metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D6, C12.50D4, Q8.14D6, C12.19C23, Dic6.12C22, D4.S3⋊6C2, C4○D4.4S3, C3⋊Q16⋊6C2, (C2×C6).10D4, (C2×C4).23D6, C6.61(C2×D4), C3⋊C8.4C22, C3⋊5(C8.C22), (C2×Dic6)⋊11C2, C4.25(C3⋊D4), C4.Dic3⋊10C2, C4.19(C22×S3), (C3×D4).9C22, (C3×Q8).9C22, (C2×C12).44C22, C22.6(C3⋊D4), (C3×C4○D4).3C2, C2.25(C2×C3⋊D4), SmallGroup(96,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.14D6
G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c-1 >
Subgroups: 122 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, Q8.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, C2×C3⋊D4, Q8.14D6
Character table of Q8.14D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 12 | 12 | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 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 | 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 | -2 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 0 | 0 | -1 | -1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 1 | 1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 0 | 0 | -1 | -1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 1 | 1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
(1 30 19 7)(2 25 20 8)(3 26 21 9)(4 27 22 10)(5 28 23 11)(6 29 24 12)(13 32 42 47)(14 33 37 48)(15 34 38 43)(16 35 39 44)(17 36 40 45)(18 31 41 46)
(1 22 19 4)(2 5 20 23)(3 24 21 6)(7 10 30 27)(8 28 25 11)(9 12 26 29)(13 44 42 35)(14 36 37 45)(15 46 38 31)(16 32 39 47)(17 48 40 33)(18 34 41 43)
(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 46 19 31)(2 45 20 36)(3 44 21 35)(4 43 22 34)(5 48 23 33)(6 47 24 32)(7 18 30 41)(8 17 25 40)(9 16 26 39)(10 15 27 38)(11 14 28 37)(12 13 29 42)
G:=sub<Sym(48)| (1,30,19,7)(2,25,20,8)(3,26,21,9)(4,27,22,10)(5,28,23,11)(6,29,24,12)(13,32,42,47)(14,33,37,48)(15,34,38,43)(16,35,39,44)(17,36,40,45)(18,31,41,46), (1,22,19,4)(2,5,20,23)(3,24,21,6)(7,10,30,27)(8,28,25,11)(9,12,26,29)(13,44,42,35)(14,36,37,45)(15,46,38,31)(16,32,39,47)(17,48,40,33)(18,34,41,43), (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), (1,46,19,31)(2,45,20,36)(3,44,21,35)(4,43,22,34)(5,48,23,33)(6,47,24,32)(7,18,30,41)(8,17,25,40)(9,16,26,39)(10,15,27,38)(11,14,28,37)(12,13,29,42)>;
G:=Group( (1,30,19,7)(2,25,20,8)(3,26,21,9)(4,27,22,10)(5,28,23,11)(6,29,24,12)(13,32,42,47)(14,33,37,48)(15,34,38,43)(16,35,39,44)(17,36,40,45)(18,31,41,46), (1,22,19,4)(2,5,20,23)(3,24,21,6)(7,10,30,27)(8,28,25,11)(9,12,26,29)(13,44,42,35)(14,36,37,45)(15,46,38,31)(16,32,39,47)(17,48,40,33)(18,34,41,43), (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), (1,46,19,31)(2,45,20,36)(3,44,21,35)(4,43,22,34)(5,48,23,33)(6,47,24,32)(7,18,30,41)(8,17,25,40)(9,16,26,39)(10,15,27,38)(11,14,28,37)(12,13,29,42) );
G=PermutationGroup([[(1,30,19,7),(2,25,20,8),(3,26,21,9),(4,27,22,10),(5,28,23,11),(6,29,24,12),(13,32,42,47),(14,33,37,48),(15,34,38,43),(16,35,39,44),(17,36,40,45),(18,31,41,46)], [(1,22,19,4),(2,5,20,23),(3,24,21,6),(7,10,30,27),(8,28,25,11),(9,12,26,29),(13,44,42,35),(14,36,37,45),(15,46,38,31),(16,32,39,47),(17,48,40,33),(18,34,41,43)], [(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)], [(1,46,19,31),(2,45,20,36),(3,44,21,35),(4,43,22,34),(5,48,23,33),(6,47,24,32),(7,18,30,41),(8,17,25,40),(9,16,26,39),(10,15,27,38),(11,14,28,37),(12,13,29,42)]])
Q8.14D6 is a maximal subgroup of
C42⋊5D6 Q8.14D12 Q8.8D12 Q8.10D12 M4(2).13D6 M4(2).16D6 2+ 1+4.4S3 2- 1+4.2S3 D8⋊11D6 D8.10D6 D8⋊4D6 S3×C8.C22 C12.C24 D12.33C23 D12.35C23 D4.D18 C12.3S4 D12.32D6 D12.29D6 Dic6.19D6 D12.11D6 C62.75D4 C12.6S4 D20.37D6 C60.63D4 C60.8C23 D20.13D6 D4.9D30
Q8.14D6 is a maximal quotient of
C4⋊C4.232D6 C4⋊C4.233D6 C4⋊C4.237D6 C12.50D8 C42.51D6 D4.2D12 Q8⋊4Dic6 C42.59D6 C12⋊7Q16 (C2×C6).D8 Dic6⋊17D4 C3⋊C8⋊5D4 (C2×Q8).49D6 Dic6.37D4 C3⋊C8.6D4 C42.61D6 C42.62D6 C42.65D6 Dic6.4Q8 C42.68D6 C42.71D6 C4○D4⋊3Dic3 (C3×D4).32D4 D4.D18 D12.32D6 D12.29D6 Dic6.19D6 D12.11D6 C62.75D4 D20.37D6 C60.63D4 C60.8C23 D20.13D6 D4.9D30
Matrix representation of Q8.14D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 29 | 72 | 0 | 0 |
| 0 | 0 | 32 | 14 | 1 | 71 |
| 0 | 0 | 16 | 14 | 1 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 51 | 3 | 0 |
| 0 | 0 | 0 | 46 | 14 | 59 |
| 0 | 0 | 23 | 13 | 2 | 19 |
| 0 | 0 | 23 | 13 | 48 | 46 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 71 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 51 | 3 | 0 |
| 0 | 0 | 48 | 46 | 0 | 14 |
| 0 | 0 | 35 | 13 | 21 | 54 |
| 0 | 0 | 29 | 13 | 21 | 27 |
| 72 | 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 14 | 0 | 36 |
| 0 | 0 | 69 | 41 | 11 | 0 |
| 0 | 0 | 56 | 38 | 32 | 33 |
| 0 | 0 | 24 | 5 | 16 | 33 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,29,32,16,0,0,10,72,14,14,0,0,0,0,1,1,0,0,0,0,71,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,52,0,23,23,0,0,51,46,13,13,0,0,3,14,2,48,0,0,0,59,19,46],[8,71,0,0,0,0,0,64,0,0,0,0,0,0,52,48,35,29,0,0,51,46,13,13,0,0,3,0,21,21,0,0,0,14,54,27],[72,0,0,0,0,0,28,1,0,0,0,0,0,0,40,69,56,24,0,0,14,41,38,5,0,0,0,11,32,16,0,0,36,0,33,33] >;
Q8.14D6 in GAP, Magma, Sage, TeX
Q_8._{14}D_6 % in TeX
G:=Group("Q8.14D6"); // GroupNames label
G:=SmallGroup(96,158);
// by ID
G=gap.SmallGroup(96,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,188,579,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export