p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4:3Q8, C42.49C22, C23.48C23, C22.48C24, C2.172+ 1+4, D4o3(C4:C4), C4:Q8:15C2, (C4xQ8):15C2, C4.17(C2xQ8), (C4xD4).12C2, C22:Q8:17C2, C4.37(C4oD4), C22.6(C2xQ8), C4:C4.37C22, (C2xC4).31C23, C42.C2:10C2, C2.10(C22xQ8), (C2xD4).81C22, (C2xQ8).32C22, C22:C4.23C22, (C22xC4).73C22, (C2xC4:C4):23C2, C2.27(C2xC4oD4), SmallGroup(64,235)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4:3Q8
G = < a,b,c,d | a4=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 157 in 114 conjugacy classes, 83 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C2xQ8, C2xC4:C4, C4xD4, C4xD4, C4xQ8, C22:Q8, C42.C2, C4:Q8, D4:3Q8
Quotients: C1, C2, C22, Q8, C23, C2xQ8, C4oD4, C24, C22xQ8, C2xC4oD4, 2+ 1+4, D4:3Q8
Character table of D4:3Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
►On 32 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)
(1 13)(2 16)(3 15)(4 14)(5 17)(6 20)(7 19)(8 18)(9 27)(10 26)(11 25)(12 28)(21 32)(22 31)(23 30)(24 29)
(1 20 13 6)(2 17 14 7)(3 18 15 8)(4 19 16 5)(9 23 25 32)(10 24 26 29)(11 21 27 30)(12 22 28 31)
(1 32 13 23)(2 29 14 24)(3 30 15 21)(4 31 16 22)(5 12 19 28)(6 9 20 25)(7 10 17 26)(8 11 18 27)
G:=sub<Sym(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), (1,13)(2,16)(3,15)(4,14)(5,17)(6,20)(7,19)(8,18)(9,27)(10,26)(11,25)(12,28)(21,32)(22,31)(23,30)(24,29), (1,20,13,6)(2,17,14,7)(3,18,15,8)(4,19,16,5)(9,23,25,32)(10,24,26,29)(11,21,27,30)(12,22,28,31), (1,32,13,23)(2,29,14,24)(3,30,15,21)(4,31,16,22)(5,12,19,28)(6,9,20,25)(7,10,17,26)(8,11,18,27)>;
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), (1,13)(2,16)(3,15)(4,14)(5,17)(6,20)(7,19)(8,18)(9,27)(10,26)(11,25)(12,28)(21,32)(22,31)(23,30)(24,29), (1,20,13,6)(2,17,14,7)(3,18,15,8)(4,19,16,5)(9,23,25,32)(10,24,26,29)(11,21,27,30)(12,22,28,31), (1,32,13,23)(2,29,14,24)(3,30,15,21)(4,31,16,22)(5,12,19,28)(6,9,20,25)(7,10,17,26)(8,11,18,27) );
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)], [(1,13),(2,16),(3,15),(4,14),(5,17),(6,20),(7,19),(8,18),(9,27),(10,26),(11,25),(12,28),(21,32),(22,31),(23,30),(24,29)], [(1,20,13,6),(2,17,14,7),(3,18,15,8),(4,19,16,5),(9,23,25,32),(10,24,26,29),(11,21,27,30),(12,22,28,31)], [(1,32,13,23),(2,29,14,24),(3,30,15,21),(4,31,16,22),(5,12,19,28),(6,9,20,25),(7,10,17,26),(8,11,18,27)]])
D4:3Q8 is a maximal subgroup of
D4:SD16 D4:Q16 D4.5SD16 D4:3Q16 D4:9SD16 D4:6Q16 C42.58C23 C42.60C23 C42.497C23 C42.498C23 C42.501C23 C42.502C23 C42.507C23 C42.511C23 C42.513C23 C42.517C23 SD16:4Q8 SD16:Q8 SD16:3Q8 C22.64C25 Q8xC4oD4 C22.81C25 C22.84C25 C22.93C25 C22.96C25 C22.102C25 C22.104C25 C22.105C25 C22.110C25 C22.111C25 C22.124C25 C22.133C25 C22.142C25 C22.144C25 C22.146C25 C22.148C25 C22.152C25 C22.153C25 C22.154C25
D4p:Q8: D8:6Q8 D8:4Q8 D8:5Q8 D12:10Q8 D12:7Q8 D12:9Q8 D20:10Q8 D20:7Q8 ...
C2p.2+ 1+4: D4:7SD16 D4:5Q16 C42.49C23 C42.51C23 C42.480C23 C42.482C23 C22.70C25 C22.90C25 ...
D4:3Q8 is a maximal quotient of
C23.227C24 D4xC4:C4 C23.231C24 C23.237C24 C23.250C24 C23.251C24 C23.252C24 C24.252C23 C23.323C24 C23.349C24 C23.350C24 C23.352C24 C24.300C23 C23.397C24 C23.402C24 C23.405C24 C23.407C24 C23.408C24 C23.409C24 C23.411C24 C23.420C24 C23.422C24 C23.449C24 C24.583C23 C24.338C23 C23.479C24 C23.483C24 C23.485C24 C23.486C24 C24.345C23 C23.488C24 C24.346C23 C23.490C24 C23.592C24 C24.421C23 C23.632C24 C24.428C23 C24.434C23 C23.655C24 C23.663C24 C23.668C24 C23.674C24 C24.448C23 C24.450C23 C23.688C24 C24.454C23 C23.691C24 C23.692C24 C23.702C24 C24.456C23 C23.705C24 C23.706C24 C23.707C24
C42.D2p: C42.166D4 C42.173D4 C42.175D4 C42.176D4 C42.178D4 C42.181D4 D4:5Dic6 D4:6Dic6 ...
C4:C4.D2p: C24.558C23 C24.568C23 C23.354C24 C24.285C23 C24.572C23 C23.392C24 C42:6Q8 C42.35Q8 ...
Matrix representation of D4:3Q8 ►in GL4(F5) generated by
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,3,4,0,0,0,2],[0,2,0,0,3,0,0,0,0,0,2,0,0,0,2,3] >;
D4:3Q8 in GAP, Magma, Sage, TeX
D_4\rtimes_3Q_8
% in TeX
G:=Group("D4:3Q8"); // GroupNames label
G:=SmallGroup(64,235);
// by ID
G=gap.SmallGroup(64,235);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,295,650,297,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
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