p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8⋊5D4, C23.45C23, C22.42C24, C42.44C22, C2.102- 1+4, (C4×D4)⋊17C2, (C4×Q8)⋊11C2, C4.38(C2×D4), C4⋊D4⋊13C2, Q8○2(C22⋊C4), C22⋊Q8⋊13C2, (C22×Q8)⋊6C2, C4.4D4⋊11C2, C22⋊3(C4○D4), C4⋊C4.74C22, (C2×C4).29C23, C2.20(C22×D4), (C2×D4).68C22, C22⋊C4.6C22, (C2×Q8).62C22, (C22×C4).69C22, C22⋊C4○(C2×Q8), (C2×C4○D4)⋊8C2, C2.22(C2×C4○D4), SmallGroup(64,229)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊5D4
G = < a,b,c,d | a4=c4=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 213 in 145 conjugacy classes, 83 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Q8⋊5D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4
Character table of Q8⋊5D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| 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 | symplectic lifted from 2- 1+4, Schur index 2 |
►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 24 3 22)(2 23 4 21)(5 14 7 16)(6 13 8 15)(9 27 11 25)(10 26 12 28)(17 31 19 29)(18 30 20 32)
(1 5 12 30)(2 6 9 31)(3 7 10 32)(4 8 11 29)(13 25 19 21)(14 26 20 22)(15 27 17 23)(16 28 18 24)
(1 26)(2 27)(3 28)(4 25)(5 14)(6 15)(7 16)(8 13)(9 23)(10 24)(11 21)(12 22)(17 31)(18 32)(19 29)(20 30)
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), (1,24,3,22)(2,23,4,21)(5,14,7,16)(6,13,8,15)(9,27,11,25)(10,26,12,28)(17,31,19,29)(18,30,20,32), (1,5,12,30)(2,6,9,31)(3,7,10,32)(4,8,11,29)(13,25,19,21)(14,26,20,22)(15,27,17,23)(16,28,18,24), (1,26)(2,27)(3,28)(4,25)(5,14)(6,15)(7,16)(8,13)(9,23)(10,24)(11,21)(12,22)(17,31)(18,32)(19,29)(20,30)>;
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), (1,24,3,22)(2,23,4,21)(5,14,7,16)(6,13,8,15)(9,27,11,25)(10,26,12,28)(17,31,19,29)(18,30,20,32), (1,5,12,30)(2,6,9,31)(3,7,10,32)(4,8,11,29)(13,25,19,21)(14,26,20,22)(15,27,17,23)(16,28,18,24), (1,26)(2,27)(3,28)(4,25)(5,14)(6,15)(7,16)(8,13)(9,23)(10,24)(11,21)(12,22)(17,31)(18,32)(19,29)(20,30) );
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)], [(1,24,3,22),(2,23,4,21),(5,14,7,16),(6,13,8,15),(9,27,11,25),(10,26,12,28),(17,31,19,29),(18,30,20,32)], [(1,5,12,30),(2,6,9,31),(3,7,10,32),(4,8,11,29),(13,25,19,21),(14,26,20,22),(15,27,17,23),(16,28,18,24)], [(1,26),(2,27),(3,28),(4,25),(5,14),(6,15),(7,16),(8,13),(9,23),(10,24),(11,21),(12,22),(17,31),(18,32),(19,29),(20,30)]])
Q8⋊5D4 is a maximal subgroup of
SD16⋊D4 SD16⋊8D4 Q16⋊9D4 Q16⋊10D4 SD16⋊2D4 Q16⋊4D4 SD16⋊10D4 Q16⋊12D4 C42.465C23 C42.466C23 C42.43C23 C42.48C23 C42.51C23 C42.56C23 C42.475C23 C42.476C23 D4×C4○D4 C22.69C25 C22.75C25 C22.76C25 C22.77C25 C22.78C25 C4⋊2- 1+4 C22.89C25 C22.94C25 C22.103C25 C22.105C25 C23.144C24 C22.113C25 C22.125C25 C22.129C25 C22.130C25 C22.140C25 C22.147C25 C22.150C25 C22.151C25 SL2(𝔽3)⋊5D4 Q8⋊4S4
C2p.2- 1+4: C22.71C25 C22.100C25 C22.107C25 C22.111C25 C22.136C25 C22.141C25 C22.143C25 Dic6⋊23D4 ...
Q8⋊5D4 is a maximal quotient of
Q8×C22⋊C4 C23.223C24 C23.233C24 C23.234C24 C24.215C23 C23.244C24 C24.220C23 C24.244C23 C23.309C24 C23.315C24 C23.316C24 C23.321C24 C24.259C23 C23.327C24 C23.329C24 C24.264C23 C24.565C23 C24.267C23 C24.569C23 C24.269C23 C23.346C24 C23.348C24 C23.352C24 C23.353C24 C24.279C23 C23.360C24 C24.282C23 C24.283C23 C23.369C24 C23.375C24 C23.377C24 C24.573C23 C24.301C23 C23.391C24 C23.419C24 C42.165D4 C42⋊19D4 C42.168D4 C42⋊6Q8 C23.458C24 C24.332C23 C23.574C24 C23.576C24 C23.581C24 C23.590C24 C24.403C23 C23.602C24 C23.607C24 C23.611C24 C23.616C24 C23.619C24 C24.418C23 C23.625C24 C24.420C23 C24.421C23 C23.630C24 C23.631C24
Q8⋊D4p: Q8⋊4D8 Q8⋊6D12 Q8⋊5D20 Q8⋊5D28 ...
C2p.2- 1+4: Q8⋊7SD16 C42.501C23 C42.502C23 Q8⋊8SD16 Q8⋊5Q16 C42.505C23 C42.506C23 C42.507C23 ...
Matrix representation of Q8⋊5D4 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
| 3 | 2 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
| 4 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0],[3,0,0,0,2,2,0,0,0,0,0,2,0,0,3,0],[4,3,0,0,0,1,0,0,0,0,0,2,0,0,3,0] >;
Q8⋊5D4 in GAP, Magma, Sage, TeX
Q_8\rtimes_5D_4
% in TeX
G:=Group("Q8:5D4"); // GroupNames label
G:=SmallGroup(64,229);
// by ID
G=gap.SmallGroup(64,229);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,297,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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