Copied to
clipboard

G = D46D4order 64 = 26

2nd semidirect product of D4 and D4 acting through Inn(D4)

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

Aliases: D46D4, C22.41C24, C23.44C23, C42.43C22, C2.92- 1+4, D42(C4⋊C4), C4⋊Q813C2, (C4×D4)⋊16C2, C42(C4○D4), C4.37(C2×D4), C4⋊D412C2, C22⋊Q812C2, C22.4(C2×D4), C4⋊C4.33C22, (C2×C4).28C23, C2.19(C22×D4), (C2×D4).79C22, C22.D49C2, C22⋊C4.5C22, (C2×Q8).61C22, (C22×C4).68C22, (C2×C4⋊C4)⋊20C2, (C2×C4○D4)⋊7C2, C2.21(C2×C4○D4), SmallGroup(64,228)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D46D4
C1C2C22C23C22×C4C2×C4○D4 — D46D4
C1C22 — D46D4
C1C22 — D46D4
C1C22 — D46D4

Generators and relations for D46D4
 G = < a,b,c,d | a4=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 213 in 146 conjugacy classes, 83 normal (17 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C2×C4 [×3], C2×C4 [×8], C2×C4 [×16], D4 [×4], D4 [×10], Q8 [×4], C23 [×4], C42, C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], D46D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2- 1+4, D46D4

Character table of D46D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 1111222244222222224444444
ρ11111111111111111111111111    trivial
ρ211111-11-1-1-1111-111-111-1-111-1-1    linear of order 2
ρ3111111111-1-1-1-11-1111-11-1-11-1-1    linear of order 2
ρ411111-11-1-11-1-1-1-1-11-11-1-11-1111    linear of order 2
ρ51111-1-1-1-11-1-1-1-11-11111-1-11-111    linear of order 2
ρ61111-11-11-11-1-1-1-1-11-111111-1-1-1    linear of order 2
ρ71111-1-1-1-11111111111-1-11-1-1-1-1    linear of order 2
ρ81111-11-11-1-1111-111-11-11-1-1-111    linear of order 2
ρ91111-1-1-1-1-1-111-11-1-11-1111-11-11    linear of order 2
ρ101111-11-111111-1-1-1-1-1-11-1-1-111-1    linear of order 2
ρ111111-1-1-1-1-11-1-1111-11-1-11-1111-1    linear of order 2
ρ121111-11-111-1-1-11-11-1-1-1-1-1111-11    linear of order 2
ρ1311111111-11-1-1111-11-11-1-1-1-1-11    linear of order 2
ρ1411111-11-11-1-1-11-11-1-1-1111-1-11-1    linear of order 2
ρ1511111111-1-111-11-1-11-1-1-111-11-1    linear of order 2
ρ1611111-11-11111-1-1-1-1-1-1-11-11-1-11    linear of order 2
ρ172-2-22-222-200000200-200000000    orthogonal lifted from D4
ρ182-2-2222-2-200000-200200000000    orthogonal lifted from D4
ρ192-2-222-2-2200000200-200000000    orthogonal lifted from D4
ρ202-2-22-2-22200000-200200000000    orthogonal lifted from D4
ρ212-22-2000000-222i0-2i-2i02i0000000    complex lifted from C4○D4
ρ222-22-2000000-22-2i02i2i0-2i0000000    complex lifted from C4○D4
ρ232-22-20000002-2-2i02i-2i02i0000000    complex lifted from C4○D4
ρ242-22-20000002-22i0-2i2i0-2i0000000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of D46D4
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 18)(2 17)(3 20)(4 19)(5 16)(6 15)(7 14)(8 13)(9 23)(10 22)(11 21)(12 24)(25 31)(26 30)(27 29)(28 32)
(1 27 13 11)(2 28 14 12)(3 25 15 9)(4 26 16 10)(5 22 19 30)(6 23 20 31)(7 24 17 32)(8 21 18 29)
(1 11)(2 12)(3 9)(4 10)(5 32)(6 29)(7 30)(8 31)(13 27)(14 28)(15 25)(16 26)(17 22)(18 23)(19 24)(20 21)

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

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

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

D46D4 is a maximal subgroup of
D4.SD16  D4.7D8  D44SD16  SD168D4  SD163D4  SD1611D4  D48SD16  C42.467C23  C42.468C23  C42.41C23  C42.42C23  C42.43C23  C42.50C23  C42.479C23  C42.480C23  C42.481C23  D4×C4○D4  C22.64C25  C22.74C25  C22.78C25  C22.81C25  C22.83C25  C4⋊2+ 1+4  C22.88C25  C22.95C25  C22.102C25  C22.106C25  C23.144C24  C22.110C25  C22.122C25  C22.124C25  C22.125C25  C22.128C25  C22.130C25  C22.131C25  C22.135C25  C22.149C25  C22.151C25
 D4p⋊D4: D810D4  D84D4  D813D4  D1224D4  D1222D4  D1212D4  D2024D4  D2022D4 ...
 D4⋊D4p: D43D8  D44D8  D45D8  D46D12  D46D20  D46D28 ...
 C2p.2- 1+4: D49SD16  C42.57C23  C42.59C23  C42.494C23  C42.495C23  C22.71C25  C22.100C25  C22.104C25 ...
D46D4 is a maximal quotient of
C24.549C23  C23.226C24  D4×C4⋊C4  C23.236C24  C23.241C24  C24.215C23  C23.244C24  C23.247C24  C24.243C23  C24.244C23  C23.313C24  C23.315C24  C23.316C24  C24.563C23  C23.322C24  C24.258C23  C23.327C24  C23.329C24  C24.262C23  C24.264C23  C24.567C23  C24.568C23  C24.268C23  C24.269C23  C24.271C23  C23.349C24  C23.353C24  C24.276C23  C24.282C23  C23.362C24  C24.286C23  C23.368C24  C23.369C24  C24.289C23  C23.374C24  C23.375C24  C24.295C23  C23.379C24  C24.299C23  C23.388C24  C24.301C23  C23.391C24  C23.396C24  C23.401C24  C23.406C24  C23.412C24  C23.419C24  C24.311C23  C4218D4  C42.167D4  C42.170D4  C42.35Q8  C24.326C23  C23.571C24  C23.572C24  C23.574C24  C23.580C24  C24.389C23  C24.393C23  C23.589C24  C23.595C24  C24.405C23  C24.407C23  C23.606C24  C23.608C24  C24.412C23  C23.615C24  C23.617C24  C23.618C24  C23.619C24  C23.620C24  C23.621C24  C23.622C24  C24.418C23  C23.624C24  C23.625C24  C23.626C24  C23.627C24
 D4⋊D4p: D45D8  D46D12  D46D20  D46D28 ...
 C2p.2- 1+4: D49SD16  C42.485C23  C42.486C23  D46Q16  C42.488C23  C42.489C23  C42.490C23  C42.491C23 ...

Matrix representation of D46D4 in GL4(𝔽5) generated by

1000
0100
0020
0033
,
4000
0400
0031
0022
,
0400
1000
0010
0001
,
0400
4000
0010
0044
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,2,3,0,0,0,3],[4,0,0,0,0,4,0,0,0,0,3,2,0,0,1,2],[0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,1,4,0,0,0,4] >;

D46D4 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_4
% in TeX

G:=Group("D4:6D4");
// GroupNames label

G:=SmallGroup(64,228);
// by ID

G=gap.SmallGroup(64,228);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,650,86,297]);
// Polycyclic

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

Export

Character table of D46D4 in TeX

׿
×
𝔽