metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.2D4, D4.5D6, C8.11D6, Q8.7D6, Dic3○SD16, SD16⋊3S3, C12.7C23, C24.11C22, Dic3.13D4, D12.3C22, Dic6.3C22, (S3×C8)⋊5C2, D4⋊S3⋊4C2, C3⋊3(C4○D8), C24⋊C2⋊6C2, C3⋊Q16⋊2C2, C6.33(C2×D4), C2.21(S3×D4), C3⋊C8.6C22, D4⋊2S3⋊3C2, Q8⋊3S3⋊2C2, (C3×SD16)⋊4C2, C4.7(C22×S3), (C3×D4).5C22, (C3×Q8).2C22, (C4×S3).10C22, SmallGroup(96,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.7D6
G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 154 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, SD16, Q16, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C4○D8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, Q8.7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S3×D4, Q8.7D6
Character table of Q8.7D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | |
| size | 1 | 1 | 4 | 6 | 12 | 2 | 2 | 3 | 3 | 4 | 12 | 2 | 8 | 2 | 2 | 6 | 6 | 4 | 8 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 2 | 0 | -1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | -2 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | 0 | -2 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | 0 | 2 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | -2 | 0 | -1 | 1 | 2 | 2 | 0 | 0 | -1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 2 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | √-2 | -√-2 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | √-2 | -√-2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | -√-2 | √-2 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | -√-2 | √-2 | complex lifted from C4○D8 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | complex faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | √-2 | -√-2 | complex faithful, Schur index 2 |
►On 48 points
(1 23 37 33)(2 34 38 24)(3 19 39 35)(4 36 40 20)(5 21 41 31)(6 32 42 22)(7 43 27 13)(8 14 28 44)(9 45 29 15)(10 16 30 46)(11 47 25 17)(12 18 26 48)
(1 43 37 13)(2 28 38 8)(3 45 39 15)(4 30 40 10)(5 47 41 17)(6 26 42 12)(7 33 27 23)(9 35 29 19)(11 31 25 21)(14 24 44 34)(16 20 46 36)(18 22 48 32)
(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 18 37 48)(2 47 38 17)(3 16 39 46)(4 45 40 15)(5 14 41 44)(6 43 42 13)(7 22 27 32)(8 31 28 21)(9 20 29 36)(10 35 30 19)(11 24 25 34)(12 33 26 23)
G:=sub<Sym(48)| (1,23,37,33)(2,34,38,24)(3,19,39,35)(4,36,40,20)(5,21,41,31)(6,32,42,22)(7,43,27,13)(8,14,28,44)(9,45,29,15)(10,16,30,46)(11,47,25,17)(12,18,26,48), (1,43,37,13)(2,28,38,8)(3,45,39,15)(4,30,40,10)(5,47,41,17)(6,26,42,12)(7,33,27,23)(9,35,29,19)(11,31,25,21)(14,24,44,34)(16,20,46,36)(18,22,48,32), (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), (1,18,37,48)(2,47,38,17)(3,16,39,46)(4,45,40,15)(5,14,41,44)(6,43,42,13)(7,22,27,32)(8,31,28,21)(9,20,29,36)(10,35,30,19)(11,24,25,34)(12,33,26,23)>;
G:=Group( (1,23,37,33)(2,34,38,24)(3,19,39,35)(4,36,40,20)(5,21,41,31)(6,32,42,22)(7,43,27,13)(8,14,28,44)(9,45,29,15)(10,16,30,46)(11,47,25,17)(12,18,26,48), (1,43,37,13)(2,28,38,8)(3,45,39,15)(4,30,40,10)(5,47,41,17)(6,26,42,12)(7,33,27,23)(9,35,29,19)(11,31,25,21)(14,24,44,34)(16,20,46,36)(18,22,48,32), (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), (1,18,37,48)(2,47,38,17)(3,16,39,46)(4,45,40,15)(5,14,41,44)(6,43,42,13)(7,22,27,32)(8,31,28,21)(9,20,29,36)(10,35,30,19)(11,24,25,34)(12,33,26,23) );
G=PermutationGroup([[(1,23,37,33),(2,34,38,24),(3,19,39,35),(4,36,40,20),(5,21,41,31),(6,32,42,22),(7,43,27,13),(8,14,28,44),(9,45,29,15),(10,16,30,46),(11,47,25,17),(12,18,26,48)], [(1,43,37,13),(2,28,38,8),(3,45,39,15),(4,30,40,10),(5,47,41,17),(6,26,42,12),(7,33,27,23),(9,35,29,19),(11,31,25,21),(14,24,44,34),(16,20,46,36),(18,22,48,32)], [(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)], [(1,18,37,48),(2,47,38,17),(3,16,39,46),(4,45,40,15),(5,14,41,44),(6,43,42,13),(7,22,27,32),(8,31,28,21),(9,20,29,36),(10,35,30,19),(11,24,25,34),(12,33,26,23)]])
Q8.7D6 is a maximal subgroup of
SD16⋊13D6 S3×C4○D8 D8⋊11D6 D8⋊4D6 D8⋊5D6 D24⋊C22 SD16.D6 SD16⋊3D9 D6.1D12 D12.2D6 Dic6.20D6 D12.8D6 D12.12D6 D12.14D6 C24.40D6 Dic3.4S4 D6.1D20 Dic6.D10 C60.19C23 D30.11D4 D20.27D6 D20.16D6 D4.5D30
Q8.7D6 is a maximal quotient of
Dic3⋊4D8 D4.Dic6 (C2×C8).200D6 D4⋊2S3⋊C4 D6⋊D8 D6⋊C8⋊11C2 D4⋊3D12 D12.D4 Dic3⋊4Q16 Dic6.11D4 Q8.4Dic6 Q8⋊C4⋊S3 C4⋊C4.150D6 Q8.11D12 D6⋊1Q16 C8⋊Dic3⋊C2 Dic3⋊8SD16 Dic6.Q8 C8.8Dic6 (S3×C8)⋊C4 C8⋊8D12 C4.Q8⋊S3 C6.(C4○D8) D12.Q8 Dic3×SD16 (C3×D4).D4 (C3×Q8).D4 C24.43D4 C24⋊14D4 D12⋊7D4 Dic6.16D4 SD16⋊3D9 D6.1D12 D12.2D6 Dic6.20D6 D12.8D6 D12.12D6 D12.14D6 C24.40D6 D6.1D20 Dic6.D10 C60.19C23 D30.11D4 D20.27D6 D20.16D6 D4.5D30
Matrix representation of Q8.7D6 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 16 | 16 | 0 | 0 |
| 16 | 57 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 6 | 67 | 0 | 0 |
| 67 | 67 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,27,0,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,16,57,0,0,0,0,0,72,0,0,1,1],[6,67,0,0,67,67,0,0,0,0,1,0,0,0,72,72] >;
Q8.7D6 in GAP, Magma, Sage, TeX
Q_8._7D_6
% in TeX
G:=Group("Q8.7D6"); // GroupNames label
G:=SmallGroup(96,123);
// by ID
G=gap.SmallGroup(96,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,362,116,86,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
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