C2xS3xC3.A4 - GroupNames

G = C₂xS₃xC₃.A₄ order 432 = 2⁴·3³

Direct product of C₂, S₃ and C₃.A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂xS₃xC₃.A₄, (S₃xC₂³):C₉, (C₂²xC₆):C₁₈, (C₂²xS₃):C₁₈, (S₃xC₆).₂A₄, C₆.₁₉(S₃xA₄), C₂³:₂(S₃xC₉), C₂²:₂(S₃xC₁₈), (C₂xC₆²).₈C₆, C₆².₁₄(C₂xC₆), C₃².₂(C₂²xA₄), C₆:(C₂xC₃.A₄), (S₃xC₂xC₆).C₆, (C₂xC₆):(C₂xC₁₈), C₃.₄(C₂xS₃xA₄), (C₃xS₃).(C₂xA₄), C₃:(C₂²xC₃.A₄), (S₃xC₂²xC₆).C₃, (C₆xC₃.A₄):₁C₂, (C₂xC₆).₁₇(S₃xC₆), (C₃xC₆).₁₄(C₂xA₄), (C₃xC₃.A₄):₂C₂², (C₂²xC₆).₁₉(C₃xS₃), SmallGroup(432,541)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂xS₃xC₃.A₄

Chief series

C₁ — C₃ — C₂xC₆ — C₆² — C₃xC₃.A₄ — S₃xC₃.A₄ — C₂xS₃xC₃.A₄

Lower central

C₂xC₆ — C₂xS₃xC₃.A₄

Upper central

C₁ — C₆

Generators and relations for C₂xS₃xC₃.A₄
G = < a,b,c,d,e,f,g | a²=b³=c²=d³=e²=f²=1, g³=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b^-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 604 in 148 conjugacy classes, 35 normal (25 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, S₃, C₆, C₆, C₂³, C₂³, C₉, C₃², D₆, D₆, C₂xC₆, C₂xC₆, C₂⁴, C₁₈, C₃xS₃, C₃xS₃, C₃xC₆, C₃xC₆, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₃xC₉, C₃.A₄, C₃.A₄, C₂xC₁₈, S₃xC₆, S₃xC₆, C₆², C₆², S₃xC₂³, C₂³xC₆, S₃xC₉, C₃xC₁₈, C₂xC₃.A₄, C₂xC₃.A₄, S₃xC₂xC₆, S₃xC₂xC₆, C₂xC₆², C₃xC₃.A₄, S₃xC₁₈, C₂²xC₃.A₄, S₃xC₂²xC₆, S₃xC₃.A₄, C₆xC₃.A₄, C₂xS₃xC₃.A₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₉, A₄, D₆, C₂xC₆, C₁₈, C₃xS₃, C₂xA₄, C₃.A₄, C₂xC₁₈, S₃xC₆, C₂²xA₄, S₃xC₉, C₂xC₃.A₄, S₃xA₄, S₃xC₁₈, C₂²xC₃.A₄, C₂xS₃xA₄, S₃xC₃.A₄, C₂xS₃xC₃.A₄

Smallest permutation representation of C₂xS₃xC₃.A₄
►On 36 points

Generators in S₃₆

(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 28)(17 29)(18 30)

(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)

(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 28)(27 29)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)

(1 20)(3 22)(4 23)(6 25)(7 26)(9 19)(10 31)(12 33)(13 34)(15 36)(16 28)(18 30)

(1 20)(2 21)(4 23)(5 24)(7 26)(8 27)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)

(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)

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

G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,28)(17,29)(18,30), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,28)(27,29), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,20)(3,22)(4,23)(6,25)(7,26)(9,19)(10,31)(12,33)(13,34)(15,36)(16,28)(18,30), (1,20)(2,21)(4,23)(5,24)(7,26)(8,27)(10,31)(11,32)(13,34)(14,35)(16,28)(17,29), (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) );

G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,19),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,28),(17,29),(18,30)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(25,36),(26,28),(27,29)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(1,20),(3,22),(4,23),(6,25),(7,26),(9,19),(10,31),(12,33),(13,34),(15,36),(16,28),(18,30)], [(1,20),(2,21),(4,23),(5,24),(7,26),(8,27),(10,31),(11,32),(13,34),(14,35),(16,28),(17,29)], [(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)]])

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	6A	6B	6C	6D	6E	6F	···	6M	6N	···	6S	6T	6U	6V	6W	9A	···	9F	9G	···	9L	18A	···	18F	18G	···	18L	18M	···	18X
order	1	2	2	2	2	2	2	2	3	3	3	3	3	6	6	6	6	6	6	···	6	6	···	6	6	6	6	6	9	···	9	9	···	9	18	···	18	18	···	18	18	···	18
size	1	1	3	3	3	3	9	9	1	1	2	2	2	1	1	2	2	2	3	···	3	6	···	6	9	9	9	9	4	···	4	8	···	8	4	···	4	8	···	8	12	···	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	3	3	6	6	6	6
type	+	+	+							+	+					+	+	+				+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	S₃	D₆	C₃xS₃	S₃xC₆	S₃xC₉	S₃xC₁₈	A₄	C₂xA₄	C₂xA₄	C₃.A₄	C₂xC₃.A₄	C₂xC₃.A₄	S₃xA₄	C₂xS₃xA₄	S₃xC₃.A₄	C₂xS₃xC₃.A₄
kernel	C₂xS₃xC₃.A₄	S₃xC₃.A₄	C₆xC₃.A₄	S₃xC₂²xC₆	S₃xC₂xC₆	C₂xC₆²	S₃xC₂³	C₂²xS₃	C₂²xC₆	C₂xC₃.A₄	C₃.A₄	C₂²xC₆	C₂xC₆	C₂³	C₂²	S₃xC₆	C₃xS₃	C₃xC₆	D₆	S₃	C₆	C₆	C₃	C₂	C₁
# reps	1	2	1	2	4	2	6	12	6	1	1	2	2	6	6	1	2	1	2	4	2	1	1	2	2

Matrix representation of C₂xS₃xC₃.A₄ ►in GL₅(F₁₉)

18	0	0	0	0
0	18	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

11	0	0	0	0
0	7	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	18	0	0	0
18	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	7	0	0
0	0	0	7	0
0	0	0	0	7

1	0	0	0	0
0	1	0	0	0
0	0	18	0	0
0	0	0	18	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	18	0	0
0	0	0	1	0
0	0	0	0	18

1	0	0	0	0
0	1	0	0	0
0	0	0	6	0
0	0	0	0	6
0	0	6	0	0

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,6,0,0,0,0,0,6,0] >;

C₂xS₃xC₃.A₄ in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3.A_4

% in TeX

G:=Group("C2xS3xC3.A4");

// GroupNames label

G:=SmallGroup(432,541);

// by ID

G=gap.SmallGroup(432,541);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-3,79,963,397,14118]);

// Polycyclic

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

// generators/relations

G = C2xS3xC3.A4 order 432 = 24·33

Direct product of C2, S3 and C3.A4

G = C₂xS₃xC₃.A₄ order 432 = 2⁴·3³

Direct product of C₂, S₃ and C₃.A₄