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## G = Dic3×3- 1+2order 324 = 22·34

### Direct product of Dic3 and 3- 1+2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — Dic3×3- 1+2
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C6×3- 1+2 — Dic3×3- 1+2
 Lower central C3 — C32 — Dic3×3- 1+2
 Upper central C1 — C6 — C2×3- 1+2

Generators and relations for Dic3×3- 1+2
G = < a,b,c,d | a6=c9=d3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 140 in 70 conjugacy classes, 35 normal (20 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C9, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C9, 3- 1+2, 3- 1+2, C33, C36, C3×Dic3, C3×Dic3, C3×C12, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C3×3- 1+2, C9×Dic3, C4×3- 1+2, C32×Dic3, C6×3- 1+2, Dic3×3- 1+2
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, 3- 1+2, C3×Dic3, C3×C12, C2×3- 1+2, S3×C32, C4×3- 1+2, C32×Dic3, S3×3- 1+2, Dic3×3- 1+2

Smallest permutation representation of Dic3×3- 1+2
On 36 points
Generators in S36
(1 21 4 24 7 27)(2 22 5 25 8 19)(3 23 6 26 9 20)(10 35 16 32 13 29)(11 36 17 33 14 30)(12 28 18 34 15 31)
(1 15 24 28)(2 16 25 29)(3 17 26 30)(4 18 27 31)(5 10 19 32)(6 11 20 33)(7 12 21 34)(8 13 22 35)(9 14 23 36)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)

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

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

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

66 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 18G ··· 18L 36A ··· 36L order 1 2 3 3 3 3 3 3 3 3 3 4 4 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 1 1 2 2 2 3 3 6 6 3 3 1 1 2 2 2 3 3 6 6 3 ··· 3 6 ··· 6 3 3 3 3 9 9 9 9 3 ··· 3 6 ··· 6 9 ··· 9

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 type + + + - image C1 C2 C3 C3 C4 C6 C6 C12 C12 S3 Dic3 C3×S3 C3×S3 C3×Dic3 C3×Dic3 3- 1+2 C2×3- 1+2 C4×3- 1+2 S3×3- 1+2 Dic3×3- 1+2 kernel Dic3×3- 1+2 C6×3- 1+2 C9×Dic3 C32×Dic3 C3×3- 1+2 C3×C18 C32×C6 C3×C9 C33 C2×3- 1+2 3- 1+2 C18 C3×C6 C9 C32 Dic3 C6 C3 C2 C1 # reps 1 1 6 2 2 6 2 12 4 1 1 6 2 6 2 2 2 4 2 2

Matrix representation of Dic3×3- 1+2 in GL5(𝔽37)

 0 1 0 0 0 36 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 6 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 26 0 0 0 0 0 26 0 0 0 0 0 0 0 26 0 0 26 0 0 0 0 0 1 0
,
 10 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 10

G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[26,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0,26,0,0],[10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,10] >;

Dic3×3- 1+2 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times 3_-^{1+2}
% in TeX

G:=Group("Dic3xES-(3,1)");
// GroupNames label

G:=SmallGroup(324,95);
// by ID

G=gap.SmallGroup(324,95);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,122,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^9=d^3=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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