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G = Q16.A4order 192 = 26·3

The non-split extension by Q16 of A4 acting through Inn(Q16)

non-abelian, soluble

Aliases: Q16.A4, SL2(𝔽3).11D4, 2+ 1+4.2C6, D4○D8⋊C3, C8○D4.C6, C8.A43C2, C8.2(C2×A4), C2.9(D4×A4), Q8.A44C2, Q8.3(C3×D4), Q8.4(C2×A4), C4.4(C22×A4), C4.A4.15C22, C4○D4.1(C2×C6), SmallGroup(192,1017)

Series: Derived Chief Lower central Upper central

C1C2C4○D4 — Q16.A4
C1C2Q8C4○D4C4.A4Q8.A4 — Q16.A4
Q8C4○D4 — Q16.A4
C1C2C4Q16

Generators and relations for Q16.A4
 G = < a,b,c,d,e | a8=e3=1, b2=c2=d2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >

Subgroups: 291 in 73 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×5], C6, C8, C8, C2×C4 [×3], D4 [×7], Q8, Q8 [×2], C23 [×2], C12 [×3], C2×C8, M4(2), D8 [×3], SD16 [×2], Q16, C2×D4 [×4], C4○D4, C4○D4 [×4], C24, SL2(𝔽3), C3×Q8 [×2], C8○D4, C2×D8, C4○D8, C8⋊C22 [×2], 2+ 1+4 [×2], C3×Q16, C4.A4, C4.A4 [×2], D4○D8, C8.A4, Q8.A4 [×2], Q16.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C22×A4, D4×A4, Q16.A4

Character table of Q16.A4

 class 12A2B2C2D3A3B4A4B4C4D6A6B8A8B8C12A12B12C12D12E12F24A24B24C24D
 size 116121244244644221288161616168888
ρ111111111111111111111111111    trivial
ρ21111-11111-1111-1-1-111-1-111-1-1-1-1    linear of order 2
ρ3111-1-1111-1-111111111-1-1-1-11111    linear of order 2
ρ4111-11111-11111-1-1-11111-1-1-1-1-1-1    linear of order 2
ρ5111-1-1ζ3ζ321-1-11ζ3ζ32111ζ3ζ32ζ6ζ65ζ6ζ65ζ32ζ32ζ3ζ3    linear of order 6
ρ6111-11ζ32ζ31-111ζ32ζ3-1-1-1ζ32ζ3ζ3ζ32ζ65ζ6ζ65ζ65ζ6ζ6    linear of order 6
ρ71111-1ζ3ζ3211-11ζ3ζ32-1-1-1ζ3ζ32ζ6ζ65ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ8111-1-1ζ32ζ31-1-11ζ32ζ3111ζ32ζ3ζ65ζ6ζ65ζ6ζ3ζ3ζ32ζ32    linear of order 6
ρ9111-11ζ3ζ321-111ζ3ζ32-1-1-1ζ3ζ32ζ32ζ3ζ6ζ65ζ6ζ6ζ65ζ65    linear of order 6
ρ101111-1ζ32ζ311-11ζ32ζ3-1-1-1ζ32ζ3ζ65ζ6ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ1111111ζ3ζ321111ζ3ζ32111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ1211111ζ32ζ31111ζ32ζ3111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ1322-20022-200222000-2-200000000    orthogonal lifted from D4
ρ1422-200-1+-3-1--3-2002-1+-3-1--30001--31+-300000000    complex lifted from C3×D4
ρ1522-200-1--3-1+-3-2002-1--3-1+-30001+-31--300000000    complex lifted from C3×D4
ρ1633-11-1003-33-100-3-310000000000    orthogonal lifted from C2×A4
ρ1733-1-1-100333-10033-10000000000    orthogonal lifted from A4
ρ1833-111003-3-3-10033-10000000000    orthogonal lifted from C2×A4
ρ1933-1-110033-3-100-3-310000000000    orthogonal lifted from C2×A4
ρ204-4000-2-2000022-22220000000-22-22    orthogonal faithful
ρ214-4000-2-200002222-2200000002-22-2    orthogonal faithful
ρ224-40001--31+-30000-1+-3-1--322-22000000083ζ328ζ3287ζ3285ζ3283ζ38ζ387ζ385ζ3    complex faithful
ρ234-40001--31+-30000-1+-3-1--3-2222000000087ζ3285ζ3283ζ328ζ3287ζ385ζ383ζ38ζ3    complex faithful
ρ244-40001+-31--30000-1--3-1+-322-22000000083ζ38ζ387ζ385ζ383ζ328ζ3287ζ3285ζ32    complex faithful
ρ254-40001+-31--30000-1--3-1+-3-2222000000087ζ385ζ383ζ38ζ387ζ3285ζ3283ζ328ζ32    complex faithful
ρ266620000-600-2000000000000000    orthogonal lifted from D4×A4

Smallest permutation representation of Q16.A4
On 48 points
Generators in S48
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 39 5 35)(2 38 6 34)(3 37 7 33)(4 36 8 40)(9 46 13 42)(10 45 14 41)(11 44 15 48)(12 43 16 47)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)
(1 3 5 7)(2 4 6 8)(9 44 13 48)(10 45 14 41)(11 46 15 42)(12 47 16 43)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)(33 39 37 35)(34 40 38 36)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)
(1 26 43)(2 27 44)(3 28 45)(4 29 46)(5 30 47)(6 31 48)(7 32 41)(8 25 42)(9 40 24)(10 33 17)(11 34 18)(12 35 19)(13 36 20)(14 37 21)(15 38 22)(16 39 23)

G:=sub<Sym(48)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,39,5,35)(2,38,6,34)(3,37,7,33)(4,36,8,40)(9,46,13,42)(10,45,14,41)(11,44,15,48)(12,43,16,47)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29), (1,3,5,7)(2,4,6,8)(9,44,13,48)(10,45,14,41)(11,46,15,42)(12,47,16,43)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29)(33,39,37,35)(34,40,38,36), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32), (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,41)(8,25,42)(9,40,24)(10,33,17)(11,34,18)(12,35,19)(13,36,20)(14,37,21)(15,38,22)(16,39,23)>;

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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,39,5,35)(2,38,6,34)(3,37,7,33)(4,36,8,40)(9,46,13,42)(10,45,14,41)(11,44,15,48)(12,43,16,47)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29), (1,3,5,7)(2,4,6,8)(9,44,13,48)(10,45,14,41)(11,46,15,42)(12,47,16,43)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29)(33,39,37,35)(34,40,38,36), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32), (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,41)(8,25,42)(9,40,24)(10,33,17)(11,34,18)(12,35,19)(13,36,20)(14,37,21)(15,38,22)(16,39,23) );

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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,39,5,35),(2,38,6,34),(3,37,7,33),(4,36,8,40),(9,46,13,42),(10,45,14,41),(11,44,15,48),(12,43,16,47),(17,28,21,32),(18,27,22,31),(19,26,23,30),(20,25,24,29)], [(1,3,5,7),(2,4,6,8),(9,44,13,48),(10,45,14,41),(11,46,15,42),(12,47,16,43),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29),(33,39,37,35),(34,40,38,36)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,46,13,42),(10,47,14,43),(11,48,15,44),(12,41,16,45),(17,23,21,19),(18,24,22,20),(25,27,29,31),(26,28,30,32)], [(1,26,43),(2,27,44),(3,28,45),(4,29,46),(5,30,47),(6,31,48),(7,32,41),(8,25,42),(9,40,24),(10,33,17),(11,34,18),(12,35,19),(13,36,20),(14,37,21),(15,38,22),(16,39,23)])

Matrix representation of Q16.A4 in GL4(𝔽7) generated by

4536
4244
3301
2562
,
6530
0062
4456
2563
,
6205
5043
2543
6644
,
3663
4032
1605
3324
,
6263
6332
0020
5511
G:=sub<GL(4,GF(7))| [4,4,3,2,5,2,3,5,3,4,0,6,6,4,1,2],[6,0,4,2,5,0,4,5,3,6,5,6,0,2,6,3],[6,5,2,6,2,0,5,6,0,4,4,4,5,3,3,4],[3,4,1,3,6,0,6,3,6,3,0,2,3,2,5,4],[6,6,0,5,2,3,0,5,6,3,2,1,3,2,0,1] >;

Q16.A4 in GAP, Magma, Sage, TeX

Q_{16}.A_4
% in TeX

G:=Group("Q16.A4");
// GroupNames label

G:=SmallGroup(192,1017);
// by ID

G=gap.SmallGroup(192,1017);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,197,680,3027,1522,248,438,172,775,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=e^3=1,b^2=c^2=d^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^4*c,e*c*e^-1=a^4*c*d,e*d*e^-1=c>;
// generators/relations

Export

Character table of Q16.A4 in TeX

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