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G = Q83Dic3order 96 = 25·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q83Dic3, D42Dic3, C12.56D4, C33C4≀C2, (C3×D4)⋊2C4, (C3×Q8)⋊2C4, (C2×C6).3D4, C12.9(C2×C4), C4○D4.3S3, (C2×C4).41D6, (C4×Dic3)⋊2C2, C4.Dic34C2, 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

C1C12 — Q83Dic3
C1C3C6C12C2×C12C4.Dic3 — Q83Dic3
C3C6C12 — Q83Dic3
C1C4C2×C4C4○D4

Generators and relations for Q83Dic3
 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 >

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

Character table of Q83Dic3

 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 Q83Dic3
On 24 points - transitive group 24T109
Generators in S24
(1 10 8 4)(2 11 9 5)(3 12 7 6)(13 19 16 22)(14 20 17 23)(15 21 18 24)
(1 19 8 22)(2 23 9 20)(3 21 7 24)(4 16 10 13)(5 14 11 17)(6 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)
(1 2)(4 5)(8 9)(10 11)(13 20 16 23)(14 19 17 22)(15 24 18 21)

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

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

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

G:=TransitiveGroup(24,109);

Q83Dic3 is a maximal subgroup of
S3×C4≀C2  C423D6  C24.100D4  C24.54D4  D85Dic3  D84Dic3  D1218D4  D12.38D4  D12.39D4  D12.40D4  (C6×D4)⋊9C4  2+ 1+46S3  2+ 1+4.4S3  2- 1+44S3  2- 1+4.2S3  Q83Dic9  C12.9S4  D124Dic3  D122Dic3  C62.39D4  C3⋊U2(𝔽3)  C60.96D4  C60.97D4  Q83Dic15  Dic10⋊Dic3  D202Dic3
Q83Dic3 is a maximal quotient of
C12.2C42  C12.57D8  C12.26Q16  (C6×D4)⋊C4  (C6×Q8)⋊C4  C42.7D6  C42.8D6  Q83Dic9  D124Dic3  D122Dic3  C62.39D4  C60.96D4  C60.97D4  Q83Dic15  Dic10⋊Dic3  D202Dic3

Matrix representation of Q83Dic3 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] >;

Q83Dic3 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 Q83Dic3 in TeX
Character table of Q83Dic3 in TeX

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