C3xC2^4:4S3 - GroupNames

G = C₃xC₂⁴:₄S₃ order 288 = 2⁵·3²

Direct product of C₃ and C₂⁴:₄S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂⁴:₄S₃, (C₂xC₆)wr C₂, C₆²:₂₃D₄, C₆².₂₁₁C₂³, (C₂³xC₆):₇S₃, C₆.₆₃(C₆xD₄), (C₂³xC₆):₁₁C₆, C₂⁴:₁₀(C₃xS₃), C₃²:₁₂C₂²wrC₂, C₂³.₃₅(S₃xC₆), (C₂²xC₆²):₃C₂, C₆.D₄:₁₃C₆, (C₂²xC₆).₁₃₁D₆, (C₆xDic₃):₁₉C₂², (C₂xC₆²).₁₀₃C₂², (C₂xC₆):₁₁(C₃xD₄), (C₂xC₃:D₄):₈C₆, (S₃xC₂xC₆):₅C₂², C₃:₃(C₃xC₂²wrC₂), (C₆xC₃:D₄):₂₂C₂, C₂.₂₆(C₆xC₃:D₄), C₂²:₅(C₃xC₃:D₄), C₂².₆₆(S₃xC₂xC₆), (C₂xC₆):₁₆(C₃:D₄), (C₂²xS₃):₂(C₂xC₆), (C₂xDic₃):₃(C₂xC₆), (C₃xC₆).₂₇₁(C₂xD₄), C₆.₁₆₄(C₂xC₃:D₄), (C₂²xC₆).₆₇(C₂xC₆), (C₂xC₆).₆₆(C₂²xC₆), (C₃xC₆.D₄):₂₉C₂, (C₂xC₆).₃₄₄(C₂²xS₃), SmallGroup(288,724)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂⁴:₄S₃

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — S₃xC₂xC₆ — C₆xC₃:D₄ — C₃xC₂⁴:₄S₃

Lower central

C₃ — C₂xC₆ — C₃xC₂⁴:₄S₃

Upper central

C₁ — C₂xC₆ — C₂³xC₆

Generators and relations for C₃xC₂⁴:₄S₃
G = < a,b,c,d,e,f,g | a³=b²=c²=d²=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f^-1 >

Subgroups: 714 in 327 conjugacy classes, 82 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂xD₄, C₂⁴, C₃xS₃, C₃xC₆, C₃xC₆, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²wrC₂, C₃xDic₃, S₃xC₆, C₆², C₆², C₆², C₆.D₄, C₃xC₂²:C₄, C₂xC₃:D₄, C₆xD₄, C₂³xC₆, C₂³xC₆, C₆xDic₃, C₃xC₃:D₄, S₃xC₂xC₆, C₂xC₆², C₂xC₆², C₂⁴:₄S₃, C₃xC₂²wrC₂, C₃xC₆.D₄, C₆xC₃:D₄, C₂²xC₆², C₃xC₂⁴:₄S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃:D₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂²wrC₂, S₃xC₆, C₂xC₃:D₄, C₆xD₄, C₃xC₃:D₄, S₃xC₂xC₆, C₂⁴:₄S₃, C₃xC₂²wrC₂, C₆xC₃:D₄, C₃xC₂⁴:₄S₃

Permutation representations of C₃xC₂⁴:₄S₃
►On 24 points - transitive group 24T626

Generators in S₂₄

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

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,626);

90 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	3A	3B	3C	3D	3E	4A	4B	4C	6A	···	6F	6G	···	6BK	6BL	6BM	12A	···	12F
order	1	2	2	2	2	···	2	2	3	3	3	3	3	4	4	4	6	···	6	6	···	6	6	6	12	···	12
size	1	1	1	1	2	···	2	12	1	1	2	2	2	12	12	12	1	···	1	2	···	2	12	12	12	···	12

90 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2
type	+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₄	D₆	C₃xS₃	C₃:D₄	C₃xD₄	S₃xC₆	C₃xC₃:D₄
kernel	C₃xC₂⁴:₄S₃	C₃xC₆.D₄	C₆xC₃:D₄	C₂²xC₆²	C₂⁴:₄S₃	C₆.D₄	C₂xC₃:D₄	C₂³xC₆	C₂³xC₆	C₆²	C₂²xC₆	C₂⁴	C₂xC₆	C₂xC₆	C₂³	C₂²
# reps	1	3	3	1	2	6	6	2	1	6	3	2	12	12	6	24

Matrix representation of C₃xC₂⁴:₄S₃ ►in GL₄(F₁₃) generated by

1	0	0	0
0	1	0	0
0	0	3	0
0	0	0	3

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

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

12	0	0	0
0	12	0	0
0	0	12	0
0	0	0	12

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

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

0	1	0	0
1	0	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C₃xC₂⁴:₄S₃ in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes_4S_3

% in TeX

G:=Group("C3xC2^4:4S3");

// GroupNames label

G:=SmallGroup(288,724);

// by ID

G=gap.SmallGroup(288,724);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,9414]);

// Polycyclic

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

// generators/relations

G = C3xC24:4S3 order 288 = 25·32

Direct product of C3 and C24:4S3

G = C₃xC₂⁴:₄S₃ order 288 = 2⁵·3²

Direct product of C₃ and C₂⁴:₄S₃