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G = D43Q8order 64 = 26

The semidirect product of D4 and Q8 acting through Inn(D4)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: D43Q8, C42.49C22, C23.48C23, C22.48C24, C2.172+ 1+4, D43(C4⋊C4), C4⋊Q815C2, (C4×Q8)⋊15C2, C4.17(C2×Q8), (C4×D4).12C2, C22⋊Q817C2, C4.37(C4○D4), C22.6(C2×Q8), C4⋊C4.37C22, (C2×C4).31C23, C42.C210C2, C2.10(C22×Q8), (C2×D4).81C22, (C2×Q8).32C22, C22⋊C4.23C22, (C22×C4).73C22, (C2×C4⋊C4)⋊23C2, C2.27(C2×C4○D4), SmallGroup(64,235)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D43Q8
C1C2C22C2×C4C22×C4C2×C4⋊C4 — D43Q8
C1C22 — D43Q8
C1C22 — D43Q8
C1C22 — D43Q8

Generators and relations for D43Q8
 G = < a,b,c,d | a4=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 157 in 114 conjugacy classes, 83 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×4], C22 [×4], C2×C4 [×3], C2×C4 [×10], C2×C4 [×8], D4 [×4], Q8 [×4], C23 [×2], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×2], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, D43Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×2], C24, C22×Q8, C2×C4○D4, 2+ 1+4, D43Q8

Character table of D43Q8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 1111222222222222444444444
ρ11111111111111111111111111    trivial
ρ211111-1-11-111-11111-11-1-1-111-1-1    linear of order 2
ρ3111111111-1-11-11-111-1-1-1-11-11-1    linear of order 2
ρ411111-1-11-1-1-1-1-11-11-1-11111-1-11    linear of order 2
ρ51111-1-1-1-11111-1-1-1-11-1-11111-1-1    linear of order 2
ρ61111-111-1-111-1-1-1-1-1-1-11-1-11111    linear of order 2
ρ71111-1-1-1-11-1-111-11-1111-1-11-1-11    linear of order 2
ρ81111-111-1-1-1-1-11-11-1-11-1111-11-1    linear of order 2
ρ9111111111-1-111-11-1-1-1-1-11-11-11    linear of order 2
ρ1011111-1-11-1-1-1-11-11-11-111-1-111-1    linear of order 2
ρ11111111111111-1-1-1-1-1111-1-1-1-1-1    linear of order 2
ρ1211111-1-11-111-1-1-1-1-111-1-11-1-111    linear of order 2
ρ131111-1-1-1-11-1-11-11-11-111-11-111-1    linear of order 2
ρ141111-111-1-1-1-1-1-11-1111-11-1-11-11    linear of order 2
ρ151111-1-1-1-111111111-1-1-11-1-1-111    linear of order 2
ρ161111-111-1-111-111111-11-11-1-1-1-1    linear of order 2
ρ172-22-22-22-2200-20000000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-222-2-2-20020000000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-2-22-22200-20000000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-2-2-222-20020000000000000    symplectic lifted from Q8, Schur index 2
ρ2122-2-2000002-20-2i-2i2i2i000000000    complex lifted from C4○D4
ρ2222-2-200000-220-2i2i2i-2i000000000    complex lifted from C4○D4
ρ2322-2-200000-2202i-2i-2i2i000000000    complex lifted from C4○D4
ρ2422-2-2000002-202i2i-2i-2i000000000    complex lifted from C4○D4
ρ254-4-44000000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of D43Q8
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 13)(2 16)(3 15)(4 14)(5 17)(6 20)(7 19)(8 18)(9 27)(10 26)(11 25)(12 28)(21 32)(22 31)(23 30)(24 29)
(1 20 13 6)(2 17 14 7)(3 18 15 8)(4 19 16 5)(9 23 25 32)(10 24 26 29)(11 21 27 30)(12 22 28 31)
(1 32 13 23)(2 29 14 24)(3 30 15 21)(4 31 16 22)(5 12 19 28)(6 9 20 25)(7 10 17 26)(8 11 18 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), (1,13)(2,16)(3,15)(4,14)(5,17)(6,20)(7,19)(8,18)(9,27)(10,26)(11,25)(12,28)(21,32)(22,31)(23,30)(24,29), (1,20,13,6)(2,17,14,7)(3,18,15,8)(4,19,16,5)(9,23,25,32)(10,24,26,29)(11,21,27,30)(12,22,28,31), (1,32,13,23)(2,29,14,24)(3,30,15,21)(4,31,16,22)(5,12,19,28)(6,9,20,25)(7,10,17,26)(8,11,18,27)>;

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

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

D43Q8 is a maximal subgroup of
D4⋊SD16  D4⋊Q16  D4.5SD16  D43Q16  D49SD16  D46Q16  C42.58C23  C42.60C23  C42.497C23  C42.498C23  C42.501C23  C42.502C23  C42.507C23  C42.511C23  C42.513C23  C42.517C23  SD164Q8  SD16⋊Q8  SD163Q8  C22.64C25  Q8×C4○D4  C22.81C25  C22.84C25  C22.93C25  C22.96C25  C22.102C25  C22.104C25  C22.105C25  C22.110C25  C22.111C25  C22.124C25  C22.133C25  C22.142C25  C22.144C25  C22.146C25  C22.148C25  C22.152C25  C22.153C25  C22.154C25
 D4p⋊Q8: D86Q8  D84Q8  D85Q8  D1210Q8  D127Q8  D129Q8  D2010Q8  D207Q8 ...
 C2p.2+ 1+4: D47SD16  D45Q16  C42.49C23  C42.51C23  C42.480C23  C42.482C23  C22.70C25  C22.90C25 ...
D43Q8 is a maximal quotient of
C23.227C24  D4×C4⋊C4  C23.231C24  C23.237C24  C23.250C24  C23.251C24  C23.252C24  C24.252C23  C23.323C24  C23.349C24  C23.350C24  C23.352C24  C24.300C23  C23.397C24  C23.402C24  C23.405C24  C23.407C24  C23.408C24  C23.409C24  C23.411C24  C23.420C24  C23.422C24  C23.449C24  C24.583C23  C24.338C23  C23.479C24  C23.483C24  C23.485C24  C23.486C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C23.592C24  C24.421C23  C23.632C24  C24.428C23  C24.434C23  C23.655C24  C23.663C24  C23.668C24  C23.674C24  C24.448C23  C24.450C23  C23.688C24  C24.454C23  C23.691C24  C23.692C24  C23.702C24  C24.456C23  C23.705C24  C23.706C24  C23.707C24
 C42.D2p: C42.166D4  C42.173D4  C42.175D4  C42.176D4  C42.178D4  C42.181D4  D45Dic6  D46Dic6 ...
 C4⋊C4.D2p: C24.558C23  C24.568C23  C23.354C24  C24.285C23  C24.572C23  C23.392C24  C426Q8  C42.35Q8 ...

Matrix representation of D43Q8 in GL4(𝔽5) generated by

0400
1000
0010
0001
,
1000
0400
0040
0004
,
1000
0100
0030
0042
,
0300
2000
0022
0003
G:=sub<GL(4,GF(5))| [0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,3,4,0,0,0,2],[0,2,0,0,3,0,0,0,0,0,2,0,0,0,2,3] >;

D43Q8 in GAP, Magma, Sage, TeX

D_4\rtimes_3Q_8
% in TeX

G:=Group("D4:3Q8");
// GroupNames label

G:=SmallGroup(64,235);
// by ID

G=gap.SmallGroup(64,235);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,295,650,297,69]);
// Polycyclic

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

Export

Character table of D43Q8 in TeX

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