direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic32, C32⋊2C42, C62.1C22, C22.3S32, (C2×C6).8D6, C6.18(C4×S3), C3⋊Dic3⋊1C4, C3⋊1(C4×Dic3), (C3×Dic3)⋊2C4, C6.3(C2×Dic3), C2.2(S3×Dic3), (C6×Dic3).5C2, (C2×Dic3).4S3, C2.2(C6.D6), (C3×C6).12(C2×C4), (C2×C3⋊Dic3).1C2, SmallGroup(144,63)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — Dic32 |
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
►On 48 points
(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)]])
Dic32 is a maximal subgroup of
Dic3≀C2 C62.6C23 Dic3⋊5Dic6 C62.8C23 C62.9C23 C62.10C23 C62.11C23 C62.13C23 Dic3⋊6Dic6 Dic3.Dic6 C62.16C23 C62.17C23 C62.18C23 C62.19C23 C62.28C23 C62.31C23 C4×S3×Dic3 C62.47C23 C62.48C23 C62.49C23 C62.51C23 C4×C6.D6 C62.74C23 C62.77C23 C62.83C23 C62.94C23 C62.95C23 C62.97C23 C62.98C23 C62.99C23 C62.101C23 C62.115C23 C62.121C23 He3⋊C42 C33⋊6C42
Dic32 is a maximal quotient of
C6.(S3×C8) C3⋊C8⋊Dic3 C2.Dic32 C62.6Q8 He3⋊C42 C33⋊6C42
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