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G = D4⋊D4order 64 = 26

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

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

Aliases: D43D4, Q83D4, C23.14D4, (C2×D8)⋊2C2, C4⋊D42C2, C22⋊C85C2, C4.21(C2×D4), D4⋊C48C2, Q8⋊C44C2, (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

C1C2×C4 — D4⋊D4
C1C2C22C2×C4C22×C4C2×C4○D4 — D4⋊D4
C1C2C2×C4 — D4⋊D4
C1C22C22×C4 — D4⋊D4
C1C2C2C2×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 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×2], D4 [×9], Q8 [×2], Q8, C23, C23 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×4], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, D4⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], 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
ρ152-2-2200002i00-2i000-22-2--2    complex lifted from C4○D8
ρ162-2-220000-2i002i0002-2-2--2    complex lifted from C4○D8
ρ172-2-220000-2i002i000-22--2-2    complex lifted from C4○D8
ρ182-2-2200002i00-2i0002-2--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 13)(6 16)(7 15)(8 14)(9 21)(10 24)(11 23)(12 22)(17 32)(18 31)(19 30)(20 29)
(1 15 9 19)(2 14 10 18)(3 13 11 17)(4 16 12 20)(5 24 32 26)(6 23 29 25)(7 22 30 28)(8 21 31 27)
(2 4)(5 31)(6 30)(7 29)(8 32)(10 12)(13 17)(14 20)(15 19)(16 18)(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)(5,13)(6,16)(7,15)(8,14)(9,21)(10,24)(11,23)(12,22)(17,32)(18,31)(19,30)(20,29), (1,15,9,19)(2,14,10,18)(3,13,11,17)(4,16,12,20)(5,24,32,26)(6,23,29,25)(7,22,30,28)(8,21,31,27), (2,4)(5,31)(6,30)(7,29)(8,32)(10,12)(13,17)(14,20)(15,19)(16,18)(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)(5,13)(6,16)(7,15)(8,14)(9,21)(10,24)(11,23)(12,22)(17,32)(18,31)(19,30)(20,29), (1,15,9,19)(2,14,10,18)(3,13,11,17)(4,16,12,20)(5,24,32,26)(6,23,29,25)(7,22,30,28)(8,21,31,27), (2,4)(5,31)(6,30)(7,29)(8,32)(10,12)(13,17)(14,20)(15,19)(16,18)(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),(5,13),(6,16),(7,15),(8,14),(9,21),(10,24),(11,23),(12,22),(17,32),(18,31),(19,30),(20,29)], [(1,15,9,19),(2,14,10,18),(3,13,11,17),(4,16,12,20),(5,24,32,26),(6,23,29,25),(7,22,30,28),(8,21,31,27)], [(2,4),(5,31),(6,30),(7,29),(8,32),(10,12),(13,17),(14,20),(15,19),(16,18),(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  SD168D4  Q164D4  SD1610D4  C42.468C23  C42.42C23  C42.479C23  SL2(𝔽3)⋊D4
 D4p⋊D4: D89D4  D810D4  D84D4  D812D4  D1214D4  Q84D12  D1217D4  D127D4 ...
 (Cp×D4)⋊D4: C24.104D4  C4○D4⋊D4  (C2×D4)⋊21D4  D43D12  Dic6⋊D4  (C3×D4)⋊14D4  D43D20  Dic10⋊D4 ...
 C8pD4⋊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: D43D8  D43D12  D43D20  D43D28 ...
 (Cp×D4)⋊D4: C24.9D4  C232D8  Dic6⋊D4  (C3×D4)⋊14D4  Dic10⋊D4  (C5×D4)⋊14D4  Dic14⋊D4  (C7×D4)⋊14D4 ...
 Q8⋊D4p: Q8⋊D8  Q84D12  D204D4  D284D4 ...
 (Cp×Q8)⋊D4: C233SD16  D127D4  D207D4  D287D4 ...
 C4⋊C4.D2p: C4⋊C4.D4  C4⋊C4.12D4  C24.15D4  C42.181C23  D4⋊SD16  C42.185C23  Q86SD16  C42.189C23 ...
 (C2×C8).D2p: (C22×D8).C2  D1214D4  D2014D4  D2814D4 ...

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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