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G = D4.D4order 64 = 26

1st non-split extension by D4 of D4 acting via D4/C4=C2

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

Aliases: D4.1D4, C42SD16, C42.18C22, C4⋊C89C2, C4⋊Q82C2, (C4×D4).6C2, C4.31(C2×D4), (C2×C4).131D4, C2.8(C2×SD16), Q8⋊C411C2, C4.41(C4○D4), C4⋊C4.58C22, (C2×C4).89C23, (C2×C8).30C22, (C2×SD16).3C2, C22.85(C2×D4), (C2×Q8).7C22, C2.13(C4⋊D4), (C2×D4).56C22, C2.9(C8.C22), SmallGroup(64,142)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.D4
C1C2C4C2×C4C2×D4C4×D4 — D4.D4
C1C2C2×C4 — D4.D4
C1C22C42 — D4.D4
C1C2C2C2×C4 — D4.D4

Generators and relations for D4.D4
 G = < a,b,c,d | a4=b2=c4=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 109 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×4], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], D4.D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8.C22, D4.D4

Character table of D4.D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111442222444884444
ρ11111111111111111111    trivial
ρ21111-1-1-111-11-11-111-1-11    linear of order 2
ρ3111111-111-1-1-1-1-11-111-1    linear of order 2
ρ41111-1-11111-11-111-1-1-1-1    linear of order 2
ρ51111-1-11111-11-1-1-11111    linear of order 2
ρ6111111-111-1-1-1-11-11-1-11    linear of order 2
ρ71111-1-1-111-11-111-1-111-1    linear of order 2
ρ81111111111111-1-1-1-1-1-1    linear of order 2
ρ92222002-2-220-20000000    orthogonal lifted from D4
ρ102-22-2-220-220000000000    orthogonal lifted from D4
ρ112-22-22-20-220000000000    orthogonal lifted from D4
ρ12222200-2-2-2-2020000000    orthogonal lifted from D4
ρ132-22-20002-202i0-2i000000    complex lifted from C4○D4
ρ142-22-20002-20-2i02i000000    complex lifted from C4○D4
ρ152-2-2200-200200000--2-2--2-2    complex lifted from SD16
ρ162-2-2200200-200000--2--2-2-2    complex lifted from SD16
ρ172-2-2200200-200000-2-2--2--2    complex lifted from SD16
ρ182-2-2200-200200000-2--2-2--2    complex lifted from SD16
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

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

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

D4.D4 is a maximal subgroup of
C42.443D4  C42.211D4  C42.445D4  C42.222D4  C42.451D4  C42.228D4  C42.234D4  C42.264D4  C42.270D4  C42.272D4  C42.276D4  C42.294D4  C42.296D4  C42.300D4  C42.302D4
 D4.D4p: D4.D8  D4.7D8  C810SD16  D4.2D8  D4.2D12  D4.2D20  D4.2D28 ...
 C4p⋊SD16: C811SD16  C83SD16  C84SD16  C12⋊SD16  C125SD16  C20⋊SD16  C205SD16  C28⋊SD16 ...
 (Cp×D4).D4: Q83SD16  D4.5SD16  C42.207C23  C42.211C23  Q84SD16  D44SD16  D4.3SD16  C42.250C23 ...
 C4⋊C4.D2p: C42.19C23  C42.354C23  C42.357C23  C42.407C23  C42.411C23  C42.25C23  C42.27C23  Q87SD16 ...
D4.D4 is a maximal quotient of
 C4p⋊SD16: C811SD16  C810SD16  C83SD16  C84SD16  C12⋊SD16  D4.2D12  C125SD16  C20⋊SD16 ...
 (Cp×D4).D4: D4.1Q16  C8.8SD16  C42.98D4  (C2×SD16)⋊14C4  (C2×C4)⋊3SD16  (C2×C4)⋊5SD16  (C2×C4).24D8  Dic33SD16 ...
 C4⋊C4.D2p: C4.68(C4×D4)  C42.30Q8  C42.117D4  C4⋊C4.95D4  (C2×Q8).8Q8  C2.(C83Q8)  Dic3⋊SD16  Dic5⋊SD16 ...

Matrix representation of D4.D4 in GL4(𝔽17) generated by

16000
01600
0001
00160
,
16000
0100
0001
0010
,
13000
0400
0010
0001
,
0400
13000
00512
001212
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,13,0,0,4,0,0,0,0,0,5,12,0,0,12,12] >;

D4.D4 in GAP, Magma, Sage, TeX

D_4.D_4
% in TeX

G:=Group("D4.D4");
// GroupNames label

G:=SmallGroup(64,142);
// by ID

G=gap.SmallGroup(64,142);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,1444,376,88]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^4=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D4.D4 in TeX

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