p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊4C22, C4.7C24, C8.2C23, Q16⋊4C22, D4.4C23, C23.21D4, Q8.4C23, SD16⋊3C22, M4(2)⋊5C22, C4○D8⋊3C2, C4○(C8⋊C22), C8⋊C22⋊6C2, (C2×C8)⋊3C22, C4.84(C2×D4), (C2×C4).136D4, C4○D4⋊5C22, C4○(C8.C22), C8.C22⋊6C2, (C2×D4)⋊16C22, (C2×M4(2))⋊5C2, (C2×C4).42C23, (C2×Q8)⋊16C22, C2.29(C22×D4), C22.25(C2×D4), (C22×C4).81C22, (C2×C4○D4)⋊12C2, SmallGroup(64,256)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊C22
G = < a,b,c,d | a8=b2=c2=d2=1, bab=a-1, cac=a5, ad=da, cbc=dbd=a4b, cd=dc >
Subgroups: 201 in 131 conjugacy classes, 79 normal (11 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, D8⋊C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, D8⋊C22
Character table of D8⋊C22
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T100
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 16)(8 15)
(2 6)(4 8)(10 14)(12 16)
(9 13)(10 14)(11 15)(12 16)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15), (2,6)(4,8)(10,14)(12,16), (9,13)(10,14)(11,15)(12,16)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15), (2,6)(4,8)(10,14)(12,16), (9,13)(10,14)(11,15)(12,16) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,16),(8,15)], [(2,6),(4,8),(10,14),(12,16)], [(9,13),(10,14),(11,15),(12,16)]])
G:=TransitiveGroup(16,100);
►On 16 points - transitive group 16T118
(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)
(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)
G:=sub<Sym(16)| (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), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)>;
G:=Group( (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), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11) );
G=PermutationGroup([[(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)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11)]])
G:=TransitiveGroup(16,118);
D8⋊C22 is a maximal subgroup of
M4(2).44D4 C42.426D4 M4(2)⋊5D4 M4(2).D4 C42⋊2D4 (C2×C8).2D4 C42.131D4 C22⋊C4.7D4 D8⋊D4 D8.D4 C42.283C23 M4(2).51D4 C42.313C23 M4(2)⋊C23 M4(2).C23 M4(2).10C23 M4(2).37D4 M4(2).38D4 GL2(𝔽3)⋊C22
C4p.C24: C8.C24 D8⋊C23 C4.C25 C24.9C23 SD16⋊D6 D8⋊4D6 D24⋊C22 C12.C24 ...
D8⋊C22 is a maximal quotient of
C24.98D4 C24.100D4 C42.383D4 C4×C8⋊C22 C4×C8.C22 C24.104D4 C24.106D4 C42.211D4 C42.212D4 C42.446D4 C24.110D4 C42.447D4 C42.219D4 C42.220D4 C42.449D4 C24.115D4 C24.116D4 C24.117D4 C24.118D4 C42.221D4 C42.222D4 C42.384D4 C42.223D4 C42.224D4 C42.229D4 C42.232D4 C42.233D4 C42.234D4 C42.235D4 C42.239D4 C42.242D4 C42.244D4 C42.247D4 C42.252D4 M4(2)⋊6Q8 C42.255D4 C42.256D4 C42.257D4 C42.258D4 C42.260D4 C23⋊3D8 C23⋊4SD16 C23⋊3Q16 C24.124D4 C24.127D4 C24.129D4 C24.130D4 C42.269D4 C42.270D4 C42.271D4 C42.272D4 C42.273D4 C42.274D4 C42.277D4 C42.284D4 C42.285D4 C42.286D4 C42.287D4 C42.288D4 C42.289D4 C42.292D4 C42.293D4 C42.294D4 C42.297D4 C42.298D4 C42.299D4 C42.300D4 C42.304D4 D8⋊10D4 Q16⋊10D4 D8⋊5D4 Q16⋊4D4 C42.42C23 C42.44C23 C42.46C23 C42.48C23 C42.50C23 C42.52C23 C42.54C23 C42.56C23 C42.471C23 C42.472C23 C42.475C23 C42.476C23 C42.61C23 C42.62C23 C42.63C23 C42.64C23 C42.492C23 C42.493C23 C42.496C23 C42.511C23 C42.512C23 C42.517C23 C42.518C23 SD16⋊3Q8 D8⋊5Q8 Q16⋊5Q8 C42.531C23 C42.532C23 C42.533C23
SD16⋊D2p: SD16⋊7D4 SD16⋊8D4 SD16⋊2D4 SD16⋊D6 D8⋊4D6 D24⋊C22 Q16⋊D10 SD16⋊D10 ...
M4(2)⋊D2p: M4(2)⋊14D4 M4(2)⋊15D4 M4(2)⋊9D4 C24.9C23 C40.9C23 C56.9C23 ...
C4○D4⋊D2p: C24.103D4 C24.105D4 C42.443D4 C12.C24 C20.C24 C28.C24 ...
Matrix representation of D8⋊C22 ►in GL4(𝔽5) generated by
| 0 | 0 | 3 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,4,0,0,0,0,0,2,3,0,0,0,0,0,1,0],[0,0,0,3,0,1,0,0,0,0,4,0,2,0,0,0],[0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0],[0,0,0,1,0,0,4,0,0,4,0,0,1,0,0,0] >;
D8⋊C22 in GAP, Magma, Sage, TeX
D_8\rtimes C_2^2
% in TeX
G:=Group("D8:C2^2"); // GroupNames label
G:=SmallGroup(64,256);
// by ID
G=gap.SmallGroup(64,256);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,217,650,117,1444,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^5,a*d=d*a,c*b*c=d*b*d=a^4*b,c*d=d*c>;
// generators/relations
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