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G = D8⋊C22order 64 = 26

4th semidirect product of D8 and C22 acting via C22/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D84C22, C4.7C24, C8.2C23, Q164C22, D4.4C23, C23.21D4, Q8.4C23, SD163C22, M4(2)⋊5C22, C4○D83C2, C4(C8⋊C22), C8⋊C226C2, (C2×C8)⋊3C22, C4.84(C2×D4), (C2×C4).136D4, C4○D45C22, C4(C8.C22), C8.C226C2, (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

C1C4 — D8⋊C22
C1C2C4C2×C4C22×C4C2×C4○D4 — D8⋊C22
C1C2C4 — D8⋊C22
C1C4C22×C4 — D8⋊C22
C1C2C2C4 — D8⋊C22

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 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], D8⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, D8⋊C22

Character table of D8⋊C22

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1122244441122244444444
ρ11111111111111111111111    trivial
ρ2111-1-11-1-1-1-1-1-111111-11-11-1    linear of order 2
ρ311-1-11-11-1-1-1-11-111-111-111-1    linear of order 2
ρ411-11-1-1-11111-1-111-11-1-1-111    linear of order 2
ρ5111-1-1-1-11-1-1-1-111-1111-11-11    linear of order 2
ρ611111-11-1111111-111-1-1-1-1-1    linear of order 2
ρ711-11-11-1-1111-1-11-1-11111-1-1    linear of order 2
ρ811-1-11111-1-1-11-11-1-11-11-1-11    linear of order 2
ρ911-1-111-111-1-11-11-11-1-1-111-1    linear of order 2
ρ1011-11-111-1-111-1-11-11-11-1-111    linear of order 2
ρ1111111-1-1-1-111111-1-1-1-11111    linear of order 2
ρ12111-1-1-1111-1-1-111-1-1-111-11-1    linear of order 2
ρ1311-11-1-111-111-1-1111-1-111-1-1    linear of order 2
ρ1411-1-11-1-1-11-1-11-1111-111-1-11    linear of order 2
ρ15111-1-111-11-1-1-1111-1-1-1-11-11    linear of order 2
ρ16111111-11-1111111-1-11-1-1-1-1    linear of order 2
ρ1722-2-22000022-22-200000000    orthogonal lifted from D4
ρ18222220000-2-2-2-2-200000000    orthogonal lifted from D4
ρ1922-22-20000-2-222-200000000    orthogonal lifted from D4
ρ20222-2-20000222-2-200000000    orthogonal lifted from D4
ρ214-40000000-4i4i00000000000    complex faithful
ρ224-400000004i-4i00000000000    complex faithful

Permutation representations of D8⋊C22
On 16 points - transitive group 16T100
Generators in S16
(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
Generators in S16
(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 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)

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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)>;

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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15) );

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)], [(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)])

G:=TransitiveGroup(16,118);

D8⋊C22 is a maximal subgroup of
M4(2).44D4  C42.426D4  M4(2)⋊5D4  M4(2).D4  C422D4  (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  D84D6  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  C233D8  C234SD16  C233Q16  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  D810D4  Q1610D4  D85D4  Q164D4  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  SD163Q8  D85Q8  Q165Q8  C42.531C23  C42.532C23  C42.533C23
 SD16⋊D2p: SD167D4  SD168D4  SD162D4  SD16⋊D6  D84D6  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

0030
4000
0001
0200
,
0002
0100
0040
3000
,
0001
0010
0100
1000
,
0001
0040
0400
1000
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

Export

Character table of D8⋊C22 in TeX

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