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G = D8.A4  order 192 = 26·3

The non-split extension by D8 of A4 acting through Inn(D8)

non-abelian, soluble

Aliases: D8.A4, 2- (1+4)⋊C6, SL2(𝔽3).13D4, Q8○D8⋊C3, C8.A4⋊4C2, C8○D4⋊2C6, C8.3(C2×A4), D4.A4⋊5C2, D4.3(C2×A4), C2.11(D4×A4), Q8.5(C3×D4), C4.6(C22×A4), C4.A4.17C22, C4○D4.3(C2×C6), SmallGroup(192,1019)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C4○D4 — D8.A4
Chief series
C1 — C2 — Q8 — C4○D4 — C4.A4 — D4.A4 — D8.A4
Lower central
Q8 — C4○D4 — D8.A4
Upper central
C1 — C2 — C4 — D8

Subgroups: 259 in 73 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C6 [×3], C8, C8, C2×C4 [×5], D4 [×2], D4 [×3], Q8, Q8 [×4], C12, C2×C6 [×2], C2×C8, M4(2), D8, SD16 [×2], Q16 [×3], C2×Q8 [×4], C4○D4, C4○D4 [×4], C24, SL2(𝔽3), C3×D4 [×2], C8○D4, C2×Q16, C4○D8, C8.C22 [×2], 2- (1+4) [×2], C3×D8, C2×SL2(𝔽3) [×2], C4.A4, Q8○D8, C8.A4, D4.A4 [×2], D8.A4

Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C22×A4, D4×A4, D8.A4

Generators and relations
 G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a-1, 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 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)

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

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

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

Matrix representation ►G ⊆ GL4(𝔽7) generated by

1136
2405
5434
2035
,
6000
6663
5645
1215
,
6214
5444
6001
6204
,
6635
2160
0053
0032
,
5563
2150
0343
5122
G:=sub<GL(4,GF(7))| [1,2,5,2,1,4,4,0,3,0,3,3,6,5,4,5],[6,6,5,1,0,6,6,2,0,6,4,1,0,3,5,5],[6,5,6,6,2,4,0,2,1,4,0,0,4,4,1,4],[6,2,0,0,6,1,0,0,3,6,5,3,5,0,3,2],[5,2,0,5,5,1,3,1,6,5,4,2,3,0,3,2] >;
 

Character table of D8.A4

 class 12A2B2C2D3A3B4A4B4C4D6A6B6C6D6E6F8A8B8C12A12B24A24B24C24D
 size 114464426121244161616162212888888
ρ111111111111111111111111111    trivial
ρ2111-1111111-111-1-111-1-1-111-1-1-1-1    linear of order 2
ρ311-1-111111-1-111-1-1-1-1111111111    linear of order 2
ρ411-1111111-111111-1-1-1-1-111-1-1-1-1    linear of order 2
ρ511-1-11ζ32ζ311-1-1ζ32ζ3ζ6ζ65ζ6ζ65111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 6
ρ611111ζ3ζ321111ζ3ζ32ζ3ζ32ζ3ζ32111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ711111ζ32ζ31111ζ32ζ3ζ32ζ3ζ32ζ3111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ8111-11ζ3ζ32111-1ζ3ζ32ζ65ζ6ζ3ζ32-1-1-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ9111-11ζ32ζ3111-1ζ32ζ3ζ6ζ65ζ32ζ3-1-1-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ1011-111ζ32ζ311-11ζ32ζ3ζ32ζ3ζ6ζ65-1-1-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ1111-111ζ3ζ3211-11ζ3ζ32ζ3ζ32ζ65ζ6-1-1-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ1211-1-11ζ3ζ3211-1-1ζ3ζ32ζ65ζ6ζ65ζ6111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 6
ρ132200-222-2200220000000-2-20000    orthogonal lifted from D4
ρ142200-2-1-√-3-1+√-3-2200-1-√-3-1+√-300000001+√-31-√-30000    complex lifted from C3×D4
ρ152200-2-1+√-3-1-√-3-2200-1+√-3-1-√-300000001-√-31+√-30000    complex lifted from C3×D4
ρ1633-3-3-1003-11100000033-1000000    orthogonal lifted from C2×A4
ρ173333-1003-1-1-100000033-1000000    orthogonal lifted from A4
ρ1833-33-1003-11-1000000-3-31000000    orthogonal lifted from C2×A4
ρ19333-3-1003-1-11000000-3-31000000    orthogonal lifted from C2×A4
ρ204-4000-2-200002200002√22√2000√2√2√2√2    symplectic faithful, Schur index 2
ρ214-4000-2-200002200002√22√2000√2√2√2√2    symplectic faithful, Schur index 2
ρ224-40001-√-31+√-30000-1+√-3-1-√-300002√22√2000-ζ83ζ3+ζ8ζ3-ζ87ζ3+ζ85ζ3-ζ83ζ32+ζ8ζ32-ζ87ζ32+ζ85ζ32    complex faithful
ρ234-40001-√-31+√-30000-1+√-3-1-√-300002√22√2000-ζ87ζ3+ζ85ζ3-ζ83ζ3+ζ8ζ3-ζ87ζ32+ζ85ζ32-ζ83ζ32+ζ8ζ32    complex faithful
ρ244-40001+√-31-√-30000-1-√-3-1+√-300002√22√2000-ζ83ζ32+ζ8ζ32-ζ87ζ32+ζ85ζ32-ζ83ζ3+ζ8ζ3-ζ87ζ3+ζ85ζ3    complex faithful
ρ254-40001+√-31-√-30000-1-√-3-1+√-300002√22√2000-ζ87ζ32+ζ85ζ32-ζ83ζ32+ζ8ζ32-ζ87ζ3+ζ85ζ3-ζ83ζ3+ζ8ζ3    complex faithful
ρ266600200-6-200000000000000000    orthogonal lifted from D4×A4

In GAP, Magma, Sage, TeX 

D_8.A_4
 
% in TeX
 
G:=Group("D8.A4");
 
// GroupNames label
 
G:=SmallGroup(192,1019);
 
// by ID
 
G=gap.SmallGroup(192,1019);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,3027,1522,248,438,172,775,285,124]);
 
// 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^-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
 

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