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G = Q83D6order 96 = 25·3

2nd semidirect product of Q8 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D6, Q83D6, D246C2, D4.3D6, D6.7D4, C243C22, SD161S3, D122C22, C12.5C23, Dic3.9D4, D4⋊S33C2, (S3×D4)⋊3C2, C3⋊C82C22, C8⋊S31C2, C33(C8⋊C22), C2.19(S3×D4), C6.31(C2×D4), Q83S31C2, Q82S32C2, (C3×SD16)⋊1C2, C4.5(C22×S3), (C3×Q8)⋊2C22, (C4×S3).2C22, (C3×D4).3C22, SmallGroup(96,121)

Series: Derived Chief Lower central Upper central

C1C12 — Q83D6
C1C3C6C12C4×S3S3×D4 — Q83D6
C3C6C12 — Q83D6
C1C2C4SD16

Generators and relations for Q83D6
 G = < a,b,c,d | a4=c6=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 202 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, Dic3, C12, C12, D6, D6 [×4], C2×C6, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, Q83D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C22×S3, C8⋊C22, S3×D4, Q83D6

Character table of Q83D6

 class 12A2B2C2D2E34A4B4C6A6B8A8B12A12B24A24B
 size 114612122246284124844
ρ1111111111111111111    trivial
ρ2111-1-1-1111-1111-11111    linear of order 2
ρ311-1-11-1111-11-1-1111-1-1    linear of order 2
ρ4111-1-1111-1-111-111-1-1-1    linear of order 2
ρ511-1-11111-1-11-11-11-111    linear of order 2
ρ611-11-1111111-1-1-111-1-1    linear of order 2
ρ711111-111-1111-1-11-1-1-1    linear of order 2
ρ811-11-1-111-111-1111-111    linear of order 2
ρ92202002-20-22000-2000    orthogonal lifted from D4
ρ1022-2000-12-20-1120-11-1-1    orthogonal lifted from D6
ρ11222000-12-20-1-1-20-1111    orthogonal lifted from D6
ρ12220-2002-2022000-2000    orthogonal lifted from D4
ρ13222000-1220-1-120-1-1-1-1    orthogonal lifted from S3
ρ1422-2000-1220-11-20-1-111    orthogonal lifted from D6
ρ15440000-2-400-20002000    orthogonal lifted from S3×D4
ρ164-400004000-40000000    orthogonal lifted from C8⋊C22
ρ174-40000-20002000006-6    orthogonal faithful
ρ184-40000-2000200000-66    orthogonal faithful

Permutation representations of Q83D6
On 24 points - transitive group 24T140
Generators in S24
(1 10 4 7)(2 8 5 11)(3 12 6 9)(13 23 20 16)(14 17 21 24)(15 19 22 18)
(1 19 4 18)(2 23 5 16)(3 21 6 14)(7 22 10 15)(8 13 11 20)(9 24 12 17)
(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 3)(4 6)(7 12)(8 11)(9 10)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)

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

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

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

G:=TransitiveGroup(24,140);

Q83D6 is a maximal subgroup of
SD1613D6  SD16⋊D6  D815D6  S3×C8⋊C22  D85D6  D24⋊C22  C24.C23  D72⋊C2  C241D6  C246D6  D12.7D6  D125D6  D126D6  D12.10D6  C247D6  GL2(𝔽3)⋊S3  C401D6  D246D5  D12.9D10  D125D10  D12⋊D10  D60⋊C22  Q83D30
Q83D6 is a maximal quotient of
C4⋊C4.D6  D4.2Dic6  C4⋊C419D6  D4⋊D12  C3⋊C8⋊D4  C241C4⋊C2  D4⋊S3⋊C4  D123D4  Q83Dic6  (C2×C8).D6  Q87(C4×S3)  Q84D12  D6.Q16  C3⋊(C8⋊D4)  Q83(C4×S3)  D12.12D4  C243Q8  C8⋊(C4×S3)  D6.4SD16  C247D4  C4.Q8⋊S3  D249C4  D12⋊Q8  D12.Q8  Dic35SD16  SD16⋊Dic3  (C3×D4).D4  D66SD16  D127D4  C248D4  C249D4  D72⋊C2  C241D6  C246D6  D12.7D6  D125D6  D126D6  D12.10D6  C247D6  C401D6  D246D5  D12.9D10  D125D10  D12⋊D10  D60⋊C22  Q83D30

Matrix representation of Q83D6 in GL4(𝔽5) generated by

0024
0114
4214
4423
,
3303
2411
3424
3341
,
2200
1400
1104
2414
,
4000
4100
4214
3404
G:=sub<GL(4,GF(5))| [0,0,4,4,0,1,2,4,2,1,1,2,4,4,4,3],[3,2,3,3,3,4,4,3,0,1,2,4,3,1,4,1],[2,1,1,2,2,4,1,4,0,0,0,1,0,0,4,4],[4,4,4,3,0,1,2,4,0,0,1,0,0,0,4,4] >;

Q83D6 in GAP, Magma, Sage, TeX

Q_8\rtimes_3D_6
% in TeX

G:=Group("Q8:3D6");
// GroupNames label

G:=SmallGroup(96,121);
// by ID

G=gap.SmallGroup(96,121);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,116,86,297,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^6=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q83D6 in TeX

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