Copied to
clipboard

G = D8.S3  order 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, C3⋊2SD32, C12.4D4, Dic12⋊3C2, C24.3C22, C3⋊C16⋊2C2, (C3×D8).1C2, C2.5(D4⋊S3), C4.2(C3⋊D4), SmallGroup(96,34)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — D8.S3
Chief series
C1 — C3 — C6 — C12 — C24 — Dic12 — D8.S3
Lower central
C3 — C6 — C12 — C24 — D8.S3
Upper central
C1 — C2 — C4 — C8 — D8

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 >

C1
C2
8C2
C3
C4
4C22
12C4
C6
8C6
C8
2D4
6Q8
C12
4Dic3
4C2×C6
D8
3C16
3Q16
C24
2Dic6
2C3×D4
3SD32
C3⋊C16
C3×D8
Dic12
D8.S3

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
ρ82202-2020000-2√2-√2√2-√200    orthogonal lifted from D8
ρ92202-2020000-2-√2√2-√2√200    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
ρ122-20200-200√2-√20ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163√2-√2    complex lifted from SD32
ρ132-20200-200-√2√20ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16-√2√2    complex lifted from SD32
ρ142-20200-200-√2√20ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169-√2√2    complex lifted from SD32
ρ152-20200-200√2-√20ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611√2-√2    complex lifted from SD32
ρ16440-2-40-200002000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ174-40-200200-2√22√200000√2-√2    symplectic faithful, Schur index 2
ρ184-40-2002002√2-2√200000-√2√2    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 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 32)(26 31)(27 30)(28 29)(33 34)(35 40)(36 39)(37 38)(42 48)(43 47)(44 46)

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

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

D8.S3 is a maximal subgroup of
 D8⋊D6  D16⋊3S3  S3×SD32  SD32⋊S3  D8.D6  Q16.D6  D8.9D6  D8.D9  D24.S3  C32⋊3SD32  C32⋊8SD32  C40.D6  D40.S3  D8.D15
D8.S3 is a maximal quotient of
 C6.SD32  C6.Q32  D8⋊1Dic3  D8.D9  D24.S3  C32⋊3SD32  C32⋊8SD32  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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁