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G = A4⋊SD16  order 192 = 26·3

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

non-abelian, soluble, monomial

Aliases: D4.1S4, A4⋊2SD16, A4⋊C8⋊1C2, C4.1(C2×S4), A4⋊Q8⋊2C2, (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, C3, C4, C4, C22, C22, C6, C8, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, A4, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C3⋊C8, Dic6, C3×D4, C2×A4, C2×A4, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, D4.S3, A4⋊C4, C4×A4, C22×A4, C22⋊SD16, A4⋊C8, A4⋊Q8, D4×A4, A4⋊SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊D4, S4, D4.S3, C2×S4, A4⋊D4, A4⋊SD16

Character table of A4⋊SD16

 class 12A2B2C2D2E34A4B4C4D6A6B6C8A8B8C8D12
 size 11334128262424816161212121216
ρ11111111111111111111    trivial
ρ21111-1-1111-1-11-1-111111    linear of order 2
ρ31111-1-1111111-1-1-1-1-1-11    linear of order 2
ρ4111111111-1-1111-1-1-1-11    linear of order 2
ρ52222-2-2-12200-1110000-1    orthogonal lifted from D6
ρ6222222-12200-1-1-10000-1    orthogonal lifted from S3
ρ72222002-2-2002000000-2    orthogonal lifted from D4
ρ8222200-1-2-200-1√-3-√-300001    complex lifted from C3⋊D4
ρ9222200-1-2-200-1-√-3√-300001    complex lifted from C3⋊D4
ρ102-22-20020000-200√-2√-2-√-2-√-20    complex lifted from SD16
ρ112-22-20020000-200-√-2-√-2√-2√-20    complex lifted from SD16
ρ1233-1-1-3103-1-11000-11-110    orthogonal lifted from C2×S4
ρ1333-1-13-103-11-1000-11-110    orthogonal lifted from S4
ρ1433-1-13-103-1-110001-11-10    orthogonal lifted from S4
ρ1533-1-1-3103-11-10001-11-10    orthogonal lifted from C2×S4
ρ164-44-400-2000020000000    symplectic lifted from D4.S3, Schur index 2
ρ1766-2-2000-620000000000    orthogonal lifted from A4⋊D4
ρ186-6-220000000000-√-2√-2√-2-√-20    complex faithful
ρ196-6-220000000000√-2-√-2-√-2√-20    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 11)(2 12 23)(3 24 13)(4 14 17)(5 18 15)(6 16 19)(7 20 9)(8 10 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 13)(10 16)(12 14)(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), (1,22,11)(2,12,23)(3,24,13)(4,14,17)(5,18,15)(6,16,19)(7,20,9)(8,10,21), (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)(9,13)(10,16)(12,14)(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), (1,22,11)(2,12,23)(3,24,13)(4,14,17)(5,18,15)(6,16,19)(7,20,9)(8,10,21), (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)(9,13)(10,16)(12,14)(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)], [(1,22,11),(2,12,23),(3,24,13),(4,14,17),(5,18,15),(6,16,19),(7,20,9),(8,10,21)], [(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),(9,13),(10,16),(12,14),(17,23),(19,21),(20,24)]])
 
G:=TransitiveGroup(24,331);
 

Matrix representation of A4⋊SD16 ►in GL5(𝔽73)

10000
01000
000721
000720
001720
,
10000
01000
000172
001072
000072
,
10000
01000
00001
00100
00010
,
611000
10000
00010
00100
00001
,
10000
1272000
00100
00010
00001

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

Character table of A4⋊SD16 in TeX

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