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G = A4×SD16  order 192 = 26·3

Direct product of A4 and SD16

direct product, metabelian, soluble, monomial

Aliases: A4×SD16, C8⋊2(C2×A4), (C8×A4)⋊6C2, Q8⋊3(C2×A4), (Q8×A4)⋊4C2, C2.7(D4×A4), (C22×C8)⋊2C6, (D4×A4).2C2, D4.1(C2×A4), (C22×D4).C6, C22⋊(C3×SD16), (C22×SD16)⋊C3, (C2×A4).15D4, C4.2(C22×A4), (C22×Q8)⋊3C6, C23.24(C3×D4), (C4×A4).18C22, (C22×C4).2(C2×C6), SmallGroup(192,1015)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C4 — A4×SD16
Chief series
C1 — C22 — C23 — C22×C4 — C4×A4 — D4×A4 — A4×SD16
Lower central
C22 — C23 — C22×C4 — A4×SD16
Upper central
C1 — C2 — C4 — SD16

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

Subgroups: 324 in 83 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, A4, C2×C6, C2×C8, SD16, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, C3×D4, C3×Q8, C2×A4, C2×A4, C22×C8, C2×SD16, C22×D4, C22×Q8, C3×SD16, C4×A4, C4×A4, C22×A4, C22×SD16, C8×A4, D4×A4, Q8×A4, A4×SD16
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, SD16, C3×D4, C2×A4, C3×SD16, C22×A4, D4×A4, A4×SD16

Character table of A4×SD16

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D8A8B8C8D12A12B12C12D24A24B24C24D
 size 1133412442461244161622668816168888
ρ11111111111111111111111111111    trivial
ρ2111111111-11-11111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ31111-1-1111-11-111-1-1111111-1-11111    linear of order 2
ρ41111-1-111111111-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ51111-1-1ζ3ζ321-11-1ζ3ζ32ζ65ζ61111ζ32ζ3ζ6ζ65ζ32ζ32ζ3ζ3    linear of order 6
ρ61111-1-1ζ32ζ31111ζ32ζ3ζ6ζ65-1-1-1-1ζ3ζ32ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ7111111ζ3ζ321-11-1ζ3ζ32ζ3ζ32-1-1-1-1ζ32ζ3ζ6ζ65ζ6ζ6ζ65ζ65    linear of order 6
ρ81111-1-1ζ32ζ31-11-1ζ32ζ3ζ6ζ651111ζ3ζ32ζ65ζ6ζ3ζ3ζ32ζ32    linear of order 6
ρ9111111ζ32ζ31111ζ32ζ3ζ32ζ31111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ10111111ζ3ζ321111ζ3ζ32ζ3ζ321111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ111111-1-1ζ3ζ321111ζ3ζ32ζ65ζ6-1-1-1-1ζ32ζ3ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ12111111ζ32ζ31-11-1ζ32ζ3ζ32ζ3-1-1-1-1ζ3ζ32ζ65ζ6ζ65ζ65ζ6ζ6    linear of order 6
ρ1322220022-20-2022000000-2-2000000    orthogonal lifted from D4
ρ14222200-1+√-3-1-√-3-20-20-1+√-3-1-√-30000001+√-31-√-3000000    complex lifted from C3×D4
ρ15222200-1-√-3-1+√-3-20-20-1-√-3-1+√-30000001-√-31+√-3000000    complex lifted from C3×D4
ρ162-2-2200220000-2-200-√-2√-2-√-2√-20000√-2-√-2√-2-√-2    complex lifted from SD16
ρ172-2-2200220000-2-200√-2-√-2√-2-√-20000-√-2√-2-√-2√-2    complex lifted from SD16
ρ182-2-2200-1-√-3-1+√-300001+√-31-√-300-√-2√-2-√-2√-20000ζ83ζ3+ζ8ζ3ζ87ζ3+ζ85ζ3ζ83ζ32+ζ8ζ32ζ87ζ32+ζ85ζ32    complex lifted from C3×SD16
ρ192-2-2200-1+√-3-1-√-300001-√-31+√-300-√-2√-2-√-2√-20000ζ83ζ32+ζ8ζ32ζ87ζ32+ζ85ζ32ζ83ζ3+ζ8ζ3ζ87ζ3+ζ85ζ3    complex lifted from C3×SD16
ρ202-2-2200-1+√-3-1-√-300001-√-31+√-300√-2-√-2√-2-√-20000ζ87ζ32+ζ85ζ32ζ83ζ32+ζ8ζ32ζ87ζ3+ζ85ζ3ζ83ζ3+ζ8ζ3    complex lifted from C3×SD16
ρ212-2-2200-1-√-3-1+√-300001+√-31-√-300√-2-√-2√-2-√-20000ζ87ζ3+ζ85ζ3ζ83ζ3+ζ8ζ3ζ87ζ32+ζ85ζ32ζ83ζ32+ζ8ζ32    complex lifted from C3×SD16
ρ2233-1-13-1003-3-110000-3-31100000000    orthogonal lifted from C2×A4
ρ2333-1-1-310033-1-10000-3-31100000000    orthogonal lifted from C2×A4
ρ2433-1-13-10033-1-1000033-1-100000000    orthogonal lifted from A4
ρ2533-1-1-31003-3-11000033-1-100000000    orthogonal lifted from C2×A4
ρ2666-2-20000-60200000000000000000    orthogonal lifted from D4×A4
ρ276-62-20000000000003√-2-3√-2-√-2√-200000000    complex faithful
ρ286-62-2000000000000-3√-23√-2√-2-√-200000000    complex faithful

Permutation representations of A4×SD16
►On 24 points - transitive group 24T329
Generators in S24
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)

(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)

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

(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)(18 20)(19 23)(22 24)
 
G:=sub<Sym(24)| (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,15,17)(2,16,18)(3,9,19)(4,10,20)(5,11,21)(6,12,22)(7,13,23)(8,14,24), (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)(18,20)(19,23)(22,24)>;
 
G:=Group( (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24), (1,15,17)(2,16,18)(3,9,19)(4,10,20)(5,11,21)(6,12,22)(7,13,23)(8,14,24), (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)(18,20)(19,23)(22,24) );
 
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24)], [(1,15,17),(2,16,18),(3,9,19),(4,10,20),(5,11,21),(6,12,22),(7,13,23),(8,14,24)], [(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),(18,20),(19,23),(22,24)]])
 
G:=TransitiveGroup(24,329);
 

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

10000
01000
007200
007201
007210
,
10000
01000
000721
000720
001720
,
10000
01000
00001
00100
00010
,
072000
7212000
00100
00010
00001
,
10000
6172000
00100
00010
00001

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

A4×SD16 in GAP, Magma, Sage, TeX 

A_4\times {\rm SD}_{16}
 
% in TeX
 
G:=Group("A4xSD16");
 
// GroupNames label
 
G:=SmallGroup(192,1015);
 
// by ID
 
G=gap.SmallGroup(192,1015);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,197,176,1011,514,80,1027,1784]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

Export

Character table of A4×SD16 in TeX

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