p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊9D4, C42.31C23, C4.1182+ (1+4), C8⋊8(C2×D4), (D42)⋊3C2, D4⋊3(C2×D4), C8⋊9D4⋊1C2, C2.48(D42), D4⋊5D4⋊1C2, C8⋊2D4⋊19C2, C8⋊3D4⋊16C2, C4⋊D8⋊33C2, C8⋊8D4⋊17C2, C4⋊C8⋊23C22, C4⋊C4.146D4, D8⋊C4⋊19C2, C22⋊D8⋊25C2, D4⋊D4⋊31C2, (C2×D4).305D4, C2.37(D4○D8), (C22×D8)⋊20C2, (C4×D4)⋊17C22, (C2×D8)⋊27C22, C4⋊1D4⋊9C22, C4⋊D4⋊7C22, (C2×C8).81C23, C4.78(C22×D4), C4.Q8⋊22C22, C8⋊C4⋊14C22, C22⋊Q8⋊7C22, D4.2D4⋊34C2, C22⋊SD16⋊14C2, C4⋊C4.203C23, C22⋊C8⋊19C22, C22⋊2(C8⋊C22), (C2×C4).462C24, C22⋊C4.156D4, (C22×C8)⋊22C22, Q8⋊C4⋊7C22, C23.459(C2×D4), D4⋊C4⋊34C22, (C2×SD16)⋊25C22, (C2×D4).202C23, C4.4D4⋊11C22, (C22×D4)⋊26C22, (C2×Q8).190C23, (C2×M4(2))⋊19C22, (C22×C4).316C23, C22.722(C22×D4), (C2×C8⋊C22)⋊26C2, (C2×C4).586(C2×D4), C2.70(C2×C8⋊C22), (C2×C4○D4)⋊11C22, SmallGroup(128,1996)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 720 in 284 conjugacy classes, 96 normal (84 characteristic)
C1, C2 [×3], C2 [×10], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×34], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×12], D4 [×4], D4 [×26], Q8 [×2], C23 [×2], C23 [×20], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4, C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], D8 [×9], SD16 [×5], C22×C4 [×2], C22×C4 [×3], C2×D4 [×6], C2×D4 [×22], C2×Q8, C4○D4 [×3], C24 [×3], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×5], Q8⋊C4, C4⋊C8, C4.Q8, C2×C22⋊C4, C4×D4 [×3], C22≀C2 [×3], C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8 [×6], C2×D8 [×4], C2×SD16 [×3], C8⋊C22 [×4], C22×D4 [×3], C22×D4, C2×C4○D4, C8⋊9D4, D8⋊C4, C22⋊D8 [×2], D4⋊D4, C22⋊SD16, C4⋊D8, D4.2D4, C8⋊8D4, C8⋊2D4, C8⋊3D4, D42, D4⋊5D4, C22×D8, C2×C8⋊C22, D8⋊9D4
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○D8, D8⋊9D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, cac-1=dad=a3, cbc-1=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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
(1 30 17 13)(2 25 18 16)(3 28 19 11)(4 31 20 14)(5 26 21 9)(6 29 22 12)(7 32 23 15)(8 27 24 10)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 30)(18 25)(19 28)(20 31)(21 26)(22 29)(23 32)(24 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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26), (1,30,17,13)(2,25,18,16)(3,28,19,11)(4,31,20,14)(5,26,21,9)(6,29,22,12)(7,32,23,15)(8,27,24,10), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26), (1,30,17,13)(2,25,18,16)(3,28,19,11)(4,31,20,14)(5,26,21,9)(6,29,22,12)(7,32,23,15)(8,27,24,10), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)(17,30)(18,25)(19,28)(20,31)(21,26)(22,29)(23,32)(24,27) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26)], [(1,30,17,13),(2,25,18,16),(3,28,19,11),(4,31,20,14),(5,26,21,9),(6,29,22,12),(7,32,23,15),(8,27,24,10)], [(1,13),(2,16),(3,11),(4,14),(5,9),(6,12),(7,15),(8,10),(17,30),(18,25),(19,28),(20,31),(21,26),(22,29),(23,32),(24,27)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 14 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 0 | 3 | 11 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 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 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,6,6,0,0,0,0,0,0,11,14,0,0,0,0,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,0,0,0,0,6,3,0,0,0,0,11,11],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
Character table of D8⋊9D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | -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 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ (1+4) |
| ρ27 | 4 | -4 | -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 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
In GAP, Magma, Sage, TeX
D_8\rtimes_9D_4
% in TeX
G:=Group("D8:9D4"); // GroupNames label
G:=SmallGroup(128,1996);
// by ID
G=gap.SmallGroup(128,1996);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations