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## G = C3×C23.3A4order 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 >

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

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