p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.8D4, Q8.8D4, C42.14C22, 2- 1+4⋊1C2, M4(2).1C22, C4≀C2⋊2C2, (C2×C4).5D4, C8⋊C22⋊2C2, C4.26(C2×D4), C4.4D4⋊1C2, (C2×C4).5C23, C2.15C22≀C2, C4.10D4⋊1C2, C4○D4.2C22, (C2×D4).7C22, C22.13(C2×D4), (C2×Q8).5C22, SmallGroup(64,135)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.8D4
G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=a2c3 >
Subgroups: 137 in 73 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, D4.8D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4.8D4
Character table of D4.8D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T162
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)
(1 11)(2 14)(3 13)(4 16)(5 15)(6 10)(7 9)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)
G:=sub<Sym(16)| (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16), (1,11)(2,14)(3,13)(4,16)(5,15)(6,10)(7,9)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)>;
G:=Group( (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16), (1,11)(2,14)(3,13)(4,16)(5,15)(6,10)(7,9)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15) );
G=PermutationGroup([[(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16)], [(1,11),(2,14),(3,13),(4,16),(5,15),(6,10),(7,9),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15)]])
G:=TransitiveGroup(16,162);
►On 16 points - transitive group 16T164
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)
(1 2)(3 4)(5 6)(7 8)(9 14)(10 13)(11 16)(12 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 16)(7 15)(8 14)
G:=sub<Sym(16)| (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16), (1,2)(3,4)(5,6)(7,8)(9,14)(10,13)(11,16)(12,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14)>;
G:=Group( (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16), (1,2)(3,4)(5,6)(7,8)(9,14)(10,13)(11,16)(12,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14) );
G=PermutationGroup([[(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16)], [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,16),(12,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,16),(7,15),(8,14)]])
G:=TransitiveGroup(16,164);
D4.8D4 is a maximal subgroup of
C42.313C23 C42.12C23 D8⋊11D4 D4.3S4 2- 1+4⋊D5
D4p.D4: D8.13D4 D8○SD16 D12.5D4 D4.10D12 D12.14D4 D12.38D4 D20.5D4 D4.10D20 ...
(Cp×D4).D4: C42.3D4 C42.15D4 C42.16D4 C42.17D4 M4(2).C23 2- 1+4⋊4S3 2- 1+4⋊2D5 2- 1+4⋊D7 ...
D4.8D4 is a maximal quotient of
(C2×C4)⋊SD16 C4⋊C4.20D4 Q8⋊6SD16 Q8.Q16 C42.4C23 C42.9C23 C4.10D4⋊2C4 C4.4D4⋊13C4 M4(2)⋊D4 (C2×D4)⋊2Q8 C42⋊3Q8
C42.D2p: C42.129D4 D4.10D12 D12.14D4 D4.10D20 D20.14D4 D4.10D28 D28.14D4 ...
M4(2).D2p: M4(2).7D4 D12.5D4 D12.38D4 D20.5D4 D20.38D4 D28.5D4 D28.38D4 ...
(Cp×D4).D4: C4⋊C4.D4 (C2×C4)⋊D8 C4⋊C4.18D4 C4⋊C4.19D4 D4⋊3D8 Q8⋊3D8 C42.189C23 D4.SD16 ...
Matrix representation of D4.8D4 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 3 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 1 |
| 4 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2],[0,0,0,2,0,0,1,1,2,1,0,0,3,0,0,0],[0,0,1,1,0,0,3,4,1,0,0,0,0,1,0,0],[0,0,4,4,0,0,0,1,4,4,0,0,0,1,0,0] >;
D4.8D4 in GAP, Magma, Sage, TeX
D_4._8D_4
% in TeX
G:=Group("D4.8D4"); // GroupNames label
G:=SmallGroup(64,135);
// by ID
G=gap.SmallGroup(64,135);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,158,963,489,255,117,730]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^4=a^2,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=a^2*c^3>;
// generators/relations
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