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G = D6⋊Q8order 96 = 25·3

1st semidirect product of D6 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D61Q8, Dic3.7D4, C4⋊C44S3, C2.6(S3×Q8), D6⋊C4.2C2, (C2×C4).13D6, C2.14(S3×D4), C6.26(C2×D4), C32(C22⋊Q8), C6.13(C2×Q8), (C2×Dic6)⋊4C2, Dic3⋊C412C2, C6.13(C4○D4), (C2×C6).37C23, C2.15(C4○D12), (C2×C12).58C22, C22.51(C22×S3), (C22×S3).21C22, (C2×Dic3).12C22, (C3×C4⋊C4)⋊7C2, (S3×C2×C4).9C2, SmallGroup(96,103)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D6⋊Q8
C1C3C6C2×C6C22×S3S3×C2×C4 — D6⋊Q8
C3C2×C6 — D6⋊Q8
C1C22C4⋊C4

Generators and relations for D6⋊Q8
 G = < a,b,c,d | a6=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 170 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×5], Q8 [×2], C23, Dic3 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×C12 [×3], C22×S3, C22⋊Q8, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, D6⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C4○D12, S3×D4, S3×Q8, D6⋊Q8

Character table of D6⋊Q8

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D12E12F
 size 11116622244661212222444444
ρ1111111111111111111111111    trivial
ρ2111111111-1-111-1-11111-1-11-1-1    linear of order 2
ρ31111-1-11-1-11-1111-1111-11-1-1-11    linear of order 2
ρ41111-1-11-1-1-1111-11111-1-11-11-1    linear of order 2
ρ51111-1-1111-1-1-1-1111111-1-11-1-1    linear of order 2
ρ61111-1-111111-1-1-1-1111111111    linear of order 2
ρ71111111-1-1-11-1-11-1111-1-11-11-1    linear of order 2
ρ81111111-1-11-1-1-1-11111-11-1-1-11    linear of order 2
ρ92-22-20020000-22002-2-2000000    orthogonal lifted from D4
ρ10222200-1-2-2-220000-1-1-111-11-11    orthogonal lifted from D6
ρ112-22-200200002-2002-2-2000000    orthogonal lifted from D4
ρ12222200-122-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ13222200-122220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ14222200-1-2-22-20000-1-1-11-1111-1    orthogonal lifted from D6
ρ152-2-22-22200000000-2-22000000    symplectic lifted from Q8, Schur index 2
ρ162-2-222-2200000000-2-22000000    symplectic lifted from Q8, Schur index 2
ρ1722-2-20022i-2i000000-22-2-2i002i00    complex lifted from C4○D4
ρ1822-2-2002-2i2i000000-22-22i00-2i00    complex lifted from C4○D4
ρ1922-2-200-1-2i2i0000001-11-i-3-3i3--3    complex lifted from C4○D12
ρ2022-2-200-12i-2i0000001-11i-33-i-3--3    complex lifted from C4○D12
ρ2122-2-200-1-2i2i0000001-11-i--33i-3-3    complex lifted from C4○D12
ρ2222-2-200-12i-2i0000001-11i--3-3-i3-3    complex lifted from C4○D12
ρ234-44-400-200000000-222000000    orthogonal lifted from S3×D4
ρ244-4-4400-20000000022-2000000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of D6⋊Q8
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 15)(2 14)(3 13)(4 18)(5 17)(6 16)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)
(1 29 13 22)(2 28 14 21)(3 27 15 20)(4 26 16 19)(5 25 17 24)(6 30 18 23)(7 37 44 36)(8 42 45 35)(9 41 46 34)(10 40 47 33)(11 39 48 32)(12 38 43 31)
(1 41 13 34)(2 40 14 33)(3 39 15 32)(4 38 16 31)(5 37 17 36)(6 42 18 35)(7 24 44 25)(8 23 45 30)(9 22 46 29)(10 21 47 28)(11 20 48 27)(12 19 43 26)

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

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

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

D6⋊Q8 is a maximal subgroup of
C6.2- 1+4  C6.102+ 1+4  C6.62- 1+4  C4212D6  C42.93D6  C42.96D6  C42.99D6  C4214D6  Dic623D4  C4218D6  C42.118D6  C42.122D6  C42.232D6  D1210Q8  C42.133D6  C6.322+ 1+4  C6.402+ 1+4  C6.422+ 1+4  C6.492+ 1+4  S3×C22⋊Q8  C6.172- 1+4  Dic621D4  Dic622D4  C6.512+ 1+4  C6.522+ 1+4  C6.222- 1+4  C6.252- 1+4  C6.592+ 1+4  C6.792- 1+4  C6.1212+ 1+4  C6.822- 1+4  C4⋊C428D6  C6.622+ 1+4  C6.632+ 1+4  C6.652+ 1+4  C6.692+ 1+4  C42.236D6  C42.148D6  D127Q8  C42.150D6  C42.151D6  C42.154D6  C42.157D6  C42.158D6  C42.160D6  C4225D6  C4226D6  C42.189D6  C42.161D6  C42.162D6  C42.164D6  C42.165D6  C42.171D6  D128Q8  C42.174D6  C42.180D6  D18⋊Q8  D6⋊Dic6  C62.35C23  C62.58C23  C62.65C23  D63Dic6  D64Dic6  C62.240C23  D6⋊Dic10  D308Q8  D303Q8  D304Q8  D63Dic10  D64Dic10  D305Q8
D6⋊Q8 is a maximal quotient of
(C2×C12)⋊Q8  C6.(C4×D4)  C6.(C4⋊Q8)  (C2×Dic3).9D4  D6⋊(C4⋊C4)  D6⋊C4⋊C4  (C22×S3)⋊Q8  (C22×C4).37D6  Dic6⋊Q8  Dic6.Q8  D12⋊Q8  D12.Q8  Dic3.Q16  Dic6.2Q8  D122Q8  D12.2Q8  Dic3⋊(C4⋊C4)  (C2×Dic3)⋊Q8  C4⋊C45Dic3  (C2×Dic3).Q8  D6⋊C46C4  (C2×C12).290D4  (C2×C12).56D4  D18⋊Q8  D6⋊Dic6  C62.35C23  C62.58C23  C62.65C23  D63Dic6  D64Dic6  C62.240C23  D6⋊Dic10  D308Q8  D303Q8  D304Q8  D63Dic10  D64Dic10  D305Q8

Matrix representation of D6⋊Q8 in GL6(𝔽13)

1200000
0120000
0001200
0011200
0000120
0000012
,
100000
0120000
0011200
0001200
0000120
000001
,
0120000
1200000
0001200
0012000
000001
0000120
,
100000
010000
000100
001000
000080
000005

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5] >;

D6⋊Q8 in GAP, Magma, Sage, TeX

D_6\rtimes Q_8
% in TeX

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

G:=SmallGroup(96,103);
// by ID

G=gap.SmallGroup(96,103);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,506,188,86,2309]);
// Polycyclic

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

Export

Character table of D6⋊Q8 in TeX

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