metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.6D4, C22⋊2Dic6, C23.17D6, (C2×C6)⋊Q8, C2.6(S3×D4), (C2×C4).5D6, C6.4(C2×Q8), C4⋊Dic3⋊2C2, C6.16(C2×D4), C3⋊1(C22⋊Q8), Dic3⋊C4⋊4C2, (C2×Dic6)⋊2C2, C22⋊C4.1S3, C2.6(C2×Dic6), C6.21(C4○D4), (C2×C12).1C22, (C2×C6).19C23, C2.6(D4⋊2S3), C6.D4.2C2, (C22×C6).8C22, C22.39(C22×S3), (C2×Dic3).5C22, (C22×Dic3).3C2, (C3×C22⋊C4).1C2, SmallGroup(96,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.D4
G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a3c-1 >
Subgroups: 154 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22⋊Q8, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic3.D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C2×Dic6, S3×D4, D4⋊2S3, Dic3.D4
Character table of Dic3.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, 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 38 4 41)(2 37 5 40)(3 42 6 39)(7 21 10 24)(8 20 11 23)(9 19 12 22)(13 36 16 33)(14 35 17 32)(15 34 18 31)(25 47 28 44)(26 46 29 43)(27 45 30 48)
(1 21 13 27)(2 22 14 28)(3 23 15 29)(4 24 16 30)(5 19 17 25)(6 20 18 26)(7 36 48 38)(8 31 43 39)(9 32 44 40)(10 33 45 41)(11 34 46 42)(12 35 47 37)
(7 45)(8 46)(9 47)(10 48)(11 43)(12 44)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)
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), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48), (1,21,13,27)(2,22,14,28)(3,23,15,29)(4,24,16,30)(5,19,17,25)(6,20,18,26)(7,36,48,38)(8,31,43,39)(9,32,44,40)(10,33,45,41)(11,34,46,42)(12,35,47,37), (7,45)(8,46)(9,47)(10,48)(11,43)(12,44)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)>;
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), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48), (1,21,13,27)(2,22,14,28)(3,23,15,29)(4,24,16,30)(5,19,17,25)(6,20,18,26)(7,36,48,38)(8,31,43,39)(9,32,44,40)(10,33,45,41)(11,34,46,42)(12,35,47,37), (7,45)(8,46)(9,47)(10,48)(11,43)(12,44)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27) );
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)], [(1,38,4,41),(2,37,5,40),(3,42,6,39),(7,21,10,24),(8,20,11,23),(9,19,12,22),(13,36,16,33),(14,35,17,32),(15,34,18,31),(25,47,28,44),(26,46,29,43),(27,45,30,48)], [(1,21,13,27),(2,22,14,28),(3,23,15,29),(4,24,16,30),(5,19,17,25),(6,20,18,26),(7,36,48,38),(8,31,43,39),(9,32,44,40),(10,33,45,41),(11,34,46,42),(12,35,47,37)], [(7,45),(8,46),(9,47),(10,48),(11,43),(12,44),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27)]])
Dic3.D4 is a maximal subgroup of
C23⋊3Dic6 C24.38D6 C24.42D6 C42.88D6 C42.90D6 C42⋊12D6 C42.96D6 D4×Dic6 C42.102D6 D4⋊5Dic6 D4⋊6Dic6 C42⋊14D6 C42⋊19D6 C42.118D6 C24.67D6 C24.43D6 C24.44D6 C24.46D6 C6.322+ 1+4 Dic6⋊19D4 C6.702- 1+4 C6.712- 1+4 C6.722- 1+4 C6.732- 1+4 C6.492+ 1+4 (Q8×Dic3)⋊C2 C6.752- 1+4 S3×C22⋊Q8 Dic6⋊21D4 C6.512+ 1+4 C6.1182+ 1+4 C6.522+ 1+4 C6.792- 1+4 C4⋊C4.197D6 C6.802- 1+4 C6.812- 1+4 C6.822- 1+4 C6.1222+ 1+4 C6.632+ 1+4 C6.652+ 1+4 C6.852- 1+4 C6.692+ 1+4 C42.137D6 C42.139D6 C42.140D6 C42.141D6 Dic6⋊10D4 C42.144D6 C42.145D6 C42.159D6 C42.160D6 C42.161D6 C42.162D6 C42.164D6 C42.165D6 C22⋊2Dic18 D6⋊1Dic6 D6⋊2Dic6 D6⋊3Dic6 D6⋊4Dic6 C62⋊3Q8 C62⋊4Q8 C62⋊6Q8 Dic3.S4 D10⋊1Dic6 D10⋊2Dic6 Dic15.D4 D10⋊4Dic6 (C2×C10)⋊8Dic6 Dic15.48D4 C22⋊2Dic30
Dic3.D4 is a maximal quotient of
(C2×C12)⋊Q8 C6.(C4×Q8) C2.(C4×Dic6) Dic3⋊C4⋊C4 (C2×C4)⋊Dic6 C6.(C4⋊Q8) (C2×C4).Dic6 (C22×C4).85D6 Dic3.D8 D4⋊Dic6 D4.Dic6 D4.2Dic6 Q8⋊2Dic6 Q8⋊3Dic6 Q8.3Dic6 Q8.4Dic6 C24.55D6 C24.57D6 C23⋊2Dic6 C24.17D6 C24.18D6 C24.58D6 C22⋊2Dic18 D6⋊1Dic6 D6⋊2Dic6 D6⋊3Dic6 D6⋊4Dic6 C62⋊3Q8 C62⋊4Q8 C62⋊6Q8 D10⋊1Dic6 D10⋊2Dic6 Dic15.D4 D10⋊4Dic6 (C2×C10)⋊8Dic6 Dic15.48D4 C22⋊2Dic30
Matrix representation of Dic3.D4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 5 | 0 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 |
| 0 | 0 | 6 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,0,5,0,0,5,0],[0,1,0,0,12,0,0,0,0,0,3,6,0,0,7,10],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
Dic3.D4 in GAP, Magma, Sage, TeX
{\rm Dic}_3.D_4 % in TeX
G:=Group("Dic3.D4"); // GroupNames label
G:=SmallGroup(96,85);
// by ID
G=gap.SmallGroup(96,85);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,218,188,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^4=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^3*c^-1>;
// generators/relations
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