metabelian, supersoluble, monomial
Aliases: Dic3⋊Dic3, C6.18D12, C6.1Dic6, C62.4C22, C22.6S32, (C3×C6).2Q8, C6.20(C4×S3), C32⋊4(C4⋊C4), (C3×C6).15D4, (C2×C6).11D6, C3⋊1(C4⋊Dic3), (C3×Dic3)⋊1C4, C6.6(C3⋊D4), C6.5(C2×Dic3), C2.5(S3×Dic3), C3⋊1(Dic3⋊C4), (C6×Dic3).3C2, (C2×Dic3).1S3, C2.3(C3⋊D12), C2.1(C32⋊2Q8), (C3×C6).15(C2×C4), (C2×C3⋊Dic3).2C2, SmallGroup(144,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊Dic3
G = < a,b,c,d | a6=c6=1, b2=a3, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 160 in 60 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C4⋊Dic3, C6×Dic3, C2×C3⋊Dic3, Dic3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C32⋊2Q8, Dic3⋊Dic3
Character table of Dic3⋊Dic3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | i | 1 | -i | i | -i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | 1 | 1 | -i | -1 | -1 | i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -i | 1 | i | -i | i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | 1 | 1 | i | -1 | -1 | -i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -1 | -i | -i | i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | -1 | -1 | -i | 1 | 1 | i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | -1 | i | i | -i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -1 | -1 | i | 1 | 1 | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | 1 | 1 | -1 | 0 | √3 | -√3 | 0 | √3 | -√3 | 0 | 0 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | -2 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | 1 | 1 | -1 | 0 | -√3 | √3 | 0 | -√3 | √3 | 0 | 0 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | -1 | 1 | 1 | -√3 | 0 | 0 | √3 | 0 | 0 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | √3 | -√3 | 0 | -√3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | -1 | 1 | 1 | √3 | 0 | 0 | -√3 | 0 | 0 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | -√3 | √3 | 0 | √3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | -2i | 0 | 2i | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | -i | 0 | 0 | -i | 0 | 0 | i | i | complex lifted from C4×S3 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | 2i | 0 | -2i | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | i | 0 | 0 | i | 0 | 0 | -i | -i | complex lifted from C4×S3 |
| ρ25 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | 1 | 1 | -1 | -√-3 | 0 | 0 | √-3 | 0 | 0 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ26 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | 1 | 1 | -1 | √-3 | 0 | 0 | -√-3 | 0 | 0 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ27 | 4 | 4 | -4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊D12 |
| ρ28 | 4 | 4 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ29 | 4 | -4 | -4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2Q8, Schur index 2 |
| ρ30 | 4 | -4 | 4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | -2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Dic3, Schur index 2 |
►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 34 4 31)(2 33 5 36)(3 32 6 35)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)
(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 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (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)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (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)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
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)], [(1,34,4,31),(2,33,5,36),(3,32,6,35),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39)], [(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),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
Dic3⋊Dic3 is a maximal subgroup of
C62.6C23 Dic3⋊5Dic6 C62.9C23 C62.10C23 Dic3×Dic6 C62.13C23 Dic3.Dic6 C62.16C23 C62.17C23 C62.18C23 D6⋊Dic6 C62.23C23 C62.25C23 D6⋊7Dic6 C62.28C23 C62.29C23 C62.32C23 Dic3⋊Dic6 C62.37C23 C62.38C23 C62.39C23 C62.40C23 C12.30D12 C62.42C23 S3×Dic3⋊C4 C62.48C23 Dic3⋊4D12 C62.53C23 D6⋊1Dic6 C62.58C23 S3×C4⋊Dic3 D6.D12 D6.9D12 D6⋊2Dic6 C62.65C23 D6⋊3Dic6 C4×C3⋊D12 C62.75C23 D6⋊D12 C4×C32⋊2Q8 C12⋊3Dic6 C62.97C23 C62.98C23 C62.100C23 C62.57D4 C62⋊3Q8 C62.60D4 C62.111C23 C62.112C23 Dic3×C3⋊D4 C62.115C23 C62.117C23 C62⋊6D4 C62⋊4Q8 Dic9⋊Dic3 Dic3⋊Dic9 C62.D6 C62.3D6 C62.80D6 C62.82D6 C62.85D6
Dic3⋊Dic3 is a maximal quotient of
C12.81D12 C12.Dic6 C6.18D24 C12.82D12 C62.6Q8 Dic9⋊Dic3 Dic3⋊Dic9 C62.D6 C62.80D6 C62.82D6 C62.85D6
Matrix representation of Dic3⋊Dic3 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
Dic3⋊Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("Dic3:Dic3"); // GroupNames label
G:=SmallGroup(144,66);
// by ID
G=gap.SmallGroup(144,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,490,3461]);
// 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,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations
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