metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.15D6, C6.10C24, D6.5C23, C3⋊12- 1+4, C12.24C23, D12.13C22, Dic3.6C23, Dic6.13C22, (C2×Q8)⋊7S3, (C6×Q8)⋊7C2, (S3×Q8)⋊4C2, Q8○(C3⋊D4), C4○D12⋊6C2, (C2×C4).24D6, Q8⋊3S3⋊4C2, (C4×S3).5C22, C2.11(S3×C23), (C2×C6).68C23, C4.24(C22×S3), C3⋊D4.2C22, (C2×C12).48C22, C22.7(C22×S3), (C3×Q8).10C22, SmallGroup(96,214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.15D6
G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c5 >
Subgroups: 274 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, Dic3, C12, D6, C2×C6, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, 2- 1+4, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, Q8.15D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S3×C23, Q8.15D6
Character table of Q8.15D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | 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 | -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 | 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 | -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 | -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 | 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 | -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 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -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 |
| ρ10 | 1 | 1 | 1 | -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 |
| ρ11 | 1 | 1 | -1 | 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 |
| ρ12 | 1 | 1 | 1 | 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 |
| ρ13 | 1 | 1 | -1 | 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 |
| ρ14 | 1 | 1 | 1 | 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 |
| ρ15 | 1 | 1 | -1 | -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 |
| ρ16 | 1 | 1 | 1 | -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 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ23 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ24 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
(1 42 7 48)(2 43 8 37)(3 44 9 38)(4 45 10 39)(5 46 11 40)(6 47 12 41)(13 25 19 31)(14 26 20 32)(15 27 21 33)(16 28 22 34)(17 29 23 35)(18 30 24 36)
(1 26 7 32)(2 33 8 27)(3 28 9 34)(4 35 10 29)(5 30 11 36)(6 25 12 31)(13 41 19 47)(14 48 20 42)(15 43 21 37)(16 38 22 44)(17 45 23 39)(18 40 24 46)
(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 19 7 13)(2 24 8 18)(3 17 9 23)(4 22 10 16)(5 15 11 21)(6 20 12 14)(25 48 31 42)(26 41 32 47)(27 46 33 40)(28 39 34 45)(29 44 35 38)(30 37 36 43)
G:=sub<Sym(48)| (1,42,7,48)(2,43,8,37)(3,44,9,38)(4,45,10,39)(5,46,11,40)(6,47,12,41)(13,25,19,31)(14,26,20,32)(15,27,21,33)(16,28,22,34)(17,29,23,35)(18,30,24,36), (1,26,7,32)(2,33,8,27)(3,28,9,34)(4,35,10,29)(5,30,11,36)(6,25,12,31)(13,41,19,47)(14,48,20,42)(15,43,21,37)(16,38,22,44)(17,45,23,39)(18,40,24,46), (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), (1,19,7,13)(2,24,8,18)(3,17,9,23)(4,22,10,16)(5,15,11,21)(6,20,12,14)(25,48,31,42)(26,41,32,47)(27,46,33,40)(28,39,34,45)(29,44,35,38)(30,37,36,43)>;
G:=Group( (1,42,7,48)(2,43,8,37)(3,44,9,38)(4,45,10,39)(5,46,11,40)(6,47,12,41)(13,25,19,31)(14,26,20,32)(15,27,21,33)(16,28,22,34)(17,29,23,35)(18,30,24,36), (1,26,7,32)(2,33,8,27)(3,28,9,34)(4,35,10,29)(5,30,11,36)(6,25,12,31)(13,41,19,47)(14,48,20,42)(15,43,21,37)(16,38,22,44)(17,45,23,39)(18,40,24,46), (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), (1,19,7,13)(2,24,8,18)(3,17,9,23)(4,22,10,16)(5,15,11,21)(6,20,12,14)(25,48,31,42)(26,41,32,47)(27,46,33,40)(28,39,34,45)(29,44,35,38)(30,37,36,43) );
G=PermutationGroup([[(1,42,7,48),(2,43,8,37),(3,44,9,38),(4,45,10,39),(5,46,11,40),(6,47,12,41),(13,25,19,31),(14,26,20,32),(15,27,21,33),(16,28,22,34),(17,29,23,35),(18,30,24,36)], [(1,26,7,32),(2,33,8,27),(3,28,9,34),(4,35,10,29),(5,30,11,36),(6,25,12,31),(13,41,19,47),(14,48,20,42),(15,43,21,37),(16,38,22,44),(17,45,23,39),(18,40,24,46)], [(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)], [(1,19,7,13),(2,24,8,18),(3,17,9,23),(4,22,10,16),(5,15,11,21),(6,20,12,14),(25,48,31,42),(26,41,32,47),(27,46,33,40),(28,39,34,45),(29,44,35,38),(30,37,36,43)]])
Q8.15D6 is a maximal subgroup of
D12.4D4 D12.5D4 D12.14D4 D12.15D4 SD16⋊13D6 D12.30D4 C24.C23 SD16.D6 C6.C25 S3×2- 1+4 D12.39C23 Q8.15D18 SL2(𝔽3).11D6 D12.33D6 D12.25D6 Dic6.26D6 C32⋊72- 1+4 C30.C24 C30.33C24 D12.29D10 Q8.15D30
Q8.15D6 is a maximal quotient of
C6.72+ 1+4 C6.82+ 1+4 C6.2- 1+4 C6.2+ 1+4 C6.52- 1+4 C6.62- 1+4 C42.122D6 Q8⋊6Dic6 C42.125D6 C42.126D6 Q8⋊6D12 C42.132D6 C42.133D6 C42.134D6 C6.152- 1+4 C6.162- 1+4 C6.172- 1+4 D12⋊22D4 Dic6⋊22D4 C6.202- 1+4 C6.212- 1+4 C6.222- 1+4 C6.232- 1+4 C6.242- 1+4 C6.252- 1+4 C6.592+ 1+4 C42.147D6 C42.150D6 C42.151D6 C42.154D6 C42.157D6 C42.158D6 Dic6⋊8Q8 C42.171D6 D12⋊12D4 D12⋊8Q8 C42.174D6 C42.176D6 C42.177D6 C42.178D6 C42.180D6 C6.422- 1+4 Q8×C3⋊D4 C6.442- 1+4 C6.452- 1+4 Q8.15D18 D12.33D6 D12.25D6 Dic6.26D6 C32⋊72- 1+4 C30.C24 C30.33C24 D12.29D10 Q8.15D30
Matrix representation of Q8.15D6 ►in GL4(𝔽7) generated by
| 4 | 3 | 6 | 6 |
| 0 | 5 | 4 | 5 |
| 1 | 0 | 4 | 3 |
| 2 | 6 | 4 | 1 |
| 6 | 3 | 6 | 4 |
| 5 | 3 | 1 | 6 |
| 2 | 2 | 6 | 3 |
| 5 | 6 | 4 | 6 |
| 3 | 6 | 2 | 4 |
| 2 | 0 | 3 | 6 |
| 2 | 1 | 2 | 2 |
| 1 | 2 | 5 | 2 |
| 3 | 0 | 0 | 2 |
| 3 | 3 | 2 | 5 |
| 0 | 2 | 4 | 4 |
| 2 | 0 | 0 | 4 |
G:=sub<GL(4,GF(7))| [4,0,1,2,3,5,0,6,6,4,4,4,6,5,3,1],[6,5,2,5,3,3,2,6,6,1,6,4,4,6,3,6],[3,2,2,1,6,0,1,2,2,3,2,5,4,6,2,2],[3,3,0,2,0,3,2,0,0,2,4,0,2,5,4,4] >;
Q8.15D6 in GAP, Magma, Sage, TeX
Q_8._{15}D_6 % in TeX
G:=Group("Q8.15D6"); // GroupNames label
G:=SmallGroup(96,214);
// by ID
G=gap.SmallGroup(96,214);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,86,579,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations
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