Copied to
clipboard

G = Q8⋊3Dic3  order 96 = 25·3

2nd semidirect product of Q8 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊3Dic3, D4⋊2Dic3, C12.56D4, C3⋊3C4≀C2, (C3×D4)⋊2C4, (C3×Q8)⋊2C4, (C2×C6).3D4, C12.9(C2×C4), C4○D4.3S3, (C2×C4).41D6, (C4×Dic3)⋊2C2, C4.Dic3⋊4C2, C4.3(C2×Dic3), C4.31(C3⋊D4), C6.18(C22⋊C4), (C2×C12).20C22, C22.3(C3⋊D4), C2.8(C6.D4), (C3×C4○D4).1C2, SmallGroup(96,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — Q8⋊3Dic3
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — Q8⋊3Dic3
Lower central
C3 — C6 — C12 — Q8⋊3Dic3
Upper central
C1 — C4 — C2×C4 — C4○D4

Generators and relations for Q8⋊3Dic3
 G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

C1
C2
2C2
4C2
C3
C4
C22
C4
2C4
2C22
6C4
6C4
C6
2C6
4C6
D4
Q8
C2×C4
2D4
2C2×C4
6C2×C4
6C8
C12
C2×C6
C12
2C2×C6
2Dic3
2C12
2Dic3
C4○D4
3C42
3M4(2)
C3×Q8
C2×C12
C3×D4
2C3⋊C8
2C2×C12
2C3×D4
2C2×Dic3
3C4≀C2
C4.Dic3
C4×Dic3
C3×C4○D4
Q8⋊3Dic3

Character table of Q8⋊3Dic3

 class 12A2B2C34A4B4C4D4E4F4G4H6A6B6C6D8A8B12A12B12C12D12E
 size 11242112466662444121222444
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-1-11111-1-111111    linear of order 2
ρ3111-11111-1-1-1-1-11-11-11111-11-1    linear of order 2
ρ4111-11111-111111-11-1-1-111-11-1    linear of order 2
ρ511-1-11-1-111-ii-ii1-1-1-1-ii-1-1111    linear of order 4
ρ611-1-11-1-111i-ii-i1-1-1-1i-i-1-1111    linear of order 4
ρ711-111-1-11-1i-ii-i11-11-ii-1-1-11-1    linear of order 4
ρ811-111-1-11-1-ii-ii11-11i-i-1-1-11-1    linear of order 4
ρ922-20222-20000020-2000220-20    orthogonal lifted from D4
ρ10222-2-1222-20000-11-1100-1-11-11    orthogonal lifted from D6
ρ1122202-2-2-200000202000-2-20-20    orthogonal lifted from D4
ρ122222-122220000-1-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ1322-2-2-1-2-2220000-11110011-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ1422-22-1-2-22-20000-1-11-100111-11    symplectic lifted from Dic3, Schur index 2
ρ1522-20-122-200000-1√-31-√-300-1-1-√-31√-3    complex lifted from C3⋊D4
ρ162220-1-2-2-200000-1√-3-1-√-30011√-31-√-3    complex lifted from C3⋊D4
ρ172220-1-2-2-200000-1-√-3-1√-30011-√-31√-3    complex lifted from C3⋊D4
ρ1822-20-122-200000-1-√-31√-300-1-1√-31-√-3    complex lifted from C3⋊D4
ρ192-2002-2i2i001-i-1-i-1+i1+i-2000002i-2i000    complex lifted from C4≀C2
ρ202-2002-2i2i00-1+i1+i1-i-1-i-2000002i-2i000    complex lifted from C4≀C2
ρ212-20022i-2i00-1-i1-i1+i-1+i-200000-2i2i000    complex lifted from C4≀C2
ρ222-20022i-2i001+i-1+i-1-i1-i-200000-2i2i000    complex lifted from C4≀C2
ρ234-400-24i-4i0000002000002i-2i000    complex faithful
ρ244-400-2-4i4i000000200000-2i2i000    complex faithful

Permutation representations of Q8⋊3Dic3
►On 24 points - transitive group 24T109
Generators in S24
(1 10 6 7)(2 11 4 8)(3 12 5 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)

(1 22 6 19)(2 20 4 23)(3 24 5 21)(7 16 10 13)(8 14 11 17)(9 18 12 15)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(2 3)(4 5)(8 9)(11 12)(13 19 16 22)(14 24 17 21)(15 23 18 20)
 
G:=sub<Sym(24)| (1,10,6,7)(2,11,4,8)(3,12,5,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,22,6,19)(2,20,4,23)(3,24,5,21)(7,16,10,13)(8,14,11,17)(9,18,12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (2,3)(4,5)(8,9)(11,12)(13,19,16,22)(14,24,17,21)(15,23,18,20)>;
 
G:=Group( (1,10,6,7)(2,11,4,8)(3,12,5,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,22,6,19)(2,20,4,23)(3,24,5,21)(7,16,10,13)(8,14,11,17)(9,18,12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (2,3)(4,5)(8,9)(11,12)(13,19,16,22)(14,24,17,21)(15,23,18,20) );
 
G=PermutationGroup([[(1,10,6,7),(2,11,4,8),(3,12,5,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,22,6,19),(2,20,4,23),(3,24,5,21),(7,16,10,13),(8,14,11,17),(9,18,12,15)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(2,3),(4,5),(8,9),(11,12),(13,19,16,22),(14,24,17,21),(15,23,18,20)]])
 
G:=TransitiveGroup(24,109);
 

Q8⋊3Dic3 is a maximal subgroup of
 S3×C4≀C2  C42⋊3D6  C24.100D4  C24.54D4  D8⋊5Dic3  D8⋊4Dic3  D12⋊18D4  D12.38D4  D12.39D4  D12.40D4  (C6×D4)⋊9C4  2+ 1+4⋊6S3  2+ 1+4.4S3  2- 1+4⋊4S3  2- 1+4.2S3  Q8⋊3Dic9  C12.9S4  D12⋊4Dic3  D12⋊2Dic3  C62.39D4  C3⋊U2(𝔽3)  C60.96D4  C60.97D4  Q8⋊3Dic15  Dic10⋊Dic3  D20⋊2Dic3
Q8⋊3Dic3 is a maximal quotient of
 C12.2C42  C12.57D8  C12.26Q16  (C6×D4)⋊C4  (C6×Q8)⋊C4  C42.7D6  C42.8D6  Q8⋊3Dic9  D12⋊4Dic3  D12⋊2Dic3  C62.39D4  C60.96D4  C60.97D4  Q8⋊3Dic15  Dic10⋊Dic3  D20⋊2Dic3

Matrix representation of Q8⋊3Dic3 ►in GL4(𝔽5) generated by

2000
0300
0030
0002
,
0300
3000
3003
0230
,
0001
0140
0100
4004
,
1001
0020
0200
0004
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[0,3,3,0,3,0,0,2,0,0,0,3,0,0,3,0],[0,0,0,4,0,1,1,0,0,4,0,0,1,0,0,4],[1,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4] >;
 

Q8⋊3Dic3 in GAP, Magma, Sage, TeX 

Q_8\rtimes_3{\rm Dic}_3
 
% in TeX
 
G:=Group("Q8:3Dic3");
 
// GroupNames label
 
G:=SmallGroup(96,44);
 
// by ID
 
G=gap.SmallGroup(96,44);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,86,579,297,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of Q8⋊3Dic3 in TeX
Character table of Q8⋊3Dic3 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁