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G = A4⋊Q16order 192 = 26·3

The semidirect product of A4 and Q16 acting via Q16/C8=C2

non-abelian, soluble, monomial

Aliases: C8.1S4, A41Q16, C22⋊Dic12, C23.8D12, C2.6(C4⋊S4), C4.16(C2×S4), (C8×A4).1C2, (C2×A4).1D4, A4⋊Q8.1C2, (C22×C8).2S3, (C4×A4).8C22, (C22×C4).13D6, SmallGroup(192,957)

Series: Derived Chief Lower central Upper central

C1C22C4×A4 — A4⋊Q16
C1C22A4C2×A4C4×A4A4⋊Q8 — A4⋊Q16
A4C2×A4C4×A4 — A4⋊Q16
C1C2C4C8

Generators and relations for A4⋊Q16
 G = < a,b,c,d,e | a2=b2=c3=d8=1, e2=d4, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 258 in 65 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, Q8, C23, Dic3, C12, A4, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×Q8, C24, Dic6, C2×A4, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, Dic12, A4⋊C4, C4×A4, C8.18D4, C8×A4, A4⋊Q8, A4⋊Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, D12, S4, Dic12, C2×S4, C4⋊S4, A4⋊Q16

Character table of A4⋊Q16

 class 12A2B2C34A4B4C4D4E4F68A8B8C8D12A12B24A24B24C24D
 size 11338262424242482266888888
ρ11111111111111111111111    trivial
ρ21111111-1-1-1-111111111111    linear of order 2
ρ311111111-11-11-1-1-1-111-1-1-1-1    linear of order 2
ρ41111111-11-111-1-1-1-111-1-1-1-1    linear of order 2
ρ52222-1220000-12222-1-1-1-1-1-1    orthogonal lifted from S3
ρ622222-2-2000020000-2-20000    orthogonal lifted from D4
ρ72222-1220000-1-2-2-2-2-1-11111    orthogonal lifted from D6
ρ82222-1-2-20000-1000011-33-33    orthogonal lifted from D12
ρ92222-1-2-20000-10000113-33-3    orthogonal lifted from D12
ρ102-22-22000000-22-2-220022-2-2    symplectic lifted from Q16, Schur index 2
ρ112-22-22000000-2-222-200-2-222    symplectic lifted from Q16, Schur index 2
ρ122-22-2-10000001-222-23-3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285    symplectic lifted from Dic12, Schur index 2
ρ132-22-2-100000012-2-223-3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3    symplectic lifted from Dic12, Schur index 2
ρ142-22-2-10000001-222-2-33ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32    symplectic lifted from Dic12, Schur index 2
ρ152-22-2-100000012-2-22-33ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38    symplectic lifted from Dic12, Schur index 2
ρ1633-1-103-11-1-110-3-311000000    orthogonal lifted from C2×S4
ρ1733-1-103-111-1-1033-1-1000000    orthogonal lifted from S4
ρ1833-1-103-1-1-111033-1-1000000    orthogonal lifted from S4
ρ1933-1-103-1-111-10-3-311000000    orthogonal lifted from C2×S4
ρ2066-2-20-62000000000000000    orthogonal lifted from C4⋊S4
ρ216-6-220000000032-322-2000000    symplectic faithful, Schur index 2
ρ226-6-2200000000-3232-22000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of A4⋊Q16 in GL5(𝔽73)

10000
01000
007200
00010
000072
,
10000
01000
00100
000720
000072
,
10000
01000
00010
00001
00100
,
016000
4132000
007200
000720
000072
,
1517000
6458000
007200
000072
000720

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[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],[0,41,0,0,0,16,32,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[15,64,0,0,0,17,58,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0] >;

A4⋊Q16 in GAP, Magma, Sage, TeX

A_4\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(192,957);
// by ID

G=gap.SmallGroup(192,957);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,85,92,254,58,1124,4037,285,2358,475]);
// Polycyclic

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

Export

Character table of A4⋊Q16 in TeX

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