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G = D4⋊Q8order 64 = 26

1st semidirect product of D4 and Q8 acting via Q8/C4=C2

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

Aliases: D41Q8, C4.13D8, C42.22C22, C4⋊C84C2, C4⋊Q83C2, C2.D84C2, C2.7(C2×D8), (C4×D4).7C2, (C2×C4).33D4, C4.11(C2×Q8), (C2×C8).5C22, D4⋊C4.3C2, C4.23(C4○D4), C4⋊C4.11C22, (C2×C4).99C23, C22.95(C2×D4), (C2×D4).57C22, C2.12(C22⋊Q8), C2.14(C8.C22), SmallGroup(64,155)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4⋊Q8
C1C2C4C2×C4C2×D4C4×D4 — D4⋊Q8
C1C2C2×C4 — D4⋊Q8
C1C22C42 — D4⋊Q8
C1C2C2C2×C4 — D4⋊Q8

Generators and relations for D4⋊Q8
 G = < a,b,c,d | a4=b2=c4=1, d2=c2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
4C2
2C4
2C22
2C22
4C22
4C4
4C4
4C22
4C4
2C8
2C23
2D4
2C2×C4
2C2×C4
2C8
2C2×C4
4Q8
4Q8
4C2×C4
4C2×C4
2C22×C4
2C22⋊C4
2C4⋊C4
2C2×Q8

Character table of D4⋊Q8

 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
ρ102-2-2200-200200000-22-22    orthogonal lifted from D8
ρ112-2-2200-2002000002-22-2    orthogonal lifted from D8
ρ12222200-2-2-2-2020000000    orthogonal lifted from D4
ρ132-2-2200200-20000022-2-2    orthogonal lifted from D8
ρ142-2-2200200-200000-2-222    orthogonal lifted from D8
ρ152-22-2-220-220000000000    symplectic lifted from Q8, Schur index 2
ρ162-22-22-20-220000000000    symplectic lifted from Q8, Schur index 2
ρ172-22-20002-202i0-2i000000    complex lifted from C4○D4
ρ182-22-20002-20-2i02i000000    complex lifted from C4○D4
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of D4⋊Q8
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 6)(2 5)(3 8)(4 7)(9 16)(10 15)(11 14)(12 13)(17 23)(18 22)(19 21)(20 24)(25 31)(26 30)(27 29)(28 32)
(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 21 7 17)(2 24 8 20)(3 23 5 19)(4 22 6 18)(9 31 13 27)(10 30 14 26)(11 29 15 25)(12 32 16 28)

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

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

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

D4⋊Q8 is a maximal subgroup of
D4.D8  D43Q16  D44Q16  D44SD16  C42.447D4  C42.219D4  C42.448D4  C42.20C23  C42.22C23  C42.221D4  C42.451D4  C42.228D4  C42.233D4  C42.352C23  C42.358C23  C42.278D4  C42.285D4  C42.293D4  C42.296D4  C42.299D4  C42.302D4  C4.2- 1+4  C42.29C23  D44D8  C42.467C23  C42.49C23  C42.54C23  C42.473C23  C42.479C23  D45D8  C42.489C23  C42.494C23  C42.495C23  Q84D8  C42.505C23  C42.508C23  C42.509C23  C42.511C23  C42.516C23  Q8×D8  SD164Q8
 D4p⋊Q8: D84Q8  D85Q8  D124Q8  D122Q8  D126Q8  D204Q8  D202Q8  D206Q8 ...
 C4p.D8: C8.28D8  C8.D8  C12.50D8  C20.50D8  C28.50D8 ...
 C4p⋊Q8⋊C2: C42.287D4  C42.291D4  C42.423C23  C42.425C23  C42.60C23  C42.61C23  SD16⋊Q8  SD162Q8 ...
 C2.(D4.pD4): Q8.D8  C42.207C23  D4.7D8  C42.213C23  C88D8  D4.3SD16  D4.Q16  C8⋊D8 ...
D4⋊Q8 is a maximal quotient of
C2.(C4×D8)  C2.D84C4  C42.29Q8  C4⋊C4⋊Q8  (C2×C4).23D8  (C2×C4).26D8  (C2×C4).27D8  (C2×C4).28D8
 D4p⋊Q8: D81Q8  D8⋊Q8  D124Q8  D122Q8  D126Q8  D204Q8  D202Q8  D206Q8 ...
 C42.D2p: C42.98D4  C42.122D4  Q16⋊Q8  C4.Q32  D8.Q8  Q16.Q8  C12.50D8  C20.50D8 ...
 (Cp×D4)⋊Q8: (C2×D4)⋊Q8  Dic3.D8  Dic5.14D8  Dic7.D8 ...

Matrix representation of D4⋊Q8 in GL4(𝔽17) generated by

16000
01600
001615
0011
,
1000
111600
001615
0001
,
13000
7400
00160
00016
,
3100
71400
001111
0036
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,1,0,0,15,1],[1,11,0,0,0,16,0,0,0,0,16,0,0,0,15,1],[13,7,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[3,7,0,0,1,14,0,0,0,0,11,3,0,0,11,6] >;

D4⋊Q8 in GAP, Magma, Sage, TeX

D_4\rtimes Q_8
% in TeX

G:=Group("D4:Q8");
// GroupNames label

G:=SmallGroup(64,155);
// by ID

G=gap.SmallGroup(64,155);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,1444,376,88]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^4=1,d^2=c^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

Subgroup lattice of D4⋊Q8 in TeX
Character table of D4⋊Q8 in TeX

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