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G = D4⋊C8  order 64 = 26

The semidirect product of D4 and C8 acting via C8/C4=C2

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

Aliases: D4⋊C8, C4.16D8, C4.13SD16, C4.1M4(2), C42.62C22, C4⋊C8⋊1C2, (C4×C8)⋊1C2, C2.1C4≀C2, C4⋊C4.3C4, C4.1(C2×C8), (C2×D4).4C4, (C4×D4).1C2, (C2×C4).93D4, C2.5(C22⋊C8), C2.1(D4⋊C4), C22.21(C22⋊C4), (C2×C4).37(C2×C4), 2-Sylow(CSO-(4,5)), SmallGroup(64,6)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — D4⋊C8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊C8
Lower central
C1 — C2 — C4 — D4⋊C8
Upper central
C1 — C2×C4 — C42 — D4⋊C8
Jennings
C1 — C22 — C22 — C42 — D4⋊C8

Generators and relations for D4⋊C8
 G = < a,b,c | a4=b2=c8=1, bab=cac-1=a-1, cbc-1=ab >

C1
C2
C2
C2
4C2
4C2
C4
C4
C22
C4
C4
2C4
2C22
2C22
4C22
4C22
4C4
C2×C4
D4
D4
C2×C4
C2×C4
2C8
2C8
2D4
2C23
2C2×C4
4C2×C4
4C8
4C2×C4
C42
C2×D4
C4⋊C4
2C2×C8
2C22×C4
2C22⋊C4
2C2×C8
C4×C8
C4×D4
C4⋊C8
D4⋊C8

Character table of D4⋊C8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111441111222244222222224444
ρ11111111111111111111111111111    trivial
ρ21111-1-111111111-1-111111111-1-1-1-1    linear of order 2
ρ31111-1-111111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ41111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-111-1-1-1i-i-i-i-iiiiii-i-i    linear of order 4
ρ61111-1-1-1-1-1-1-111-111i-i-i-i-iiii-i-iii    linear of order 4
ρ7111111-1-1-1-1-111-1-1-1-iiiii-i-i-i-i-iii    linear of order 4
ρ81111-1-1-1-1-1-1-111-111-iiiii-i-i-iii-i-i    linear of order 4
ρ91-11-1-11i-ii-i-i1-1i-iiζ8ζ83ζ87ζ83ζ87ζ8ζ85ζ85ζ8ζ85ζ87ζ83    linear of order 8
ρ101-11-11-1i-ii-i-i1-1ii-iζ8ζ83ζ87ζ83ζ87ζ8ζ85ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ111-11-1-11-ii-iii1-1-ii-iζ83ζ8ζ85ζ8ζ85ζ83ζ87ζ87ζ83ζ87ζ85ζ8    linear of order 8
ρ121-11-11-1-ii-iii1-1-i-iiζ83ζ8ζ85ζ8ζ85ζ83ζ87ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ131-11-1-11-ii-iii1-1-ii-iζ87ζ85ζ8ζ85ζ8ζ87ζ83ζ83ζ87ζ83ζ8ζ85    linear of order 8
ρ141-11-11-1-ii-iii1-1-i-iiζ87ζ85ζ8ζ85ζ8ζ87ζ83ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ151-11-1-11i-ii-i-i1-1i-iiζ85ζ87ζ83ζ87ζ83ζ85ζ8ζ8ζ85ζ8ζ83ζ87    linear of order 8
ρ161-11-11-1i-ii-i-i1-1ii-iζ85ζ87ζ83ζ87ζ83ζ85ζ8ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ172222002222-2-2-2-200000000000000    orthogonal lifted from D4
ρ18222200-2-2-2-22-2-2200000000000000    orthogonal lifted from D4
ρ192-2-2200-222-2000000-√2√2-√2-√2√2√2-√2√20000    orthogonal lifted from D8
ρ202-2-2200-222-2000000√2-√2√2√2-√2-√2√2-√20000    orthogonal lifted from D8
ρ212-22-200-2i2i-2i2i-2i-222i00000000000000    complex lifted from M4(2)
ρ2222-2-2002i2i-2i-2i000000-1+i-1-i-1-i1+i1+i1-i1-i-1+i0000    complex lifted from C4≀C2
ρ2322-2-200-2i-2i2i2i0000001+i1-i1-i-1+i-1+i-1-i-1-i1+i0000    complex lifted from C4≀C2
ρ2422-2-2002i2i-2i-2i0000001-i1+i1+i-1-i-1-i-1+i-1+i1-i0000    complex lifted from C4≀C2
ρ252-22-2002i-2i2i-2i2i-22-2i00000000000000    complex lifted from M4(2)
ρ262-2-22002-2-22000000-√-2-√-2√-2√-2-√-2√-2-√-2√-20000    complex lifted from SD16
ρ2722-2-200-2i-2i2i2i000000-1-i-1+i-1+i1-i1-i1+i1+i-1-i0000    complex lifted from C4≀C2
ρ282-2-22002-2-22000000√-2√-2-√-2-√-2√-2-√-2√-2-√-20000    complex lifted from SD16

Smallest permutation representation of D4⋊C8
►On 32 points
Generators in S32
(1 21 31 14)(2 15 32 22)(3 23 25 16)(4 9 26 24)(5 17 27 10)(6 11 28 18)(7 19 29 12)(8 13 30 20)

(1 10)(2 28)(3 12)(4 30)(5 14)(6 32)(7 16)(8 26)(9 13)(11 15)(17 31)(18 22)(19 25)(20 24)(21 27)(23 29)

(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)
 
G:=sub<Sym(32)| (1,21,31,14)(2,15,32,22)(3,23,25,16)(4,9,26,24)(5,17,27,10)(6,11,28,18)(7,19,29,12)(8,13,30,20), (1,10)(2,28)(3,12)(4,30)(5,14)(6,32)(7,16)(8,26)(9,13)(11,15)(17,31)(18,22)(19,25)(20,24)(21,27)(23,29), (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)>;
 
G:=Group( (1,21,31,14)(2,15,32,22)(3,23,25,16)(4,9,26,24)(5,17,27,10)(6,11,28,18)(7,19,29,12)(8,13,30,20), (1,10)(2,28)(3,12)(4,30)(5,14)(6,32)(7,16)(8,26)(9,13)(11,15)(17,31)(18,22)(19,25)(20,24)(21,27)(23,29), (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) );
 
G=PermutationGroup([[(1,21,31,14),(2,15,32,22),(3,23,25,16),(4,9,26,24),(5,17,27,10),(6,11,28,18),(7,19,29,12),(8,13,30,20)], [(1,10),(2,28),(3,12),(4,30),(5,14),(6,32),(7,16),(8,26),(9,13),(11,15),(17,31),(18,22),(19,25),(20,24),(21,27),(23,29)], [(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)]])
 

D4⋊C8 is a maximal subgroup of
 C8×D8  C8×SD16  SD16⋊C8  C8⋊9D8  C8⋊12SD16  C8⋊15SD16  D4.M4(2)  D4⋊2M4(2)  Q8⋊2M4(2)  C8⋊6D8  C8⋊9SD16  C8⋊M4(2)  C8⋊3M4(2)  D4⋊D8  Q8⋊D8  D4⋊SD16  C42.185C23  D4⋊3D8  Q8⋊3D8  C42.189C23  D4⋊2SD16  Q8⋊2SD16  D4⋊Q16  C42.195C23  D4.SD16  D4.3Q16  C42.199C23  D4.D8  Q8.D8  Q8⋊3SD16  D4.5SD16  D4⋊3Q16  C42.207C23  D4.7D8  D4⋊4Q16  C42.211C23  Q8⋊4SD16  C42.213C23  D4⋊4SD16  Dic5.23D8
 D4p⋊C8: D8⋊5C8  C4.17D24  D12⋊2C8  D20⋊3C8  D20⋊4C8  D20⋊C8  C4.17D56  D28⋊2C8 ...
 C42.D2p: C42.455D4  C42.397D4  C42.398D4  C42.45D4  C42.373D4  C42.47D4  C42.400D4  D4⋊M4(2) ...
D4⋊C8 is a maximal quotient of
 (C2×C4).98D8  C4⋊C4⋊C8  C42.46Q8  C4.16D16  Q16⋊1C8  D8⋊C8  Q16⋊C8  D20⋊C8  Dic5.23D8
 C4p.D8: D4⋊C16  C8.31D8  C8≀C2  C8.32D8  C4.17D24  D12⋊2C8  C12.57D8  D20⋊3C8 ...

Matrix representation of D4⋊C8 ►in GL3(𝔽17) generated by

100
001
0160
,
100
001
010
,
800
0314
01414
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[1,0,0,0,0,1,0,1,0],[8,0,0,0,3,14,0,14,14] >;
 

D4⋊C8 in GAP, Magma, Sage, TeX 

D_4\rtimes C_8
 
% in TeX
 
G:=Group("D4:C8");
 
// GroupNames label
 
G:=SmallGroup(64,6);
 
// by ID
 
G=gap.SmallGroup(64,6);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,188,86,117]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^2=c^8=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a*b>;
 
// generators/relations
 

Export

Subgroup lattice of D4⋊C8 in TeX
Character table of D4⋊C8 in TeX

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