p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).51D4, C42.284C23, M4(2).33C23, C8○D8⋊6C2, D8⋊6(C2×C4), C8⋊C22⋊6C4, M4(2)○C4≀C2, Q16⋊6(C2×C4), C8.88(C2×D4), C4.79(C4×D4), C8.26D4⋊2C2, (C4×C8)⋊23C22, SD16⋊5(C2×C4), C8○D4⋊9C22, C8.C22⋊6C4, C8.5(C22×C4), C4≀C2⋊21C22, (C4×M4(2))⋊3C2, C4.33(C23×C4), C22.18(C4×D4), C8⋊C4⋊42C22, Q8○M4(2)⋊14C2, (C2×C8).421C23, (C2×C4).213C24, C4○D4.25C23, C4○D8.25C22, D4.15(C22×C4), C4.204(C22×D4), Q8.15(C22×C4), C8.C4⋊13C22, D8⋊C22.7C2, M4(2).14(C2×C4), M4(2).C4⋊12C2, C23.195(C4○D4), (C22×C4).932C23, (C2×C42).769C22, (C2×M4(2)).244C22, C2.73(C2×C4×D4), (C2×C4≀C2)⋊32C2, C4○D4.11(C2×C4), C22.4(C2×C4○D4), (C2×D4).139(C2×C4), (C2×C4).1087(C2×D4), (C2×C4).73(C22×C4), (C2×Q8).116(C2×C4), (C2×C4).476(C4○D4), (C2×C4○D4).91C22, SmallGroup(128,1688)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).51D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a6c-1 >
Subgroups: 348 in 229 conjugacy classes, 138 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C8⋊C4, C4≀C2, C8.C4, C2×C42, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C4×M4(2), C2×C4≀C2, M4(2).C4, C8○D8, C8.26D4, Q8○M4(2), D8⋊C22, M4(2).51D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, M4(2).51D4
►On 16 points - transitive group 16T205
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(9 15 13 11)(10 16 14 12)
(1 11 7 9 5 15 3 13)(2 12 8 10 6 16 4 14)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (9,15,13,11)(10,16,14,12), (1,11,7,9,5,15,3,13)(2,12,8,10,6,16,4,14)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (9,15,13,11)(10,16,14,12), (1,11,7,9,5,15,3,13)(2,12,8,10,6,16,4,14) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(9,15,13,11),(10,16,14,12)], [(1,11,7,9,5,15,3,13),(2,12,8,10,6,16,4,14)]])
G:=TransitiveGroup(16,205);
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | ··· | 4I | 4J | ··· | 4O | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | C4○D4 | C4○D4 | M4(2).51D4 |
| kernel | M4(2).51D4 | C4×M4(2) | C2×C4≀C2 | M4(2).C4 | C8○D8 | C8.26D4 | Q8○M4(2) | D8⋊C22 | C8⋊C22 | C8.C22 | M4(2) | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 8 | 8 | 4 | 2 | 2 | 4 |
Matrix representation of M4(2).51D4 ►in GL4(𝔽5) generated by
| 0 | 3 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,1,0,0,3,0,0,0,0,0,0,4,0,0,2,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[0,0,2,0,0,0,0,3,1,0,0,0,0,4,0,0] >;
M4(2).51D4 in GAP, Magma, Sage, TeX
M_4(2)._{51}D_4 % in TeX
G:=Group("M4(2).51D4"); // GroupNames label
G:=SmallGroup(128,1688);
// by ID
G=gap.SmallGroup(128,1688);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^-1>;
// generators/relations