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G = SD16.A4  order 192 = 26·3

The non-split extension by SD16 of A4 acting through Inn(SD16)

non-abelian, soluble

Aliases: SD16.A4, 2- (1+4).C6, 2+ (1+4)⋊2C6, SL2(𝔽3).12D4, D4○SD16⋊C3, C8○D4⋊1C6, C8.A4⋊7C2, C8.4(C2×A4), D4.A4⋊4C2, D4.2(C2×A4), C2.10(D4×A4), Q8.A4⋊5C2, Q8.5(C2×A4), Q8.4(C3×D4), C4.5(C22×A4), C4.A4.16C22, C4○D4.2(C2×C6), SmallGroup(192,1018)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C4○D4 — SD16.A4
Chief series
C1 — C2 — Q8 — C4○D4 — C4.A4 — D4.A4 — SD16.A4
Lower central
Q8 — C4○D4 — SD16.A4

Subgroups: 275 in 73 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×4], C6 [×2], C8, C8, C2×C4 [×4], D4, D4 [×5], Q8 [×2], Q8 [×2], C23, C12 [×2], C2×C6, C2×C8, M4(2), D8, SD16, SD16 [×3], Q16, C2×D4 [×2], C2×Q8 [×2], C4○D4, C4○D4 [×4], C24, SL2(𝔽3), C3×D4, C3×Q8, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ (1+4), 2- (1+4), C3×SD16, C2×SL2(𝔽3), C4.A4, C4.A4, D4○SD16, C8.A4, Q8.A4, D4.A4, SD16.A4

Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C22×A4, D4×A4, SD16.A4

Generators and relations
 G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >

Smallest permutation representation
►On 32 points
Generators in S32
(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)

(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)

(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)

(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)

(9 19 32)(10 20 25)(11 21 26)(12 22 27)(13 23 28)(14 24 29)(15 17 30)(16 18 31)
 
G:=sub<Sym(32)| (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32), (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (1,20,5,24)(2,21,6,17)(3,22,7,18)(4,23,8,19)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27), (9,19,32)(10,20,25)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,17,30)(16,18,31)>;
 
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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32), (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (1,20,5,24)(2,21,6,17)(3,22,7,18)(4,23,8,19)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27), (9,19,32)(10,20,25)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,17,30)(16,18,31) );
 
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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,19),(18,22),(21,23),(26,28),(27,31),(30,32)], [(1,10,5,14),(2,11,6,15),(3,12,7,16),(4,13,8,9),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29)], [(1,20,5,24),(2,21,6,17),(3,22,7,18),(4,23,8,19),(9,28,13,32),(10,29,14,25),(11,30,15,26),(12,31,16,27)], [(9,19,32),(10,20,25),(11,21,26),(12,22,27),(13,23,28),(14,24,29),(15,17,30),(16,18,31)])
 

Matrix representation ►G ⊆ GL4(𝔽3) generated by

2112
0020
0220
2210
,
1201
1211
2100
1010
,
1002
0110
0120
2002
,
1112
1222
0011
0012
,
1211
2002
1022
1221
G:=sub<GL(4,GF(3))| [2,0,0,2,1,0,2,2,1,2,2,1,2,0,0,0],[1,1,2,1,2,2,1,0,0,1,0,1,1,1,0,0],[1,0,0,2,0,1,1,0,0,1,2,0,2,0,0,2],[1,1,0,0,1,2,0,0,1,2,1,1,2,2,1,2],[1,2,1,1,2,0,0,2,1,0,2,2,1,2,2,1] >;
 

Character table of SD16.A4

 class 12A2B2C2D3A3B4A4B4C4D6A6B6C6D8A8B8C12A12B12C12D24A24B24C24D
 size 114612442461244161622128816168888
ρ111111111111111111111111111    trivial
ρ211-11-1111-11-111-1-111111-1-11111    linear of order 2
ρ311-11111111-111-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ41111-1111-1111111-1-1-111-1-1-1-1-1-1    linear of order 2
ρ51111-1ζ32ζ31-111ζ32ζ3ζ3ζ32-1-1-1ζ3ζ32ζ6ζ65ζ6ζ6ζ65ζ65    linear of order 6
ρ611-11-1ζ32ζ31-11-1ζ32ζ3ζ65ζ6111ζ3ζ32ζ6ζ65ζ32ζ32ζ3ζ3    linear of order 6
ρ71111-1ζ3ζ321-111ζ3ζ32ζ32ζ3-1-1-1ζ32ζ3ζ65ζ6ζ65ζ65ζ6ζ6    linear of order 6
ρ811111ζ3ζ321111ζ3ζ32ζ32ζ3111ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ911111ζ32ζ31111ζ32ζ3ζ3ζ32111ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ1011-111ζ32ζ3111-1ζ32ζ3ζ65ζ6-1-1-1ζ3ζ32ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ1111-11-1ζ3ζ321-11-1ζ3ζ32ζ6ζ65111ζ32ζ3ζ65ζ6ζ3ζ3ζ32ζ32    linear of order 6
ρ1211-111ζ3ζ32111-1ζ3ζ32ζ6ζ65-1-1-1ζ32ζ3ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ13220-2022-20202200000-2-2000000    orthogonal lifted from D4
ρ14220-20-1-√-3-1+√-3-2020-1-√-3-1+√-3000001-√-31+√-3000000    complex lifted from C3×D4
ρ15220-20-1+√-3-1-√-3-2020-1+√-3-1-√-3000001+√-31-√-3000000    complex lifted from C3×D4
ρ16333-11003-3-1-10000-3-3100000000    orthogonal lifted from C2×A4
ρ1733-3-1-10033-110000-3-3100000000    orthogonal lifted from C2×A4
ρ1833-3-11003-3-11000033-100000000    orthogonal lifted from C2×A4
ρ19333-1-10033-1-1000033-100000000    orthogonal lifted from A4
ρ204-4000-2-2000022002√-22√-200000√-2√-2√-2√-2    complex faithful
ρ214-4000-2-2000022002√-22√-200000√-2√-2√-2√-2    complex faithful
ρ224-40001-√-31+√-30000-1+√-3-1-√-3002√-22√-200000ζ87ζ3+ζ85ζ3ζ83ζ3+ζ8ζ3ζ87ζ32+ζ85ζ32ζ83ζ32+ζ8ζ32    complex faithful
ρ234-40001+√-31-√-30000-1-√-3-1+√-3002√-22√-200000ζ83ζ32+ζ8ζ32ζ87ζ32+ζ85ζ32ζ83ζ3+ζ8ζ3ζ87ζ3+ζ85ζ3    complex faithful
ρ244-40001-√-31+√-30000-1+√-3-1-√-3002√-22√-200000ζ83ζ3+ζ8ζ3ζ87ζ3+ζ85ζ3ζ83ζ32+ζ8ζ32ζ87ζ32+ζ85ζ32    complex faithful
ρ254-40001+√-31-√-30000-1-√-3-1+√-3002√-22√-200000ζ87ζ32+ζ85ζ32ζ83ζ32+ζ8ζ32ζ87ζ3+ζ85ζ3ζ83ζ3+ζ8ζ3    complex faithful
ρ266602000-60-20000000000000000    orthogonal lifted from D4×A4

In GAP, Magma, Sage, TeX 

SD_{16}.A_4
 
% in TeX
 
G:=Group("SD16.A4");
 
// GroupNames label
 
G:=SmallGroup(192,1018);
 
// by ID
 
G=gap.SmallGroup(192,1018);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,680,3027,1522,248,438,172,775,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^8=b^2=e^3=1,c^2=d^2=a^4,b*a*b=a^3,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
 

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