metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊3Q8, C12.22D4, (C6×Q8)⋊3C2, (C2×Q8)⋊5S3, C2.9(S3×Q8), D6⋊C4.6C2, C6.57(C2×D4), (C2×C4).21D6, C3⋊5(C22⋊Q8), C6.17(C2×Q8), C4⋊Dic3⋊15C2, Dic3⋊C4⋊16C2, C6.36(C4○D4), C4.18(C3⋊D4), (C2×C6).58C23, (C2×C12).64C22, C2.8(Q8⋊3S3), C22.64(C22×S3), (C22×S3).27C22, (C2×Dic3).21C22, (S3×C2×C4).5C2, C2.21(C2×C3⋊D4), SmallGroup(96,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊3Q8
G = < a,b,c,d | a6=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 162 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊Q8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, D6⋊3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, S3×Q8, Q8⋊3S3, C2×C3⋊D4, D6⋊3Q8
Character table of D6⋊3Q8
| 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 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 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 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| 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 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√-3 | -√-3 | 1 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √-3 | √-3 | 1 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| 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 Q8⋊3S3, Schur index 2 |
| 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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 46)(8 45)(9 44)(10 43)(11 48)(12 47)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(31 39)(32 38)(33 37)(34 42)(35 41)(36 40)
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 34 46 39)(8 35 47 40)(9 36 48 41)(10 31 43 42)(11 32 44 37)(12 33 45 38)
(1 41 17 36)(2 42 18 31)(3 37 13 32)(4 38 14 33)(5 39 15 34)(6 40 16 35)(7 27 46 22)(8 28 47 23)(9 29 48 24)(10 30 43 19)(11 25 44 20)(12 26 45 21)
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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,46)(8,45)(9,44)(10,43)(11,48)(12,47)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(31,39)(32,38)(33,37)(34,42)(35,41)(36,40), (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,34,46,39)(8,35,47,40)(9,36,48,41)(10,31,43,42)(11,32,44,37)(12,33,45,38), (1,41,17,36)(2,42,18,31)(3,37,13,32)(4,38,14,33)(5,39,15,34)(6,40,16,35)(7,27,46,22)(8,28,47,23)(9,29,48,24)(10,30,43,19)(11,25,44,20)(12,26,45,21)>;
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), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,46)(8,45)(9,44)(10,43)(11,48)(12,47)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(31,39)(32,38)(33,37)(34,42)(35,41)(36,40), (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,34,46,39)(8,35,47,40)(9,36,48,41)(10,31,43,42)(11,32,44,37)(12,33,45,38), (1,41,17,36)(2,42,18,31)(3,37,13,32)(4,38,14,33)(5,39,15,34)(6,40,16,35)(7,27,46,22)(8,28,47,23)(9,29,48,24)(10,30,43,19)(11,25,44,20)(12,26,45,21) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,46),(8,45),(9,44),(10,43),(11,48),(12,47),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(31,39),(32,38),(33,37),(34,42),(35,41),(36,40)], [(1,29,17,24),(2,30,18,19),(3,25,13,20),(4,26,14,21),(5,27,15,22),(6,28,16,23),(7,34,46,39),(8,35,47,40),(9,36,48,41),(10,31,43,42),(11,32,44,37),(12,33,45,38)], [(1,41,17,36),(2,42,18,31),(3,37,13,32),(4,38,14,33),(5,39,15,34),(6,40,16,35),(7,27,46,22),(8,28,47,23),(9,29,48,24),(10,30,43,19),(11,25,44,20),(12,26,45,21)]])
D6⋊3Q8 is a maximal subgroup of
D6.1SD16 D6⋊2SD16 D6.Q16 C3⋊(C8⋊D4) D6⋊1Q16 D6⋊C8.C2 C8⋊Dic3⋊C2 C3⋊C8.D4 D6⋊6SD16 C24⋊14D4 Dic6.16D4 C24⋊8D4 D6⋊5Q16 D12.17D4 D6⋊3Q16 C24.36D4 C42.232D6 D12⋊10Q8 C42.131D6 C42.132D6 C42.133D6 C42.134D6 C42.135D6 S3×C22⋊Q8 C4⋊C4⋊26D6 C6.162- 1+4 C6.172- 1+4 C6.512+ 1+4 C6.1182+ 1+4 C6.522+ 1+4 C6.532+ 1+4 C6.202- 1+4 C6.212- 1+4 C6.232- 1+4 C6.772- 1+4 C6.782- 1+4 C6.252- 1+4 C6.592+ 1+4 C42.137D6 D12⋊10D4 Dic6⋊10D4 C42⋊22D6 C42⋊23D6 C42.234D6 C42.144D6 C42.145D6 D12⋊12D4 D12⋊8Q8 C42.241D6 C42.174D6 D12⋊9Q8 C42.176D6 C42.178D6 C42.180D6 Q8×C3⋊D4 C6.442- 1+4 C6.452- 1+4 C6.1042- 1+4 (C2×D4)⋊43D6 C6.1452+ 1+4 C6.1082- 1+4 D18⋊3Q8 D6⋊6Dic6 C12.30D12 D6⋊1Dic6 C62.58C23 C62.261C23 C60.46D4 D30⋊10Q8 D6⋊1Dic10 D30⋊Q8 D30⋊7Q8
D6⋊3Q8 is a maximal quotient of
C12⋊(C4⋊C4) C6.67(C4×D4) (C2×C12).54D4 (C2×C12).288D4 C4⋊(D6⋊C4) D6⋊C4⋊6C4 (C2×C12).290D4 (C2×C12).56D4 Dic6.4Q8 D12.4Q8 D12⋊5Q8 D12⋊6Q8 Dic6⋊5Q8 Dic6⋊6Q8 (C6×Q8)⋊7C4 C22.52(S3×Q8) (C22×Q8)⋊9S3 D18⋊3Q8 D6⋊6Dic6 C12.30D12 D6⋊1Dic6 C62.58C23 C62.261C23 C60.46D4 D30⋊10Q8 D6⋊1Dic10 D30⋊Q8 D30⋊7Q8
Matrix representation of D6⋊3Q8 ►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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 12 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 3 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 9 | 8 |
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],[12,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,7,0,0,0,0,0,12],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,9,0,0,0,0,0,8] >;
D6⋊3Q8 in GAP, Magma, Sage, TeX
D_6\rtimes_3Q_8
% in TeX
G:=Group("D6:3Q8"); // GroupNames label
G:=SmallGroup(96,153);
// by ID
G=gap.SmallGroup(96,153);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,86,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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