C3xC2^3.3A4 - GroupNames

G = C₃×C₂³.₃A₄ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₃A₄

Aliases: C₃×C₂³.₃A₄, C₆.₃(C₄²⋊C₃), (C₂²×C₆).₉A₄, C₂³.₄(C₃×A₄), C₂.C₄²⋊C₃², (C₂×C₆).₂SL₂(𝔽₃), C₂².(C₃×SL₂(𝔽₃)), C₂.(C₃×C₄²⋊C₃), (C₃×C₂.C₄²)⋊C₃, SmallGroup(288,230)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂.C₄² — C₃×C₂³.₃A₄

Chief series

C₁ — C₂ — C₂³ — C₂.C₄² — C₂³.₃A₄ — C₃×C₂³.₃A₄

Lower central

C₂.C₄² — C₃×C₂³.₃A₄

Upper central

C₁ — C₆

Generators and relations for C₃×C₂³.₃A₄
G = < a,b,c,d,e,f,g | a³=b²=c²=d²=g³=1, e²=gbg^-1=bcd, f²=gcg^-1=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef^-1=de=ed, df=fd, dg=gd, geg^-1=bef, gfg^-1=cde >

C₁

C₂

₃C₂

C₃

₁₆C₃

C₂²

₃C₂²

₆C₄

C₆

₃C₆

₁₆C₆

₁₆C₃²

C₂³

₃C₂×C₄

₆C₂×C₄

C₂×C₆

₃C₂×C₆

₄A₄

₆C₁₂

₁₆C₃×C₆

₃C₂²×C₄

C₂²×C₆

₃C₂×C₁₂

₄C₂×A₄

₆C₂×C₁₂

₄C₃×A₄

C₂.C₄²

₃C₂²×C₁₂

₄C₆×A₄

C₃×C₂.C₄²

C₂³.₃A₄

C₃×C₂³.₃A₄

Smallest permutation representation of C₃×C₂³.₃A₄
►On 36 points

Generators in S₃₆

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

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

(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)

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

(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)

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	···	6L	12A	···	12H
order	1	2	2	2	3	3	3	···	3	4	4	4	4	6	6	6	6	6	6	6	···	6	12	···	12
size	1	1	3	3	1	1	16	···	16	6	6	6	6	1	1	3	3	3	3	16	···	16	6	···	6

36 irreducible representations

dim	1	1	1	2	2	2	3	3	3	3	6	6
type	+			-			+				+
image	C₁	C₃	C₃	SL₂(𝔽₃)	SL₂(𝔽₃)	C₃×SL₂(𝔽₃)	A₄	C₃×A₄	C₄²⋊C₃	C₃×C₄²⋊C₃	C₂³.₃A₄	C₃×C₂³.₃A₄
kernel	C₃×C₂³.₃A₄	C₂³.₃A₄	C₃×C₂.C₄²	C₂×C₆	C₂×C₆	C₂²	C₂²×C₆	C₂³	C₆	C₂	C₃	C₁
# reps	1	6	2	1	2	6	1	2	4	8	1	2

Matrix representation of C₃×C₂³.₃A₄ ►in GL₅(𝔽₁₃)

1	0	0	0	0
0	1	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

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

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

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

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

4	3	0	0	0
3	9	0	0	0
0	0	8	0	0
0	0	0	1	0
0	0	0	0	5

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

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,12],[4,3,0,0,0,3,9,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,5],[1,3,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C₃×C₂³.₃A₄ in GAP, Magma, Sage, TeX

C_3\times C_2^3._3A_4

% in TeX

G:=Group("C3xC2^3.3A4");

// GroupNames label

G:=SmallGroup(288,230);

// by ID

G=gap.SmallGroup(288,230);

# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,380,268,2775,521,80,7564,10589]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃×C₂³.₃A₄ in TeX

G = C3×C23.3A4 order 288 = 25·32

Direct product of C3 and C23.3A4

G = C₃×C₂³.₃A₄ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₃A₄