Copied to
clipboard

G = Q8:3Dic3order 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, C3:3C4wrC2, (C3xD4):2C4, (C3xQ8):2C4, (C2xC6).3D4, C12.9(C2xC4), C4oD4.3S3, (C2xC4).41D6, (C4xDic3):2C2, C4.Dic3:4C2, C4.3(C2xDic3), C4.31(C3:D4), C6.18(C22:C4), (C2xC12).20C22, C22.3(C3:D4), C2.8(C6.D4), (C3xC4oD4).1C2, SmallGroup(96,44)

Series: Derived Chief Lower central Upper central

C1C12 — Q8:3Dic3
C1C3C6C12C2xC12C4.Dic3 — Q8:3Dic3
C3C6C12 — Q8:3Dic3
C1C4C2xC4C4oD4

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 >

Subgroups: 90 in 44 conjugacy classes, 21 normal (all characteristic)
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C2xDic3, C3:D4, C4wrC2, C6.D4, Q8:3Dic3
2C2
4C2
2C4
2C22
6C4
6C4
2C6
4C6
2D4
2C2xC4
6C2xC4
6C8
2C2xC6
2Dic3
2C12
2Dic3
3C42
3M4(2)
2C3:C8
2C2xC12
2C3xD4
2C2xDic3
3C4wrC2

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 C4wrC2
ρ202-2002-2i2i00-1+i1+i1-i-1-i-2000002i-2i000    complex lifted from C4wrC2
ρ212-20022i-2i00-1-i1-i1+i-1+i-200000-2i2i000    complex lifted from C4wrC2
ρ222-20022i-2i001+i-1+i-1-i1-i-200000-2i2i000    complex lifted from C4wrC2
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
S3xC4wrC2  C42:3D6  C24.100D4  C24.54D4  D8:5Dic3  D8:4Dic3  D12:18D4  D12.38D4  D12.39D4  D12.40D4  (C6xD4):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  (C6xD4):C4  (C6xQ8):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

׿
x
:
Z
F
o
wr
Q
<