metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1Q8, Dic3.7D4, C4⋊C4⋊4S3, C2.6(S3×Q8), D6⋊C4.2C2, (C2×C4).13D6, C2.14(S3×D4), C6.26(C2×D4), C3⋊2(C22⋊Q8), C6.13(C2×Q8), (C2×Dic6)⋊4C2, Dic3⋊C4⋊12C2, 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
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, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊Q8, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, D6⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C4○D12, S3×D4, S3×Q8, D6⋊Q8
Character table of D6⋊Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2i | 0 | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2i | 0 | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | √-3 | -√3 | i | √3 | -√-3 | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | √-3 | √3 | -i | -√3 | -√-3 | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | -√-3 | √3 | i | -√3 | √-3 | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | -√-3 | -√3 | -i | √3 | √-3 | complex lifted from C4○D12 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
►On 48 points
(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 17)(2 16)(3 15)(4 14)(5 13)(6 18)(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 15 22)(2 28 16 21)(3 27 17 20)(4 26 18 19)(5 25 13 24)(6 30 14 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 15 34)(2 40 16 33)(3 39 17 32)(4 38 18 31)(5 37 13 36)(6 42 14 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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(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), (1,29,15,22)(2,28,16,21)(3,27,17,20)(4,26,18,19)(5,25,13,24)(6,30,14,23)(7,37,44,36)(8,42,45,35)(9,41,46,34)(10,40,47,33)(11,39,48,32)(12,38,43,31), (1,41,15,34)(2,40,16,33)(3,39,17,32)(4,38,18,31)(5,37,13,36)(6,42,14,35)(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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(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), (1,29,15,22)(2,28,16,21)(3,27,17,20)(4,26,18,19)(5,25,13,24)(6,30,14,23)(7,37,44,36)(8,42,45,35)(9,41,46,34)(10,40,47,33)(11,39,48,32)(12,38,43,31), (1,41,15,34)(2,40,16,33)(3,39,17,32)(4,38,18,31)(5,37,13,36)(6,42,14,35)(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,18),(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)], [(1,29,15,22),(2,28,16,21),(3,27,17,20),(4,26,18,19),(5,25,13,24),(6,30,14,23),(7,37,44,36),(8,42,45,35),(9,41,46,34),(10,40,47,33),(11,39,48,32),(12,38,43,31)], [(1,41,15,34),(2,40,16,33),(3,39,17,32),(4,38,18,31),(5,37,13,36),(6,42,14,35),(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 C42⋊12D6 C42.93D6 C42.96D6 C42.99D6 C42⋊14D6 Dic6⋊23D4 C42⋊18D6 C42.118D6 C42.122D6 C42.232D6 D12⋊10Q8 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 Dic6⋊21D4 Dic6⋊22D4 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⋊C4⋊28D6 C6.622+ 1+4 C6.632+ 1+4 C6.652+ 1+4 C6.692+ 1+4 C42.236D6 C42.148D6 D12⋊7Q8 C42.150D6 C42.151D6 C42.154D6 C42.157D6 C42.158D6 C42.160D6 C42⋊25D6 C42⋊26D6 C42.189D6 C42.161D6 C42.162D6 C42.164D6 C42.165D6 C42.171D6 D12⋊8Q8 C42.174D6 C42.180D6 D18⋊Q8 D6⋊Dic6 C62.35C23 C62.58C23 C62.65C23 D6⋊3Dic6 D6⋊4Dic6 C62.240C23 D6⋊Dic10 D30⋊8Q8 D30⋊3Q8 D30⋊4Q8 D6⋊3Dic10 D6⋊4Dic10 D30⋊5Q8
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 D12⋊2Q8 D12.2Q8 Dic3⋊(C4⋊C4) (C2×Dic3)⋊Q8 C4⋊C4⋊5Dic3 (C2×Dic3).Q8 D6⋊C4⋊6C4 (C2×C12).290D4 (C2×C12).56D4 D18⋊Q8 D6⋊Dic6 C62.35C23 C62.58C23 C62.65C23 D6⋊3Dic6 D6⋊4Dic6 C62.240C23 D6⋊Dic10 D30⋊8Q8 D30⋊3Q8 D30⋊4Q8 D6⋊3Dic10 D6⋊4Dic10 D30⋊5Q8
Matrix representation of D6⋊Q8 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 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 | 12 | 0 | 0 |
| 0 | 0 | 0 | 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 | 1 |
| 0 | 0 | 0 | 0 | 12 | 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 |
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
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