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## G = C2×S3×C3.A4order 432 = 24·33

### Direct product of C2, S3 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×S3×C3.A4
 Chief series C1 — C3 — C2×C6 — C62 — C3×C3.A4 — S3×C3.A4 — C2×S3×C3.A4
 Lower central C2×C6 — C2×S3×C3.A4
 Upper central C1 — C6

Generators and relations for C2×S3×C3.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, C2×C6, C2×C6, C24, C18, C3×S3, C3×S3, C3×C6, C3×C6, C22×S3, C22×S3, C22×C6, C22×C6, C3×C9, C3.A4, C3.A4, C2×C18, S3×C6, S3×C6, C62, C62, S3×C23, C23×C6, S3×C9, C3×C18, C2×C3.A4, C2×C3.A4, S3×C2×C6, S3×C2×C6, C2×C62, C3×C3.A4, S3×C18, C22×C3.A4, S3×C22×C6, S3×C3.A4, C6×C3.A4, C2×S3×C3.A4
Quotients: C1, C2, C3, C22, S3, C6, C9, A4, D6, C2×C6, C18, C3×S3, C2×A4, C3.A4, C2×C18, S3×C6, C22×A4, S3×C9, C2×C3.A4, S3×A4, S3×C18, C22×C3.A4, C2×S3×A4, S3×C3.A4, C2×S3×C3.A4

Smallest permutation representation of C2×S3×C3.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 C3×S3 S3×C6 S3×C9 S3×C18 A4 C2×A4 C2×A4 C3.A4 C2×C3.A4 C2×C3.A4 S3×A4 C2×S3×A4 S3×C3.A4 C2×S3×C3.A4 kernel C2×S3×C3.A4 S3×C3.A4 C6×C3.A4 S3×C22×C6 S3×C2×C6 C2×C62 S3×C23 C22×S3 C22×C6 C2×C3.A4 C3.A4 C22×C6 C2×C6 C23 C22 S3×C6 C3×S3 C3×C6 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 C2×S3×C3.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] >;

C2×S3×C3.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

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