metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.Dic3, Q8.2Dic3, C12.42C23, D4○(C3⋊C8), Q8○(C3⋊C8), (C3×D4).C4, (C3×Q8).C4, C3⋊3(C8○D4), C4○D4.5S3, (C2×C4).58D6, C12.15(C2×C4), C3⋊C8.13C22, C4.Dic3⋊8C2, C4.5(C2×Dic3), C4.42(C22×S3), C6.27(C22×C4), (C2×C12).41C22, C2.8(C22×Dic3), C22.1(C2×Dic3), (C2×C3⋊C8)⋊7C2, C4○D4○(C3⋊C8), (C2×C6).7(C2×C4), (C3×C4○D4).2C2, SmallGroup(96,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.Dic3
G = < a,b,c,d | a4=b2=1, c6=a2, d2=a2c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 90 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C3⋊C8, C3⋊C8, C2×C12, C3×D4, C3×Q8, C8○D4, C2×C3⋊C8, C4.Dic3, C3×C4○D4, D4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C2×Dic3, C22×S3, C8○D4, C22×Dic3, D4.Dic3
Character table of D4.Dic3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 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 | 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 | -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 | 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 | -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 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -i | i | -i | i | -i | -i | i | -i | i | i | -1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -i | i | -i | i | i | -i | -i | i | -i | i | -1 | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | i | -i | i | -i | i | -i | -i | i | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -i | -i | -i | i | i | i | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | i | -i | i | -i | -i | i | i | -i | i | -i | -1 | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | i | -i | i | -i | i | i | -i | i | -i | -i | -1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | i | i | i | -i | -i | -i | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | -i | i | -i | i | -i | i | i | -i | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 2 | -2 | -1 | 2 | 2 | -2 | 2 | -2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -2 | -2 | 2 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | -1 | 2 | 2 | 2 | -2 | -2 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | -2 | -2 | 2 | -2 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ22 | 2 | 2 | -2 | -2 | -2 | -1 | -2 | -2 | 2 | 2 | 2 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ23 | 2 | 2 | -2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 2 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ24 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | -2 | -2 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2ζ83 | 2ζ8 | 2ζ87 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ26 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2ζ85 | 2ζ87 | 2ζ8 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ27 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2ζ8 | 2ζ83 | 2ζ85 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ28 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2ζ87 | 2ζ85 | 2ζ83 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex faithful |
►On 48 points
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 10 25 7 34 4 31)(2 33 11 30 8 27 5 36)(3 26 12 35 9 32 6 29)(13 43 22 40 19 37 16 46)(14 48 23 45 20 42 17 39)(15 41 24 38 21 47 18 44)
G:=sub<Sym(48)| (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,43,22,40,19,37,16,46)(14,48,23,45,20,42,17,39)(15,41,24,38,21,47,18,44)>;
G:=Group( (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,43,22,40,19,37,16,46)(14,48,23,45,20,42,17,39)(15,41,24,38,21,47,18,44) );
G=PermutationGroup([[(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,28,10,25,7,34,4,31),(2,33,11,30,8,27,5,36),(3,26,12,35,9,32,6,29),(13,43,22,40,19,37,16,46),(14,48,23,45,20,42,17,39),(15,41,24,38,21,47,18,44)]])
D4.Dic3 is a maximal subgroup of
M4(2).22D6 C42.196D6 D8⋊5Dic3 D8⋊4Dic3 M4(2).D6 M4(2).13D6 M4(2).15D6 M4(2).16D6 S3×C8○D4 M4(2)⋊28D6 C12.76C24 D12.32C23 D12.33C23 D12.34C23 D12.35C23 D4.Dic9 SL2(𝔽3).Dic3 D12.2Dic3 D12.Dic3 D4.(C3⋊Dic3) D20.3Dic3 D20.2Dic3 D4.Dic15 Dic10.Dic3 D20.Dic3
D4.Dic3 is a maximal quotient of
C12.5C42 C42.43D6 C42.187D6 D4×C3⋊C8 C42.47D6 C12⋊3M4(2) Q8×C3⋊C8 C42.210D6 (C6×D4).11C4 D4.Dic9 D12.2Dic3 D12.Dic3 D4.(C3⋊Dic3) D20.3Dic3 D20.2Dic3 D4.Dic15 Dic10.Dic3 D20.Dic3
Matrix representation of D4.Dic3 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 0 | 4 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 1 | 0 | 4 |
| 2 | 0 | 3 | 0 |
| 0 | 2 | 0 | 2 |
| 3 | 0 | 4 | 0 |
| 0 | 1 | 0 | 1 |
| 2 | 0 | 2 | 0 |
| 0 | 1 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,1,0,4,0,0,4,0,0,1,0,0,1,0,4,0],[0,1,0,4,0,0,1,0,0,1,0,0,4,0,1,0],[0,0,2,0,0,1,0,2,3,0,3,0,0,4,0,2],[3,0,2,0,0,1,0,1,4,0,2,0,0,1,0,4] >;
D4.Dic3 in GAP, Magma, Sage, TeX
D_4.{\rm Dic}_3 % in TeX
G:=Group("D4.Dic3"); // GroupNames label
G:=SmallGroup(96,155);
// by ID
G=gap.SmallGroup(96,155);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,188,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^6=a^2,d^2=a^2*c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations
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