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G = D8.S3order 96 = 25·3

The non-split extension by D8 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.S3, C8.5D6, C6.9D8, C32SD32, C12.4D4, Dic123C2, C24.3C22, C3⋊C162C2, (C3×D8).1C2, C2.5(D4⋊S3), C4.2(C3⋊D4), SmallGroup(96,34)

Series: Derived Chief Lower central Upper central

C1C24 — D8.S3
C1C3C6C12C24Dic12 — D8.S3
C3C6C12C24 — D8.S3
C1C2C4C8D8

Generators and relations for D8.S3
 G = < a,b,c,d | a8=b2=c3=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

8C2
4C22
12C4
8C6
2D4
6Q8
4Dic3
4C2×C6
3C16
3Q16
2Dic6
2C3×D4
3SD32

Character table of D8.S3

 class 12A2B34A4B6A6B6C8A8B1216A16B16C16D24A24B
 size 1182224288224666644
ρ1111111111111111111    trivial
ρ211-111-11-1-1111111111    linear of order 2
ρ311-11111-1-1111-1-1-1-111    linear of order 2
ρ411111-1111111-1-1-1-111    linear of order 2
ρ5222-120-1-1-122-10000-1-1    orthogonal lifted from S3
ρ6220220200-2-220000-2-2    orthogonal lifted from D4
ρ722-2-120-11122-10000-1-1    orthogonal lifted from D6
ρ82202-2020000-22-22-200    orthogonal lifted from D8
ρ92202-2020000-2-22-2200    orthogonal lifted from D8
ρ10220-120-1--3-3-2-2-1000011    complex lifted from C3⋊D4
ρ11220-120-1-3--3-2-2-1000011    complex lifted from C3⋊D4
ρ122-20200-2002-20ζ1615169ζ16131611ζ16716ζ1651632-2    complex lifted from SD32
ρ132-20200-200-220ζ165163ζ1615169ζ16131611ζ16716-22    complex lifted from SD32
ρ142-20200-200-220ζ16131611ζ16716ζ165163ζ1615169-22    complex lifted from SD32
ρ152-20200-2002-20ζ16716ζ165163ζ1615169ζ161316112-2    complex lifted from SD32
ρ16440-2-40-200002000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ174-40-200200-2222000002-2    symplectic faithful, Schur index 2
ρ184-40-20020022-2200000-22    symplectic faithful, Schur index 2

Smallest permutation representation of D8.S3
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 8)(2 7)(3 6)(4 5)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 32)(26 31)(27 30)(28 29)(33 40)(34 39)(35 38)(36 37)(42 48)(43 47)(44 46)
(1 33 25)(2 34 26)(3 35 27)(4 36 28)(5 37 29)(6 38 30)(7 39 31)(8 40 32)(9 20 46)(10 21 47)(11 22 48)(12 23 41)(13 24 42)(14 17 43)(15 18 44)(16 19 45)
(1 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 46 37 42)(34 45 38 41)(35 44 39 48)(36 43 40 47)

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

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

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

D8.S3 is a maximal subgroup of
D8⋊D6  D163S3  S3×SD32  SD32⋊S3  D8.D6  Q16.D6  D8.9D6  D8.D9  D24.S3  C323SD32  C328SD32  C40.D6  D40.S3  D8.D15
D8.S3 is a maximal quotient of
C6.SD32  C6.Q32  D81Dic3  D8.D9  D24.S3  C323SD32  C328SD32  C40.D6  D40.S3  D8.D15

Matrix representation of D8.S3 in GL4(𝔽7) generated by

3302
1311
2263
3423
,
5463
4104
1555
1313
,
3046
1265
4024
1035
,
4131
4412
5150
6301
G:=sub<GL(4,GF(7))| [3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[5,4,1,1,4,1,5,3,6,0,5,1,3,4,5,3],[3,1,4,1,0,2,0,0,4,6,2,3,6,5,4,5],[4,4,5,6,1,4,1,3,3,1,5,0,1,2,0,1] >;

D8.S3 in GAP, Magma, Sage, TeX

D_8.S_3
% in TeX

G:=Group("D8.S3");
// GroupNames label

G:=SmallGroup(96,34);
// by ID

G=gap.SmallGroup(96,34);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,73,218,116,122,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^3=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D8.S3 in TeX
Character table of D8.S3 in TeX

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