p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: D8.13D4, Q16.12D4, SD16.2D4, C42.40C23, M4(2).14C23, 2- 1+4⋊3C22, C22.22+ 1+4, 2+ 1+4.6C22, C2.73D42, Q8○D8⋊2C2, C8.36(C2×D4), C4≀C2⋊6C22, C8.26D4⋊5C2, D4○SD16⋊2C2, D4.32(C2×D4), C8○D4⋊3C22, C4⋊Q8⋊18C22, Q8.32(C2×D4), D4.5D4⋊5C2, D4.3D4⋊4C2, D4.9D4⋊6C2, D4.8D4⋊6C2, C8.2D4⋊20C2, (C2×C4).22C24, (C2×C8).93C23, (C2×D4).8C23, C8⋊C4⋊20C22, D4.10D4⋊5C2, (C2×Q8).6C23, C8.C4⋊6C22, (C2×Q16)⋊30C22, C4○D4.11C23, C4○D8.10C22, C4.103(C22×D4), C8⋊C22.1C22, C8.C22⋊2C22, C4.10D4⋊3C22, C4.D4.6C22, (C2×SD16).51C22, C4.4D4.57C22, SmallGroup(128,2021)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8.13D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=dad-1=a-1, cac-1=a3, cbc-1=a2b, dbd-1=a6b, dcd-1=c3 >
Subgroups: 444 in 224 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C8⋊C4, C4.D4, C4.10D4, C4.10D4, C4≀C2, C8.C4, C4.4D4, C4⋊Q8, C8○D4, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, 2- 1+4, C8.26D4, D4.8D4, D4.9D4, D4.10D4, D4.3D4, D4.5D4, C8.2D4, D4○SD16, Q8○D8, D8.13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8.13D4
Character table of D8.13D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 0 | -2 | 0 | -2 | 0 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | -2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, 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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 22)(18 21)(19 20)(23 24)(25 26)(27 32)(28 31)(29 30)
(1 30 24 13 5 26 20 9)(2 25 17 16 6 29 21 12)(3 28 18 11 7 32 22 15)(4 31 19 14 8 27 23 10)
(1 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 29 21 25)(18 28 22 32)(19 27 23 31)(20 26 24 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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,20)(23,24)(25,26)(27,32)(28,31)(29,30), (1,30,24,13,5,26,20,9)(2,25,17,16,6,29,21,12)(3,28,18,11,7,32,22,15)(4,31,19,14,8,27,23,10), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,29,21,25)(18,28,22,32)(19,27,23,31)(20,26,24,30)>;
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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,20)(23,24)(25,26)(27,32)(28,31)(29,30), (1,30,24,13,5,26,20,9)(2,25,17,16,6,29,21,12)(3,28,18,11,7,32,22,15)(4,31,19,14,8,27,23,10), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,29,21,25)(18,28,22,32)(19,27,23,31)(20,26,24,30) );
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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,22),(18,21),(19,20),(23,24),(25,26),(27,32),(28,31),(29,30)], [(1,30,24,13,5,26,20,9),(2,25,17,16,6,29,21,12),(3,28,18,11,7,32,22,15),(4,31,19,14,8,27,23,10)], [(1,13,5,9),(2,12,6,16),(3,11,7,15),(4,10,8,14),(17,29,21,25),(18,28,22,32),(19,27,23,31),(20,26,24,30)]])
Matrix representation of D8.13D4 ►in GL8(𝔽17)
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0],[12,12,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0],[12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0] >;
D8.13D4 in GAP, Magma, Sage, TeX
D_8._{13}D_4 % in TeX
G:=Group("D8.13D4"); // GroupNames label
G:=SmallGroup(128,2021);
// by ID
G=gap.SmallGroup(128,2021);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,346,2804,1411,375,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b,d*b*d^-1=a^6*b,d*c*d^-1=c^3>;
// generators/relations
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