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G = A4⋊2Q16  order 192 = 26·3

The semidirect product of A4 and Q16 acting via Q16/Q8=C2

non-abelian, soluble, monomial

Aliases: A4⋊2Q16, Q8.3S4, A4⋊C8.C2, C4.3(C2×S4), (C2×A4).9D4, A4⋊Q8.2C2, (Q8×A4).1C2, C22⋊(C3⋊Q16), (C22×C4).3D6, (C4×A4).3C22, (C22×Q8).1S3, C2.6(A4⋊D4), C23.19(C3⋊D4), SmallGroup(192,975)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C4×A4 — A4⋊2Q16
Chief series
C1 — C22 — A4 — C2×A4 — C4×A4 — A4⋊Q8 — A4⋊2Q16
Lower central
A4 — C2×A4 — C4×A4 — A4⋊2Q16
Upper central
C1 — C2 — C4 — Q8

Generators and relations for A4⋊2Q16
 G = < a,b,c,d,e | a2=b2=c3=d8=1, e2=d4, cac-1=dad-1=ab=ba, ae=ea, cbc-1=a, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 262 in 70 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, Q8, Q8, C23, Dic3, C12, A4, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C3⋊C8, Dic6, C3×Q8, C2×A4, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C3⋊Q16, A4⋊C4, C4×A4, C4×A4, C22⋊Q16, A4⋊C8, A4⋊Q8, Q8×A4, A4⋊2Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊D4, S4, C3⋊Q16, C2×S4, A4⋊D4, A4⋊2Q16

Character table of A4⋊2Q16

 class 12A2B2C34A4B4C4D4E4F68A8B8C8D12A12B12C
 size 11338246122424812121212161616
ρ11111111111111111111    trivial
ρ2111111-11-1111-1-1-1-11-1-1    linear of order 2
ρ3111111111-1-11-1-1-1-1111    linear of order 2
ρ4111111-11-1-1-1111111-1-1    linear of order 2
ρ522222-20-200020000-200    orthogonal lifted from D4
ρ62222-1222200-10000-1-1-1    orthogonal lifted from S3
ρ72222-12-22-200-10000-111    orthogonal lifted from D6
ρ82-22-22000000-2-√2√2√2-√2000    symplectic lifted from Q16, Schur index 2
ρ92-22-22000000-2√2-√2-√2√2000    symplectic lifted from Q16, Schur index 2
ρ102222-1-20-2000-100001-√-3√-3    complex lifted from C3⋊D4
ρ112222-1-20-2000-100001√-3-√-3    complex lifted from C3⋊D4
ρ1233-1-103-3-111-101-11-1000    orthogonal lifted from C2×S4
ρ1333-1-103-3-11-110-11-11000    orthogonal lifted from C2×S4
ρ1433-1-1033-1-11-10-11-11000    orthogonal lifted from S4
ρ1533-1-1033-1-1-1101-11-1000    orthogonal lifted from S4
ρ164-44-4-200000020000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ1766-2-20-60200000000000    orthogonal lifted from A4⋊D4
ρ186-6-2200000000√2√2-√2-√2000    symplectic faithful, Schur index 2
ρ196-6-2200000000-√2-√2√2√2000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of A4⋊2Q16 ►in GL5(𝔽73)

10000
01000
007200
00010
000072
,
10000
01000
00100
000720
000072
,
10000
01000
00010
00001
00100
,
3164000
211000
00100
00001
00010
,
3924000
4034000
007200
000720
000072

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],[31,21,0,0,0,64,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[39,40,0,0,0,24,34,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72] >;
 

A4⋊2Q16 in GAP, Magma, Sage, TeX 

A_4\rtimes_2Q_{16}
 
% in TeX
 
G:=Group("A4:2Q16");
 
// GroupNames label
 
G:=SmallGroup(192,975);
 
// by ID
 
G=gap.SmallGroup(192,975);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,85,64,254,135,58,1124,4037,285,2358,475]);
 
// 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=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^-1=d^-1>;
 
// generators/relations
 

Export

Character table of A4⋊2Q16 in TeX

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