p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4⋊6D4, C22.41C24, C23.44C23, C42.43C22, C2.92- 1+4, D4○2(C4⋊C4), C4⋊Q8⋊13C2, (C4×D4)⋊16C2, C4⋊2(C4○D4), C4.37(C2×D4), C4⋊D4⋊12C2, C22⋊Q8⋊12C2, C22.4(C2×D4), C4⋊C4.33C22, (C2×C4).28C23, C2.19(C22×D4), (C2×D4).79C22, C22.D4⋊9C2, C22⋊C4.5C22, (C2×Q8).61C22, (C22×C4).68C22, (C2×C4⋊C4)⋊20C2, (C2×C4○D4)⋊7C2, C2.21(C2×C4○D4), SmallGroup(64,228)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊6D4
G = < a,b,c,d | a4=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 213 in 146 conjugacy classes, 83 normal (17 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C2×C4 [×3], C2×C4 [×8], C2×C4 [×16], D4 [×4], D4 [×10], Q8 [×4], C23 [×4], C42, C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], D4⋊6D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4
Character table of D4⋊6D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2i | 0 | -2i | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2i | 0 | 2i | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2i | 0 | 2i | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2i | 0 | -2i | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 32 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)
(1 18)(2 17)(3 20)(4 19)(5 16)(6 15)(7 14)(8 13)(9 23)(10 22)(11 21)(12 24)(25 31)(26 30)(27 29)(28 32)
(1 27 13 11)(2 28 14 12)(3 25 15 9)(4 26 16 10)(5 22 19 30)(6 23 20 31)(7 24 17 32)(8 21 18 29)
(1 11)(2 12)(3 9)(4 10)(5 32)(6 29)(7 30)(8 31)(13 27)(14 28)(15 25)(16 26)(17 22)(18 23)(19 24)(20 21)
G:=sub<Sym(32)| (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), (1,18)(2,17)(3,20)(4,19)(5,16)(6,15)(7,14)(8,13)(9,23)(10,22)(11,21)(12,24)(25,31)(26,30)(27,29)(28,32), (1,27,13,11)(2,28,14,12)(3,25,15,9)(4,26,16,10)(5,22,19,30)(6,23,20,31)(7,24,17,32)(8,21,18,29), (1,11)(2,12)(3,9)(4,10)(5,32)(6,29)(7,30)(8,31)(13,27)(14,28)(15,25)(16,26)(17,22)(18,23)(19,24)(20,21)>;
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), (1,18)(2,17)(3,20)(4,19)(5,16)(6,15)(7,14)(8,13)(9,23)(10,22)(11,21)(12,24)(25,31)(26,30)(27,29)(28,32), (1,27,13,11)(2,28,14,12)(3,25,15,9)(4,26,16,10)(5,22,19,30)(6,23,20,31)(7,24,17,32)(8,21,18,29), (1,11)(2,12)(3,9)(4,10)(5,32)(6,29)(7,30)(8,31)(13,27)(14,28)(15,25)(16,26)(17,22)(18,23)(19,24)(20,21) );
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)], [(1,18),(2,17),(3,20),(4,19),(5,16),(6,15),(7,14),(8,13),(9,23),(10,22),(11,21),(12,24),(25,31),(26,30),(27,29),(28,32)], [(1,27,13,11),(2,28,14,12),(3,25,15,9),(4,26,16,10),(5,22,19,30),(6,23,20,31),(7,24,17,32),(8,21,18,29)], [(1,11),(2,12),(3,9),(4,10),(5,32),(6,29),(7,30),(8,31),(13,27),(14,28),(15,25),(16,26),(17,22),(18,23),(19,24),(20,21)])
D4⋊6D4 is a maximal subgroup of
D4.SD16 D4.7D8 D4⋊4SD16 SD16⋊8D4 SD16⋊3D4 SD16⋊11D4 D4⋊8SD16 C42.467C23 C42.468C23 C42.41C23 C42.42C23 C42.43C23 C42.50C23 C42.479C23 C42.480C23 C42.481C23 D4×C4○D4 C22.64C25 C22.74C25 C22.78C25 C22.81C25 C22.83C25 C4⋊2+ 1+4 C22.88C25 C22.95C25 C22.102C25 C22.106C25 C23.144C24 C22.110C25 C22.122C25 C22.124C25 C22.125C25 C22.128C25 C22.130C25 C22.131C25 C22.135C25 C22.149C25 C22.151C25
D4p⋊D4: D8⋊10D4 D8⋊4D4 D8⋊13D4 D12⋊24D4 D12⋊22D4 D12⋊12D4 D20⋊24D4 D20⋊22D4 ...
D4⋊D4p: D4⋊3D8 D4⋊4D8 D4⋊5D8 D4⋊6D12 D4⋊6D20 D4⋊6D28 ...
C2p.2- 1+4: D4⋊9SD16 C42.57C23 C42.59C23 C42.494C23 C42.495C23 C22.71C25 C22.100C25 C22.104C25 ...
D4⋊6D4 is a maximal quotient of
C24.549C23 C23.226C24 D4×C4⋊C4 C23.236C24 C23.241C24 C24.215C23 C23.244C24 C23.247C24 C24.243C23 C24.244C23 C23.313C24 C23.315C24 C23.316C24 C24.563C23 C23.322C24 C24.258C23 C23.327C24 C23.329C24 C24.262C23 C24.264C23 C24.567C23 C24.568C23 C24.268C23 C24.269C23 C24.271C23 C23.349C24 C23.353C24 C24.276C23 C24.282C23 C23.362C24 C24.286C23 C23.368C24 C23.369C24 C24.289C23 C23.374C24 C23.375C24 C24.295C23 C23.379C24 C24.299C23 C23.388C24 C24.301C23 C23.391C24 C23.396C24 C23.401C24 C23.406C24 C23.412C24 C23.419C24 C24.311C23 C42⋊18D4 C42.167D4 C42.170D4 C42.35Q8 C24.326C23 C23.571C24 C23.572C24 C23.574C24 C23.580C24 C24.389C23 C24.393C23 C23.589C24 C23.595C24 C24.405C23 C24.407C23 C23.606C24 C23.608C24 C24.412C23 C23.615C24 C23.617C24 C23.618C24 C23.619C24 C23.620C24 C23.621C24 C23.622C24 C24.418C23 C23.624C24 C23.625C24 C23.626C24 C23.627C24
D4⋊D4p: D4⋊5D8 D4⋊6D12 D4⋊6D20 D4⋊6D28 ...
C2p.2- 1+4: D4⋊9SD16 C42.485C23 C42.486C23 D4⋊6Q16 C42.488C23 C42.489C23 C42.490C23 C42.491C23 ...
Matrix representation of D4⋊6D4 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 3 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 1 |
| 0 | 0 | 2 | 2 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 4 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,2,3,0,0,0,3],[4,0,0,0,0,4,0,0,0,0,3,2,0,0,1,2],[0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,1,4,0,0,0,4] >;
D4⋊6D4 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_4
% in TeX
G:=Group("D4:6D4"); // GroupNames label
G:=SmallGroup(64,228);
// by ID
G=gap.SmallGroup(64,228);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,650,86,297]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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