC₃:S₃):₆C₂², (C₃xC₁₂:S₃):₇C₂, (C₃xDic₃):₃(C₂xC₆), (C₃xC₃:D₁₂):₁₃C₂, (C₃xC₆).₁₈(C₂²xC₆), (C₃xC₆).₁₃₂(C₂²xS₃), (C₂xC₃:S₃):₄(C₂xC₆), SmallGroup(432,649)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xS₃xD₁₂

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — S₃²xC₆ — C₃xS₃xD₁₂

Lower central

C₃² — C₃xC₆ — C₃xS₃xD₁₂

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xS₃xD₁₂
G = < a,b,c,d,e | a³=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: 1096 in 270 conjugacy classes, 72 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₄xS₃, D₁₂, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃xDic₃, C₃xC₁₂, C₃xC₁₂, S₃², S₃xC₆, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₆², C₂xD₁₂, S₃xD₄, C₆xD₄, S₃xC₃², S₃xC₃², C₃xC₃:S₃, C₃²xC₆, C₃:D₁₂, S₃xC₁₂, S₃xC₁₂, C₃xD₁₂, C₃xD₁₂, C₃xC₃:D₄, C₁₂:S₃, C₆xC₁₂, D₄xC₃², C₂xS₃², S₃xC₂xC₆, C₃²xDic₃, C₃²xC₁₂, C₃xS₃², S₃xC₃xC₆, S₃xC₃xC₆, C₆xC₃:S₃, S₃xD₁₂, C₆xD₁₂, C₃xS₃xD₄, C₃xC₃:D₁₂, S₃xC₃xC₁₂, C₃²xD₁₂, C₃xC₁₂:S₃, S₃²xC₆, C₃xS₃xD₁₂
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, D₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, C₂xD₁₂, S₃xD₄, C₆xD₄, C₃xD₁₂, C₂xS₃², S₃xC₂xC₆, C₃xS₃², S₃xD₁₂, C₆xD₁₂, C₃xS₃xD₄, S₃²xC₆, C₃xS₃xD₁₂

Smallest permutation representation of C₃xS₃xD₁₂
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

81 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	6B	6C	···	6H	6I	6J	6K	6L	6M	6N	6O	6P	···	6Y	6Z	···	6AE	6AF	6AG	6AH	6AI	12A	···	12H	12I	···	12Q	12R	···	12Y
order	1	2	2	2	2	2	2	2	3	3	3	···	3	3	3	3	4	4	6	6	6	···	6	6	6	6	6	6	6	6	6	···	6	6	···	6	6	6	6	6	12	···	12	12	···	12	12	···	12
size	1	1	3	3	6	6	18	18	1	1	2	···	2	4	4	4	2	6	1	1	2	···	2	3	3	3	3	4	4	4	6	···	6	12	···	12	18	18	18	18	2	···	2	4	···	4	6	···	6

81 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	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+							+	+	+	+	+	+			+						+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	S₃	D₄	D₆	D₆	D₆	C₃xS₃	C₃xS₃	D₁₂	C₃xD₄	S₃xC₆	S₃xC₆	S₃xC₆	C₃xD₁₂	S₃²	S₃xD₄	C₂xS₃²	C₃xS₃²	S₃xD₁₂	C₃xS₃xD₄	S₃²xC₆	C₃xS₃xD₁₂
kernel	C₃xS₃xD₁₂	C₃xC₃:D₁₂	S₃xC₃xC₁₂	C₃²xD₁₂	C₃xC₁₂:S₃	S₃²xC₆	S₃xD₁₂	C₃:D₁₂	S₃xC₁₂	C₃xD₁₂	C₁₂:S₃	C₂xS₃²	S₃xC₁₂	C₃xD₁₂	S₃xC₃²	C₃xDic₃	C₃xC₁₂	S₃xC₆	C₄xS₃	D₁₂	C₃xS₃	C₃xS₃	Dic₃	C₁₂	D₆	S₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	1	1	1	2	2	4	2	2	2	4	1	1	2	1	2	3	2	2	4	4	2	4	6	8	1	1	1	2	2	2	2	4

Matrix representation of C₃xS₃xD₁₂ ►in GL₆(F₁₃)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	3	0	0	0
0	0	0	3	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	0	1	0	0
0	0	12	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

8	0	0	0	0	0
0	5	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	1
0	0	0	0	12	0

0	5	0	0	0	0
8	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	12	1

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

C₃xS₃xD₁₂ in GAP, Magma, Sage, TeX

C_3\times S_3\times D_{12}

% in TeX

G:=Group("C3xS3xD12");

// GroupNames label

G:=SmallGroup(432,649);

// by ID

G=gap.SmallGroup(432,649);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,142,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=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 = C3xS3xD12 order 432 = 24·33

Direct product of C3, S3 and D12

G = C₃xS₃xD₁₂ order 432 = 2⁴·3³

Direct product of C₃, S₃ and D₁₂