p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: M4(2)⋊7D4, C42.248D4, C42.373C23, C8⋊1(C2×D4), C8⋊3D4⋊3C2, C8⋊5D4⋊3C2, C8⋊4D4⋊17C2, C4⋊1(C8⋊C22), (C4×C8)⋊25C22, C4⋊Q8⋊67C22, C4.9(C22×D4), (C2×D8)⋊20C22, (C4×M4(2))⋊4C2, C8⋊C4⋊46C22, C4.38(C4⋊1D4), C4⋊1D4⋊38C22, (C2×C4).349C24, (C2×C8).267C23, C23.683(C2×D4), (C22×C4).468D4, (C2×SD16)⋊15C22, (C2×D4).115C23, C4.4D4⋊58C22, (C2×Q8).103C23, C22.15(C4⋊1D4), (C2×C42).855C22, C22.609(C22×D4), (C22×C4).1039C23, C22.26C24⋊10C2, (C22×D4).376C22, (C2×M4(2)).269C22, (C2×C4⋊1D4)⋊19C2, (C2×C8⋊C22)⋊22C2, (C2×C4).858(C2×D4), C2.28(C2×C4⋊1D4), C2.42(C2×C8⋊C22), (C2×C4○D4).155C22, SmallGroup(128,1883)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 772 in 328 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×8], C4 [×4], C22, C22 [×2], C22 [×28], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], D4 [×40], Q8 [×4], C23, C23 [×18], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], D8 [×16], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×6], C2×D4 [×30], C2×Q8 [×2], C4○D4 [×8], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊1D4 [×4], C4⋊1D4 [×2], C4⋊Q8, C2×M4(2) [×2], C2×D8 [×8], C2×SD16 [×8], C8⋊C22 [×16], C22×D4 [×2], C22×D4 [×2], C2×C4○D4 [×2], C4×M4(2), C8⋊5D4 [×2], C8⋊4D4 [×2], C8⋊3D4 [×4], C2×C4⋊1D4, C22.26C24, C2×C8⋊C22 [×4], M4(2)⋊7D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C8⋊C22 [×4], C22×D4 [×3], C2×C4⋊1D4, C2×C8⋊C22 [×2], M4(2)⋊7D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=a-1, bc=cb, bd=db, 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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 16 32 19)(2 9 25 20)(3 10 26 21)(4 11 27 22)(5 12 28 23)(6 13 29 24)(7 14 30 17)(8 15 31 18)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,16,32,19)(2,9,25,20)(3,10,26,21)(4,11,27,22)(5,12,28,23)(6,13,29,24)(7,14,30,17)(8,15,31,18), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)>;
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,16,32,19)(2,9,25,20)(3,10,26,21)(4,11,27,22)(5,12,28,23)(6,13,29,24)(7,14,30,17)(8,15,31,18), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24) );
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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31)], [(1,16,32,19),(2,9,25,20),(3,10,26,21),(4,11,27,22),(5,12,28,23),(6,13,29,24),(7,14,30,17),(8,15,31,18)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24)])
Matrix representation ►G ⊆ GL6(ℤ)
| -1 | -2 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | -2 |
| 0 | 0 | -1 | 0 | 0 | 2 |
| 0 | 0 | -1 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | -2 | -1 | 0 |
| 0 | 0 | 1 | -1 | 0 | -1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | -2 |
| 0 | 0 | 1 | 0 | 0 | -1 |
G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,-1,0,-1,-1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,-2,2,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,2,1,0,0,0,1,-2,-1,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,-2,-1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C8⋊C22 |
| kernel | M4(2)⋊7D4 | C4×M4(2) | C8⋊5D4 | C8⋊4D4 | C8⋊3D4 | C2×C4⋊1D4 | C22.26C24 | C2×C8⋊C22 | C42 | M4(2) | C22×C4 | C4 |
| # reps | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 4 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_7D_4 % in TeX
G:=Group("M4(2):7D4"); // GroupNames label
G:=SmallGroup(128,1883);
// by ID
G=gap.SmallGroup(128,1883);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,1018,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations