Copied to
clipboard

## G = A4⋊SD16order 192 = 26·3

### The semidirect product of A4 and SD16 acting via SD16/D4=C2

Aliases: D4.1S4, A42SD16, A4⋊C81C2, C4.1(C2×S4), A4⋊Q82C2, (C2×A4).7D4, (D4×A4).1C2, (C22×D4).S3, C22⋊(D4.S3), (C22×C4).1D6, (C4×A4).1C22, C2.4(A4⋊D4), C23.17(C3⋊D4), SmallGroup(192,973)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4×A4 — A4⋊SD16
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — A4⋊Q8 — A4⋊SD16
 Lower central A4 — C2×A4 — C4×A4 — A4⋊SD16
 Upper central C1 — C2 — C4 — D4

Generators and relations for A4⋊SD16
G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=dad-1=ab=ba, ae=ea, cbc-1=a, bd=db, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 362 in 80 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22, C22 [×8], C6 [×2], C8 [×2], C2×C4 [×4], D4, D4 [×4], Q8 [×2], C23, C23 [×4], Dic3, C12, A4, C2×C6, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×3], C2×Q8, C24, C3⋊C8, Dic6, C3×D4, C2×A4, C2×A4, C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, D4.S3, A4⋊C4, C4×A4, C22×A4, C22⋊SD16, A4⋊C8, A4⋊Q8, D4×A4, A4⋊SD16
Quotients: C1, C2 [×3], C22, S3, D4, D6, SD16, C3⋊D4, S4, D4.S3, C2×S4, A4⋊D4, A4⋊SD16

Character table of A4⋊SD16

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12 size 1 1 3 3 4 12 8 2 6 24 24 8 16 16 12 12 12 12 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 2 2 2 2 -2 -2 -1 2 2 0 0 -1 1 1 0 0 0 0 -1 orthogonal lifted from D6 ρ6 2 2 2 2 2 2 -1 2 2 0 0 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ7 2 2 2 2 0 0 2 -2 -2 0 0 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ8 2 2 2 2 0 0 -1 -2 -2 0 0 -1 √-3 -√-3 0 0 0 0 1 complex lifted from C3⋊D4 ρ9 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -√-3 √-3 0 0 0 0 1 complex lifted from C3⋊D4 ρ10 2 -2 2 -2 0 0 2 0 0 0 0 -2 0 0 √-2 √-2 -√-2 -√-2 0 complex lifted from SD16 ρ11 2 -2 2 -2 0 0 2 0 0 0 0 -2 0 0 -√-2 -√-2 √-2 √-2 0 complex lifted from SD16 ρ12 3 3 -1 -1 -3 1 0 3 -1 -1 1 0 0 0 -1 1 -1 1 0 orthogonal lifted from C2×S4 ρ13 3 3 -1 -1 3 -1 0 3 -1 1 -1 0 0 0 -1 1 -1 1 0 orthogonal lifted from S4 ρ14 3 3 -1 -1 3 -1 0 3 -1 -1 1 0 0 0 1 -1 1 -1 0 orthogonal lifted from S4 ρ15 3 3 -1 -1 -3 1 0 3 -1 1 -1 0 0 0 1 -1 1 -1 0 orthogonal lifted from C2×S4 ρ16 4 -4 4 -4 0 0 -2 0 0 0 0 2 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ17 6 6 -2 -2 0 0 0 -6 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ18 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 0 complex faithful ρ19 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 0 complex faithful

Permutation representations of A4⋊SD16
On 24 points - transitive group 24T331
Generators in S24
(1 5)(3 7)(9 13)(10 14)(11 15)(12 16)(17 21)(19 23)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 22 10)(2 11 23)(3 24 12)(4 13 17)(5 18 14)(6 15 19)(7 20 16)(8 9 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)

G:=sub<Sym(24)| (1,5)(3,7)(9,13)(10,14)(11,15)(12,16)(17,21)(19,23), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,22,10)(2,11,23)(3,24,12)(4,13,17)(5,18,14)(6,15,19)(7,20,16)(8,9,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)>;

G:=Group( (1,5)(3,7)(9,13)(10,14)(11,15)(12,16)(17,21)(19,23), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,22,10)(2,11,23)(3,24,12)(4,13,17)(5,18,14)(6,15,19)(7,20,16)(8,9,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24) );

G=PermutationGroup([(1,5),(3,7),(9,13),(10,14),(11,15),(12,16),(17,21),(19,23)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(1,22,10),(2,11,23),(3,24,12),(4,13,17),(5,18,14),(6,15,19),(7,20,16),(8,9,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24)])

G:=TransitiveGroup(24,331);

Matrix representation of A4⋊SD16 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 72 0 0 0 1 72 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 1 0 72 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 61 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 12 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[61,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4⋊SD16 in GAP, Magma, Sage, TeX

A_4\rtimes {\rm SD}_{16}
% in TeX

G:=Group("A4:SD16");
// GroupNames label

G:=SmallGroup(192,973);
// by ID

G=gap.SmallGroup(192,973);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,85,254,135,58,1124,4037,285,2358,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

Export

׿
×
𝔽