p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.9D4, Q8.9D4, C23.6D4, C42⋊3C22, M4(2)⋊2C22, 2+ 1+4.1C2, C4≀C2⋊3C2, C4.27(C2×D4), C4.4D4⋊2C2, C4.D4⋊2C2, C8.C22⋊1C2, (C2×C4).6C23, (C2×Q8)⋊2C22, C2.16C22≀C2, C4○D4.3C22, (C2×D4).8C22, C22.14(C2×D4), SmallGroup(64,136)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.9D4
G = < a,b,c,d | a4=b2=c4=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 153 in 76 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, D4.9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4.9D4
Character table of D4.9D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 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 | 0 | 2 | -2 | 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 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T138
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 14)(6 13)(7 16)(8 15)
(5 7)(6 8)(9 12 11 10)(13 14 15 16)
(1 5 3 7)(2 8 4 6)(9 16 11 14)(10 15 12 13)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12)(2,11)(3,10)(4,9)(5,14)(6,13)(7,16)(8,15), (5,7)(6,8)(9,12,11,10)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,16,11,14)(10,15,12,13)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12)(2,11)(3,10)(4,9)(5,14)(6,13)(7,16)(8,15), (5,7)(6,8)(9,12,11,10)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,16,11,14)(10,15,12,13) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12),(2,11),(3,10),(4,9),(5,14),(6,13),(7,16),(8,15)], [(5,7),(6,8),(9,12,11,10),(13,14,15,16)], [(1,5,3,7),(2,8,4,6),(9,16,11,14),(10,15,12,13)]])
G:=TransitiveGroup(16,138);
►On 16 points - transitive group 16T145
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(5 6)(7 8)(9 11)(13 14)(15 16)
(1 6 12 15)(2 7 9 16)(3 8 10 13)(4 5 11 14)
(1 16 3 14)(2 15 4 13)(5 12 7 10)(6 11 8 9)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,6)(7,8)(9,11)(13,14)(15,16), (1,6,12,15)(2,7,9,16)(3,8,10,13)(4,5,11,14), (1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,6)(7,8)(9,11)(13,14)(15,16), (1,6,12,15)(2,7,9,16)(3,8,10,13)(4,5,11,14), (1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(5,6),(7,8),(9,11),(13,14),(15,16)], [(1,6,12,15),(2,7,9,16),(3,8,10,13),(4,5,11,14)], [(1,16,3,14),(2,15,4,13),(5,12,7,10),(6,11,8,9)]])
G:=TransitiveGroup(16,145);
►On 16 points - transitive group 16T175
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 13)(10 16)(11 15)(12 14)
(1 12 3 10)(2 9 4 11)(5 14)(6 15)(7 16)(8 13)
(1 11 3 9)(2 10 4 12)(5 13 7 15)(6 16 8 14)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,13)(10,16)(11,15)(12,14), (1,12,3,10)(2,9,4,11)(5,14)(6,15)(7,16)(8,13), (1,11,3,9)(2,10,4,12)(5,13,7,15)(6,16,8,14)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,13)(10,16)(11,15)(12,14), (1,12,3,10)(2,9,4,11)(5,14)(6,15)(7,16)(8,13), (1,11,3,9)(2,10,4,12)(5,13,7,15)(6,16,8,14) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,13),(10,16),(11,15),(12,14)], [(1,12,3,10),(2,9,4,11),(5,14),(6,15),(7,16),(8,13)], [(1,11,3,9),(2,10,4,12),(5,13,7,15),(6,16,8,14)]])
G:=TransitiveGroup(16,175);
D4.9D4 is a maximal subgroup of
C42.313C23 M4(2)⋊C23 D8⋊11D4 Q8.4S4 Q8.S4
C42⋊D2p: C42⋊4D4 C42⋊5D4 C42⋊5D6 C42⋊7D6 M4(2)⋊D10 C42⋊5D10 M4(2)⋊D14 C42⋊5D14 ...
D4p.D4: D8.13D4 D8○SD16 M4(2)⋊D6 D12.39D4 D20.1D4 D20.39D4 D28.1D4 D28.39D4 ...
(Cp×Q8).D4: C42.2D4 C42.14D4 C42.13C23 2+ 1+4.4S3 2+ 1+4.D5 2+ 1+4.D7 ...
D4.9D4 is a maximal quotient of
C23⋊2SD16 C23⋊Q16 C24.12D4 C23.5D8 C24.14D4 C24.15D4 D4.D8 C42.201C23 Q8.D8 Q8⋊3SD16 D4.5SD16 D4⋊3Q16 Q8⋊3Q16 C42.207C23 C42.5C23 C42.7C23 C42.8C23 C42.10C23 C24.21D4 C42⋊9(C2×C4) C24.23D4 (C2×Q8)⋊2Q8 C42⋊3Q8
C42⋊D2p: C42⋊10D4 C42⋊12D4 C42⋊5D6 C42⋊7D6 M4(2)⋊D10 C42⋊5D10 M4(2)⋊D14 C42⋊5D14 ...
M4(2)⋊D2p: M4(2)⋊4D4 M4(2)⋊6D4 M4(2)⋊D6 D12.39D4 D20.1D4 D20.39D4 D28.1D4 D28.39D4 ...
C4○D4.D2p: 2+ 1+4⋊3C4 C8.C22⋊C4 2+ 1+4.4S3 2+ 1+4.D5 2+ 1+4.D7 ...
Matrix representation of D4.9D4 ►in GL4(𝔽5) generated by
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 0 | 0 |
| 4 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[0,2,0,0,3,0,0,0,0,0,0,2,0,0,3,0],[3,0,0,0,0,1,0,0,0,0,4,0,0,0,0,3],[0,0,0,4,0,0,3,0,0,3,0,0,1,0,0,0] >;
D4.9D4 in GAP, Magma, Sage, TeX
D_4._9D_4
% in TeX
G:=Group("D4.9D4"); // GroupNames label
G:=SmallGroup(64,136);
// by ID
G=gap.SmallGroup(64,136);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,963,489,255,117,730]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
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