p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).43D4, (C2×D8)⋊5C4, (C2×Q16)⋊5C4, C4.80(C4×D4), C4○D4.24D4, (C2×C8).321D4, (C2×SD16)⋊13C4, C2.14(C8○D8), C4.124C22≀C2, C4.C42⋊3C2, D4.1(C22⋊C4), C22.150(C4×D4), Q8.1(C22⋊C4), C2.14(C8.26D4), C4.187(C4⋊D4), C22.1(C4⋊D4), C23.204(C4○D4), (C22×C8).388C22, (C2×C42).276C22, (C22×C4).1363C23, C22.7C42⋊24C2, C2.15(C23.23D4), (C2×M4(2)).177C22, C22.20(C22.D4), (C2×C4≀C2)⋊13C2, (C2×C8○D4)⋊14C2, (C2×C4○D8).1C2, (C2×C8).108(C2×C4), (C2×D4).72(C2×C4), (C2×C4).995(C2×D4), C4.10(C2×C22⋊C4), (C2×Q8).63(C2×C4), (C22×C8)⋊C2⋊22C2, (C2×C4).865(C4○D4), (C2×C4).381(C22×C4), (C2×C4○D4).263C22, SmallGroup(128,608)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).43D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=b, bab=a5, cac-1=a-1b, dad-1=a3b, cbc-1=a4b, bd=db, dcd-1=a4bc3 >
Subgroups: 308 in 162 conjugacy classes, 56 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, 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, C22⋊C8, C4≀C2, C2×C42, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C22.7C42, C4.C42, (C22×C8)⋊C2, C2×C4≀C2, C2×C8○D4, C2×C4○D8, M4(2).43D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C8○D8, C8.26D4, M4(2).43D4
►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)(26 30)(28 32)
(1 9 27 20 5 13 31 24)(2 16 28 19 6 12 32 23)(3 11 29 22 7 15 25 18)(4 10 30 21 8 14 26 17)
(2 4 6 8)(9 20 13 24)(10 19)(11 22 15 18)(12 21)(14 23)(16 17)(26 28 30 32)
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)(26,30)(28,32), (1,9,27,20,5,13,31,24)(2,16,28,19,6,12,32,23)(3,11,29,22,7,15,25,18)(4,10,30,21,8,14,26,17), (2,4,6,8)(9,20,13,24)(10,19)(11,22,15,18)(12,21)(14,23)(16,17)(26,28,30,32)>;
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)(26,30)(28,32), (1,9,27,20,5,13,31,24)(2,16,28,19,6,12,32,23)(3,11,29,22,7,15,25,18)(4,10,30,21,8,14,26,17), (2,4,6,8)(9,20,13,24)(10,19)(11,22,15,18)(12,21)(14,23)(16,17)(26,28,30,32) );
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),(26,30),(28,32)], [(1,9,27,20,5,13,31,24),(2,16,28,19,6,12,32,23),(3,11,29,22,7,15,25,18),(4,10,30,21,8,14,26,17)], [(2,4,6,8),(9,20,13,24),(10,19),(11,22,15,18),(12,21),(14,23),(16,17),(26,28,30,32)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8N | 8O | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | C4○D4 | C4○D4 | C8○D8 | C8.26D4 |
| kernel | M4(2).43D4 | C22.7C42 | C4.C42 | (C22×C8)⋊C2 | C2×C4≀C2 | C2×C8○D4 | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C2×C8 | M4(2) | C4○D4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 8 | 2 |
Matrix representation of M4(2).43D4 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 2 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 6 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 13 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,2,2,0,0,13,15],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,13,6,0,0,8,4],[1,0,0,0,0,16,0,0,0,0,1,11,0,0,0,13] >;
M4(2).43D4 in GAP, Magma, Sage, TeX
M_4(2)._{43}D_4 % in TeX
G:=Group("M4(2).43D4"); // GroupNames label
G:=SmallGroup(128,608);
// by ID
G=gap.SmallGroup(128,608);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,1411,718,172,2028,1027]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*b*c^3>;
// generators/relations