p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊4D4, C42.439C23, C4.1262+ (1+4), C8⋊2(C2×D4), (D42)⋊5C2, D4⋊4(C2×D4), (C4×D8)⋊31C2, C8⋊6D4⋊3C2, C8⋊5D4⋊8C2, C2.56(D42), D4⋊6D4⋊4C2, C4⋊D8⋊35C2, C8⋊D4⋊30C2, C4⋊C8⋊28C22, C4⋊2(C8⋊C22), C4⋊C4.359D4, (C4×C8)⋊29C22, C4⋊Q8⋊16C22, D4⋊D4⋊37C2, (C2×D4).309D4, (C4×D4)⋊19C22, (C2×D8)⋊29C22, C22⋊C4.42D4, C4.86(C22×D4), C2.D8⋊71C22, D4.D4⋊19C2, C22⋊SD16⋊18C2, C4⋊C4.211C23, C22⋊C8⋊24C22, (C2×C4).470C24, (C2×C8).283C23, C23.312(C2×D4), C22⋊Q8⋊11C22, D4⋊C4⋊81C22, C2.60(D4○SD16), Q8⋊C4⋊40C22, (C2×SD16)⋊46C22, (C2×D4).210C23, C4⋊D4.61C22, C4⋊1D4.76C22, (C2×Q8).194C23, (C2×M4(2))⋊22C22, (C22×C4).320C23, C22.730(C22×D4), (C22×D4).399C22, (C2×C8⋊C22)⋊30C2, (C2×C4).154(C2×D4), C2.72(C2×C8⋊C22), (C2×C4○D4)⋊15C22, SmallGroup(128,2004)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 648 in 273 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×28], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], D4 [×23], Q8 [×4], C23 [×2], C23 [×14], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×4], D8 [×4], SD16 [×10], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×2], C2×D4 [×18], C2×Q8 [×2], C4○D4 [×6], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C2.D8, C2×C4⋊C4, C4×D4 [×3], C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×D8, C2×D8 [×2], C2×SD16 [×6], C8⋊C22 [×8], C22×D4 [×2], C22×D4, C2×C4○D4 [×2], C8⋊6D4, C4×D8, D4⋊D4 [×2], C22⋊SD16 [×2], C4⋊D8, D4.D4, C8⋊D4 [×2], C8⋊5D4, D42, D4⋊6D4, C2×C8⋊C22 [×2], D8⋊4D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8⋊C22 [×2], C22×D4 [×2], 2+ (1+4), D42, C2×C8⋊C22, D4○SD16, D8⋊4D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a-1, dad=a3, cbc-1=a6b, dbd=a2b, dcd=c-1 >
►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 20)(2 19)(3 18)(4 17)(5 24)(6 23)(7 22)(8 21)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
(1 30 21 13)(2 29 22 12)(3 28 23 11)(4 27 24 10)(5 26 17 9)(6 25 18 16)(7 32 19 15)(8 31 20 14)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)
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), (1,20)(2,19)(3,18)(4,17)(5,24)(6,23)(7,22)(8,21)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26), (1,30,21,13)(2,29,22,12)(3,28,23,11)(4,27,24,10)(5,26,17,9)(6,25,18,16)(7,32,19,15)(8,31,20,14), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)(17,26)(18,29)(19,32)(20,27)(21,30)(22,25)(23,28)(24,31)>;
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), (1,20)(2,19)(3,18)(4,17)(5,24)(6,23)(7,22)(8,21)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26), (1,30,21,13)(2,29,22,12)(3,28,23,11)(4,27,24,10)(5,26,17,9)(6,25,18,16)(7,32,19,15)(8,31,20,14), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)(17,26)(18,29)(19,32)(20,27)(21,30)(22,25)(23,28)(24,31) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,24),(6,23),(7,22),(8,21),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26)], [(1,30,21,13),(2,29,22,12),(3,28,23,11),(4,27,24,10),(5,26,17,9),(6,25,18,16),(7,32,19,15),(8,31,20,14)], [(1,13),(2,16),(3,11),(4,14),(5,9),(6,12),(7,15),(8,10),(17,26),(18,29),(19,32),(20,27),(21,30),(22,25),(23,28),(24,31)])
Matrix representation ►G ⊆ GL6(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0] >;
Character table of D8⋊4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ (1+4) |
| ρ28 | 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 | 2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 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 | 2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
In GAP, Magma, Sage, TeX
D_8\rtimes_4D_4
% in TeX
G:=Group("D8:4D4"); // GroupNames label
G:=SmallGroup(128,2004);
// by ID
G=gap.SmallGroup(128,2004);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^3,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations