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## G = Dic32order 144 = 24·32

### Direct product of Dic3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — Dic32
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32
 Lower central C32 — Dic32
 Upper central C1 — C22

Generators and relations for Dic32
G = < a,b,c,d | a6=c6=1, b2=a3, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 160 in 68 conjugacy classes, 36 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, C62, C4×Dic3, C6×Dic3, C2×C3⋊Dic3, Dic32
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, S32, C4×Dic3, S3×Dic3, C6.D6, Dic32

Smallest permutation representation of Dic32
On 48 points
Generators in S48
(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)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 28 4 25)(2 27 5 30)(3 26 6 29)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 7 5 11 3 9)(2 8 6 12 4 10)(13 23 15 19 17 21)(14 24 16 20 18 22)(25 35 27 31 29 33)(26 36 28 32 30 34)(37 45 41 43 39 47)(38 46 42 44 40 48)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)

G:=sub<Sym(48)| (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,28,4,25)(2,27,5,30)(3,26,6,29)(7,32,10,35)(8,31,11,34)(9,36,12,33)(13,38,16,41)(14,37,17,40)(15,42,18,39)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,23,15,19,17,21)(14,24,16,20,18,22)(25,35,27,31,29,33)(26,36,28,32,30,34)(37,45,41,43,39,47)(38,46,42,44,40,48), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)>;

G:=Group( (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,28,4,25)(2,27,5,30)(3,26,6,29)(7,32,10,35)(8,31,11,34)(9,36,12,33)(13,38,16,41)(14,37,17,40)(15,42,18,39)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,23,15,19,17,21)(14,24,16,20,18,22)(25,35,27,31,29,33)(26,36,28,32,30,34)(37,45,41,43,39,47)(38,46,42,44,40,48), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42) );

G=PermutationGroup([[(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,28,4,25),(2,27,5,30),(3,26,6,29),(7,32,10,35),(8,31,11,34),(9,36,12,33),(13,38,16,41),(14,37,17,40),(15,42,18,39),(19,44,22,47),(20,43,23,46),(21,48,24,45)], [(1,7,5,11,3,9),(2,8,6,12,4,10),(13,23,15,19,17,21),(14,24,16,20,18,22),(25,35,27,31,29,33),(26,36,28,32,30,34),(37,45,41,43,39,47),(38,46,42,44,40,48)], [(1,23,11,17),(2,24,12,18),(3,19,7,13),(4,20,8,14),(5,21,9,15),(6,22,10,16),(25,43,31,37),(26,44,32,38),(27,45,33,39),(28,46,34,40),(29,47,35,41),(30,48,36,42)]])

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A ··· 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 12A ··· 12H order 1 2 2 2 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 size 1 1 1 1 2 2 4 3 ··· 3 9 9 9 9 2 ··· 2 4 4 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + - + + - + image C1 C2 C2 C4 C4 S3 Dic3 D6 C4×S3 S32 S3×Dic3 C6.D6 kernel Dic32 C6×Dic3 C2×C3⋊Dic3 C3×Dic3 C3⋊Dic3 C2×Dic3 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 2 1 8 4 2 4 2 8 1 2 1

Matrix representation of Dic32 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

Dic32 in GAP, Magma, Sage, TeX

{\rm Dic}_3^2
% in TeX

G:=Group("Dic3^2");
// GroupNames label

G:=SmallGroup(144,63);
// by ID

G=gap.SmallGroup(144,63);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,31,490,3461]);
// Polycyclic

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

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