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## G = Dic3×C3.A4order 432 = 24·33

### Direct product of Dic3 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3×C3.A4
 Chief series C1 — C3 — C2×C6 — C62 — C2×C62 — C6×C3.A4 — Dic3×C3.A4
 Lower central C2×C6 — Dic3×C3.A4
 Upper central C1 — C6

Generators and relations for Dic3×C3.A4
G = < a,b,c,d,e,f | a6=c3=d2=e2=1, b2=a3, f3=c, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 244 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, C36, C3.A4, C3.A4, C3×Dic3, C3×Dic3, C62, C62, C22×Dic3, C22×C12, C3×C18, C2×C3.A4, C2×C3.A4, C6×Dic3, C2×C62, C9×Dic3, C3×C3.A4, C4×C3.A4, Dic3×C2×C6, C6×C3.A4, Dic3×C3.A4
Quotients: C1, C2, C3, C4, S3, C6, C9, Dic3, C12, A4, C18, C3×S3, C2×A4, C36, C3.A4, C3×Dic3, C4×A4, S3×C9, C2×C3.A4, S3×A4, C9×Dic3, C4×C3.A4, Dic3×A4, S3×C3.A4, Dic3×C3.A4

Smallest permutation representation of Dic3×C3.A4
On 36 points
Generators in S36
(1 25 4 19 7 22)(2 26 5 20 8 23)(3 27 6 21 9 24)(10 34 16 31 13 28)(11 35 17 32 14 29)(12 36 18 33 15 30)
(1 10 19 31)(2 11 20 32)(3 12 21 33)(4 13 22 34)(5 14 23 35)(6 15 24 36)(7 16 25 28)(8 17 26 29)(9 18 27 30)
(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 19)(2 20)(4 22)(5 23)(7 25)(8 26)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)
(2 20)(3 21)(5 23)(6 24)(8 26)(9 27)(11 32)(12 33)(14 35)(15 36)(17 29)(18 30)
(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)| (1,25,4,19,7,22)(2,26,5,20,8,23)(3,27,6,21,9,24)(10,34,16,31,13,28)(11,35,17,32,14,29)(12,36,18,33,15,30), (1,10,19,31)(2,11,20,32)(3,12,21,33)(4,13,22,34)(5,14,23,35)(6,15,24,36)(7,16,25,28)(8,17,26,29)(9,18,27,30), (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), (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,31)(11,32)(13,34)(14,35)(16,28)(17,29), (2,20)(3,21)(5,23)(6,24)(8,26)(9,27)(11,32)(12,33)(14,35)(15,36)(17,29)(18,30), (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( (1,25,4,19,7,22)(2,26,5,20,8,23)(3,27,6,21,9,24)(10,34,16,31,13,28)(11,35,17,32,14,29)(12,36,18,33,15,30), (1,10,19,31)(2,11,20,32)(3,12,21,33)(4,13,22,34)(5,14,23,35)(6,15,24,36)(7,16,25,28)(8,17,26,29)(9,18,27,30), (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), (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,31)(11,32)(13,34)(14,35)(16,28)(17,29), (2,20)(3,21)(5,23)(6,24)(8,26)(9,27)(11,32)(12,33)(14,35)(15,36)(17,29)(18,30), (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([[(1,25,4,19,7,22),(2,26,5,20,8,23),(3,27,6,21,9,24),(10,34,16,31,13,28),(11,35,17,32,14,29),(12,36,18,33,15,30)], [(1,10,19,31),(2,11,20,32),(3,12,21,33),(4,13,22,34),(5,14,23,35),(6,15,24,36),(7,16,25,28),(8,17,26,29),(9,18,27,30)], [(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)], [(1,19),(2,20),(4,22),(5,23),(7,25),(8,26),(10,31),(11,32),(13,34),(14,35),(16,28),(17,29)], [(2,20),(3,21),(5,23),(6,24),(8,26),(9,27),(11,32),(12,33),(14,35),(15,36),(17,29),(18,30)], [(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 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 18G ··· 18L 36A ··· 36L order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 ··· 6 9 ··· 9 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 3 3 1 1 2 2 2 3 3 9 9 1 1 2 2 2 3 3 3 3 6 ··· 6 4 ··· 4 8 ··· 8 3 3 3 3 9 9 9 9 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 C3 C4 C6 C9 C12 C18 C36 S3 Dic3 C3×S3 C3×Dic3 S3×C9 C9×Dic3 A4 C2×A4 C3.A4 C4×A4 C2×C3.A4 C4×C3.A4 S3×A4 Dic3×A4 S3×C3.A4 Dic3×C3.A4 kernel Dic3×C3.A4 C6×C3.A4 Dic3×C2×C6 C3×C3.A4 C2×C62 C22×Dic3 C62 C22×C6 C2×C6 C2×C3.A4 C3.A4 C22×C6 C2×C6 C23 C22 C3×Dic3 C3×C6 Dic3 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 1 1 2 2 6 6 1 1 2 2 2 4 1 1 2 2

Matrix representation of Dic3×C3.A4 in GL5(𝔽37)

 11 5 0 0 0 0 27 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 3 19 0 0 0 17 34 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 26 0 0 0 0 0 26
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 1 12 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 12 0 0 0 36 25 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 21 21 0 0 0 0 15 16

G:=sub<GL(5,GF(37))| [11,0,0,0,0,5,27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,17,0,0,0,19,34,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,12,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,12,25,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,21,0,0,0,16,21,15,0,0,0,0,16] >;

Dic3×C3.A4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3.A_4
% in TeX

G:=Group("Dic3xC3.A4");
// GroupNames label

G:=SmallGroup(432,271);
// by ID

G=gap.SmallGroup(432,271);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,2,-3,42,92,1901,768,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=c^3=d^2=e^2=1,b^2=a^3,f^3=c,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*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*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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