p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.5D4, C8.20D4, Q8.5D4, M4(2).5C22, (C2×Q16)⋊8C2, C8○D4.1C2, C4.60(C2×D4), C8.C22.C2, C8.C4⋊7C2, C4.10D4⋊4C2, (C2×C4).10C23, (C2×C8).19C22, C2.25(C4⋊D4), C4○D4.11C22, C22.8(C4○D4), (C2×Q8).16C22, SmallGroup(64,154)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.5D4
G = < a,b,c,d | a4=b2=1, c4=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c3 >
Character table of D4.5D4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 32 points
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 27 13 31)(10 28 14 32)(11 29 15 25)(12 30 16 26)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(25 29)(26 30)(27 31)(28 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 27 5 31)(2 26 6 30)(3 25 7 29)(4 32 8 28)(9 23 13 19)(10 22 14 18)(11 21 15 17)(12 20 16 24)
G:=sub<Sym(32)| (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,27,13,31)(10,28,14,32)(11,29,15,25)(12,30,16,26), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(25,29)(26,30)(27,31)(28,32), (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), (1,27,5,31)(2,26,6,30)(3,25,7,29)(4,32,8,28)(9,23,13,19)(10,22,14,18)(11,21,15,17)(12,20,16,24)>;
G:=Group( (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,27,13,31)(10,28,14,32)(11,29,15,25)(12,30,16,26), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(25,29)(26,30)(27,31)(28,32), (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), (1,27,5,31)(2,26,6,30)(3,25,7,29)(4,32,8,28)(9,23,13,19)(10,22,14,18)(11,21,15,17)(12,20,16,24) );
G=PermutationGroup([[(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,27,13,31),(10,28,14,32),(11,29,15,25),(12,30,16,26)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,27,5,31),(2,26,6,30),(3,25,7,29),(4,32,8,28),(9,23,13,19),(10,22,14,18),(11,21,15,17),(12,20,16,24)]])
D4.5D4 is a maximal subgroup of
M4(2).10C23 C8.S4
D4p.D4: D8.3D4 D8.12D4 D8.13D4 D8○SD16 D8○Q16 D12.7D4 C24.18D4 C24.29D4 ...
M4(2).D2p: D4.4D8 D4.5D8 M4(2).38D4 Q8.10D12 M4(2).16D6 D4.5D20 M4(2).16D10 D4.5D28 ...
D4.5D4 is a maximal quotient of
M4(2).D2p: M4(2).49D4 M4(2).33D4 M4(2).6D4 M4(2).11D4 M4(2).13D4 D12.7D4 C24.18D4 Q8.10D12 ...
(C2×C8).D2p: C8.28D8 Q8⋊1Q16 C8⋊10SD16 C8⋊7Q16 D4.3SD16 Q8.3SD16 C8.D8 C8.SD16 ...
Matrix representation of D4.5D4 ►in GL4(𝔽7) generated by
| 0 | 6 | 5 | 1 |
| 3 | 0 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 1 | 6 | 3 | 1 |
| 1 | 2 | 5 | 3 |
| 4 | 0 | 2 | 2 |
| 6 | 6 | 0 | 2 |
| 6 | 1 | 4 | 6 |
| 5 | 0 | 2 | 6 |
| 6 | 5 | 5 | 6 |
| 6 | 1 | 1 | 2 |
| 2 | 2 | 6 | 2 |
| 4 | 2 | 4 | 5 |
| 6 | 3 | 3 | 5 |
| 6 | 1 | 4 | 1 |
| 2 | 2 | 5 | 3 |
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[1,4,6,6,2,0,6,1,5,2,0,4,3,2,2,6],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[4,6,6,2,2,3,1,2,4,3,4,5,5,5,1,3] >;
D4.5D4 in GAP, Magma, Sage, TeX
D_4._5D_4
% in TeX
G:=Group("D4.5D4"); // GroupNames label
G:=SmallGroup(64,154);
// by ID
G=gap.SmallGroup(64,154);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,963,117,1444,376,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^4=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^3>;
// generators/relations
Export
Subgroup lattice of D4.5D4 in TeX
Character table of D4.5D4 in TeX