p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊9D4, C42.250D4, C42.375C23, C8.2(C2×D4), C8⋊3D4⋊4C2, (C4×C8)⋊26C22, C8.2D4⋊4C2, C4⋊Q8⋊68C22, (C4×M4(2))⋊6C2, C4.11(C22×D4), C8⋊C4⋊47C22, C8.12D4⋊14C2, C4.40(C4⋊1D4), C4⋊1D4⋊39C22, (C2×C4).351C24, (C2×C8).269C23, (C2×Q16)⋊20C22, (C2×D8).62C22, C23.685(C2×D4), (C22×C4).470D4, (C2×SD16)⋊16C22, (C2×D4).117C23, C4.4D4⋊59C22, (C2×Q8).105C23, C22.17(C4⋊1D4), (C2×C42).857C22, C22.611(C22×D4), C2.39(D8⋊C22), (C22×C4).1041C23, C22.26C24⋊11C2, (C22×D4).377C22, (C22×Q8).310C22, (C2×M4(2)).271C22, (C2×C8⋊C22)⋊23C2, (C2×C4).695(C2×D4), C2.30(C2×C4⋊1D4), (C2×C4.4D4)⋊42C2, (C2×C8.C22)⋊23C2, (C2×C4○D4).157C22, SmallGroup(128,1885)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 596 in 286 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×18], Q8 [×10], C23, C23 [×10], C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×4], C2×D4 [×9], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×8], C24, C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×6], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C8⋊C22 [×8], C8.C22 [×8], C22×D4, C22×Q8, C2×C4○D4 [×2], C4×M4(2), C8.12D4 [×4], C8⋊3D4 [×2], C8.2D4 [×2], C2×C4.4D4, C22.26C24, C2×C8⋊C22 [×2], C2×C8.C22 [×2], M4(2)⋊9D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C22×D4 [×3], C2×C4⋊1D4, D8⋊C22 [×2], M4(2)⋊9D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3, bc=cb, dbd=a4b, 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)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 15 25 19)(2 12 26 24)(3 9 27 21)(4 14 28 18)(5 11 29 23)(6 16 30 20)(7 13 31 17)(8 10 32 22)
(1 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)
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)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32), (1,15,25,19)(2,12,26,24)(3,9,27,21)(4,14,28,18)(5,11,29,23)(6,16,30,20)(7,13,31,17)(8,10,32,22), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)>;
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)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32), (1,15,25,19)(2,12,26,24)(3,9,27,21)(4,14,28,18)(5,11,29,23)(6,16,30,20)(7,13,31,17)(8,10,32,22), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25) );
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),(10,14),(12,16),(18,22),(20,24),(26,30),(28,32)], [(1,15,25,19),(2,12,26,24),(3,9,27,21),(4,14,28,18),(5,11,29,23),(6,16,30,20),(7,13,31,17),(8,10,32,22)], [(1,20),(2,23),(3,18),(4,21),(5,24),(6,19),(7,22),(8,17),(9,28),(10,31),(11,26),(12,29),(13,32),(14,27),(15,30),(16,25)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 13 | 13 | 9 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 16 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 13 | 13 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,13,13,4,0,0,4,13,0,0,0,0,0,9,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,13,0,0,0,4,0,13,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,0,15,0,0,1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8⋊C22 |
| kernel | M4(2)⋊9D4 | C4×M4(2) | C8.12D4 | C8⋊3D4 | C8.2D4 | C2×C4.4D4 | C22.26C24 | C2×C8⋊C22 | C2×C8.C22 | C42 | M4(2) | C22×C4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_9D_4 % in TeX
G:=Group("M4(2):9D4"); // GroupNames label
G:=SmallGroup(128,1885);
// by ID
G=gap.SmallGroup(128,1885);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,520,1018,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations