metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2D6, D4.4D6, D6.8D4, Q8.6D6, SD16⋊2S3, Dic12⋊6C2, C12.6C23, C24.9C22, Dic3.10D4, Dic6.2C22, (S3×Q8)⋊2C2, C8⋊S3⋊2C2, D4⋊2S3.C2, D4.S3⋊4C2, C3⋊Q16⋊1C2, C6.32(C2×D4), C2.20(S3×D4), C3⋊C8.1C22, (C3×SD16)⋊2C2, C3⋊2(C8.C22), C4.6(C22×S3), (C4×S3).3C22, (C3×D4).4C22, (C3×Q8).1C22, SmallGroup(96,122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D6
G = < a,b,c,d | a4=b2=1, c6=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c5 >
Subgroups: 138 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, D4.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S3×D4, D4.D6
Character table of D4.D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 12A | 12B | 24A | 24B | |
| size | 1 | 1 | 4 | 6 | 2 | 2 | 4 | 6 | 12 | 12 | 2 | 8 | 4 | 12 | 4 | 8 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | 1 | -2 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -2 | 0 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | 2 | 0 | -1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ16 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | symplectic faithful, Schur index 2 |
►On 48 points
(1 40 7 46)(2 47 8 41)(3 42 9 48)(4 37 10 43)(5 44 11 38)(6 39 12 45)(13 30 19 36)(14 25 20 31)(15 32 21 26)(16 27 22 33)(17 34 23 28)(18 29 24 35)
(1 32)(2 16)(3 34)(4 18)(5 36)(6 20)(7 26)(8 22)(9 28)(10 24)(11 30)(12 14)(13 38)(15 40)(17 42)(19 44)(21 46)(23 48)(25 39)(27 41)(29 43)(31 45)(33 47)(35 37)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 42 43 48)(38 47 44 41)(39 40 45 46)
G:=sub<Sym(48)| (1,40,7,46)(2,47,8,41)(3,42,9,48)(4,37,10,43)(5,44,11,38)(6,39,12,45)(13,30,19,36)(14,25,20,31)(15,32,21,26)(16,27,22,33)(17,34,23,28)(18,29,24,35), (1,32)(2,16)(3,34)(4,18)(5,36)(6,20)(7,26)(8,22)(9,28)(10,24)(11,30)(12,14)(13,38)(15,40)(17,42)(19,44)(21,46)(23,48)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,42,43,48)(38,47,44,41)(39,40,45,46)>;
G:=Group( (1,40,7,46)(2,47,8,41)(3,42,9,48)(4,37,10,43)(5,44,11,38)(6,39,12,45)(13,30,19,36)(14,25,20,31)(15,32,21,26)(16,27,22,33)(17,34,23,28)(18,29,24,35), (1,32)(2,16)(3,34)(4,18)(5,36)(6,20)(7,26)(8,22)(9,28)(10,24)(11,30)(12,14)(13,38)(15,40)(17,42)(19,44)(21,46)(23,48)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,42,43,48)(38,47,44,41)(39,40,45,46) );
G=PermutationGroup([[(1,40,7,46),(2,47,8,41),(3,42,9,48),(4,37,10,43),(5,44,11,38),(6,39,12,45),(13,30,19,36),(14,25,20,31),(15,32,21,26),(16,27,22,33),(17,34,23,28),(18,29,24,35)], [(1,32),(2,16),(3,34),(4,18),(5,36),(6,20),(7,26),(8,22),(9,28),(10,24),(11,30),(12,14),(13,38),(15,40),(17,42),(19,44),(21,46),(23,48),(25,39),(27,41),(29,43),(31,45),(33,47),(35,37)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,42,43,48),(38,47,44,41),(39,40,45,46)]])
D4.D6 is a maximal subgroup of
SD16⋊13D6 SD16⋊D6 D8.10D6 D8⋊4D6 D8⋊6D6 S3×C8.C22 SD16.D6 SD16⋊D9 C24.3D6 D12.4D6 Dic6.19D6 Dic6.D6 D12.24D6 D12.15D6 C24.32D6 D6.S4 C40.2D6 D30.3D4 C60.10C23 D30.9D4 D20.28D6 D20.17D6 SD16⋊D15
D4.D6 is a maximal quotient of
D4.S3⋊C4 Dic3.D8 C12⋊Q8⋊C2 Dic6.D4 D4⋊(C4×S3) D6.D8 C3⋊C8⋊1D4 D4.D12 C3⋊Q16⋊C4 Dic3⋊Q16 Q8.3Dic6 (C2×Q8).36D6 (S3×Q8)⋊C4 D6⋊Q16 D6⋊C8.C2 C3⋊C8.D4 Dic12⋊9C4 Dic6⋊Q8 C24⋊3Q8 Dic6.Q8 C8⋊(C4×S3) D6.2SD16 C8.2D12 C6.(C4○D8) Dic3⋊3SD16 SD16⋊Dic3 (C3×Q8).D4 C24.31D4 D6⋊8SD16 Dic6.16D4 C24⋊8D4 SD16⋊D9 C24.3D6 D12.4D6 Dic6.19D6 Dic6.D6 D12.24D6 D12.15D6 C24.32D6 C40.2D6 D30.3D4 C60.10C23 D30.9D4 D20.28D6 D20.17D6 SD16⋊D15
Matrix representation of D4.D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 3 |
| 4 | 0 | 2 | 2 |
| 2 | 2 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 1 | 0 | 1 | 3 |
| 4 | 4 | 2 | 1 |
| 1 | 2 | 4 | 4 |
| 3 | 1 | 0 | 1 |
| 4 | 3 | 4 | 0 |
| 3 | 2 | 3 | 1 |
| 1 | 2 | 2 | 3 |
| 0 | 4 | 3 | 2 |
| 2 | 0 | 0 | 0 |
| 3 | 0 | 4 | 4 |
| 1 | 1 | 0 | 2 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [0,4,2,3,0,0,2,0,0,2,0,0,3,2,4,0],[1,4,1,3,0,4,2,1,1,2,4,0,3,1,4,1],[4,3,1,0,3,2,2,4,4,3,2,3,0,1,3,2],[2,3,1,0,0,0,1,0,0,4,0,0,0,4,2,3] >;
D4.D6 in GAP, Magma, Sage, TeX
D_4.D_6
% in TeX
G:=Group("D4.D6"); // GroupNames label
G:=SmallGroup(96,122);
// by ID
G=gap.SmallGroup(96,122);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,116,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^6=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^5>;
// generators/relations
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