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G = D4⋊D4  order 64 = 26

2nd semidirect product of D4 and D4 acting via D4/C22=C2

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

Aliases: D4⋊3D4, Q8⋊3D4, C23.14D4, (C2×D8)⋊2C2, C4⋊D4⋊2C2, C22⋊C8⋊5C2, C4.21(C2×D4), D4⋊C4⋊8C2, Q8⋊C4⋊4C2, (C2×SD16)⋊8C2, C2.6(C4○D8), (C2×C4).104D4, C4⋊C4.2C22, (C2×C8).1C22, C2.10C22≀C2, C2.7(C8⋊C22), (C2×C4).83C23, (C2×D4).6C22, C22.79(C2×D4), (C2×Q8).49C22, (C22×C4).44C22, (C2×C4○D4)⋊1C2, SmallGroup(64,130)

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

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

Generators and relations for D4⋊D4
 G = < a,b,c,d | a4=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 161 in 81 conjugacy classes, 29 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, D4⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8⋊C22, D4⋊D4

Character table of D4⋊D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D
 size 1111444822224484444
ρ11111111111111111111    trivial
ρ21111-1-1-11-111-111-1-1-111    linear of order 2
ρ31111111-1111111-1-1-1-1-1    linear of order 2
ρ41111-1-1-1-1-111-111111-1-1    linear of order 2
ρ511111-11-1-111-1-1-11-1-111    linear of order 2
ρ61111-11-1-11111-1-1-11111    linear of order 2
ρ711111-111-111-1-1-1-111-1-1    linear of order 2
ρ81111-11-111111-1-11-1-1-1-1    linear of order 2
ρ92-22-220-2002-200000000    orthogonal lifted from D4
ρ1022220200-2-2-2-20000000    orthogonal lifted from D4
ρ112-22-2-202002-200000000    orthogonal lifted from D4
ρ122-22-200000-220-2200000    orthogonal lifted from D4
ρ1322220-2002-2-220000000    orthogonal lifted from D4
ρ142-22-200000-2202-200000    orthogonal lifted from D4
ρ152-2-2200002i00-2i000-√2√2√-2-√-2    complex lifted from C4○D8
ρ162-2-220000-2i002i000√2-√2√-2-√-2    complex lifted from C4○D8
ρ172-2-220000-2i002i000-√2√2-√-2√-2    complex lifted from C4○D8
ρ182-2-2200002i00-2i000√2-√2-√-2√-2    complex lifted from C4○D8
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of D4⋊D4
►On 32 points
Generators in S32
(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)

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

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

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

D4⋊D4 is a maximal subgroup of
 C24.103D4  C24.105D4  (C2×Q8)⋊16D4  (C2×Q8)⋊17D4  C42.384D4  C42.450D4  C42.229D4  C42.233D4  C42.353C23  C42.358C23  C42.359C23  C42.270D4  C42.411C23  SD16⋊8D4  Q16⋊4D4  SD16⋊10D4  C42.468C23  C42.42C23  C42.479C23  SL2(𝔽3)⋊D4
 D4p⋊D4: D8⋊9D4  D8⋊10D4  D8⋊4D4  D8⋊12D4  D12⋊14D4  Q8⋊4D12  D12⋊17D4  D12⋊7D4 ...
 (Cp×D4)⋊D4: C24.104D4  C4○D4⋊D4  (C2×D4)⋊21D4  D4⋊3D12  Dic6⋊D4  (C3×D4)⋊14D4  D4⋊3D20  Dic10⋊D4 ...
 C8⋊pD4⋊C2: C24.121D4  C24.124D4  C24.127D4  C24.130D4  C4.2+ 1+4  C4.182+ 1+4  C4.192+ 1+4  C42.265D4 ...
D4⋊D4 is a maximal quotient of
 C24.65D4  C24.74D4
 D4⋊D4p: D4⋊3D8  D4⋊3D12  D4⋊3D20  D4⋊3D28 ...
 (Cp×D4)⋊D4: C24.9D4  C23⋊2D8  Dic6⋊D4  (C3×D4)⋊14D4  Dic10⋊D4  (C5×D4)⋊14D4  Dic14⋊D4  (C7×D4)⋊14D4 ...
 Q8⋊D4p: Q8⋊D8  Q8⋊4D12  D20⋊4D4  D28⋊4D4 ...
 (Cp×Q8)⋊D4: C23⋊3SD16  D12⋊7D4  D20⋊7D4  D28⋊7D4 ...
 C4⋊C4.D2p: C4⋊C4.D4  C4⋊C4.12D4  C24.15D4  C42.181C23  D4⋊SD16  C42.185C23  Q8⋊6SD16  C42.189C23 ...
 (C2×C8).D2p: (C22×D8).C2  D12⋊14D4  D20⋊14D4  D28⋊14D4 ...

Matrix representation of D4⋊D4 ►in GL4(𝔽17) generated by

16000
01600
0001
00160
,
0100
1000
00314
001414
,
0100
16000
00013
00130
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,3,14,0,0,14,14],[0,16,0,0,1,0,0,0,0,0,0,13,0,0,13,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
 

D4⋊D4 in GAP, Magma, Sage, TeX 

D_4\rtimes D_4
 
% in TeX
 
G:=Group("D4:D4");
 
// GroupNames label
 
G:=SmallGroup(64,130);
 
// by ID
 
G=gap.SmallGroup(64,130);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,158,963,489,117]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D4⋊D4 in TeX

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