p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)○D8, M4(2)○Q16, M4(2)○SD16, M4(2).52D4, C42.285C23, M4(2).34C23, C8○D8⋊7C2, C4○D8⋊9C4, (C2×D8)⋊20C4, C8.89(C2×D4), C8.26D4⋊3C2, (C4×C8)⋊24C22, (C2×Q16)⋊20C4, D8.14(C2×C4), C4.104(C4×D4), C4≀C2⋊19C22, (C2×SD16)⋊12C4, C8○D4⋊10C22, M4(2)○(C4○D8), C4.34(C23×C4), C8.26(C22×C4), Q16.15(C2×C4), SD16.2(C2×C4), C22.23(C4×D4), C8⋊C4⋊43C22, Q8○M4(2)⋊15C2, (C2×C8).422C23, (C2×C4).214C24, C4○D8.26C22, C4○D4.26C23, D4.16(C22×C4), C4.205(C22×D4), Q8.16(C22×C4), C8○2M4(2)⋊11C2, C8.C4⋊14C22, M4(2)○(C8.C4), C42⋊C22⋊20C2, C23.111(C4○D4), (C22×C8).254C22, (C22×C4).933C23, C42⋊C2.302C22, (C2×M4(2)).245C22, C2.74(C2×C4×D4), (C2×C8).99(C2×C4), (C2×C4○D8).18C2, C4○D4.12(C2×C4), (C2×C8.C4)⋊24C2, C22.5(C2×C4○D4), (C2×D4).140(C2×C4), (C2×C4).1088(C2×D4), (C2×Q8).117(C2×C4), (C2×C4).270(C4○D4), (C2×C4).269(C22×C4), (C2×C4○D4).92C22, SmallGroup(128,1689)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)○D8
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=a4c3 >
Subgroups: 340 in 226 conjugacy classes, 138 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, 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, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C8○2M4(2), C42⋊C22, C2×C8.C4, C8○D8, C8.26D4, Q8○M4(2), C2×C4○D8, M4(2)○D8
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)○D8
►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 5)(3 7)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 11 18 25 5 15 22 29)(2 12 19 26 6 16 23 30)(3 13 20 27 7 9 24 31)(4 14 21 28 8 10 17 32)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 13)(10 14)(11 15)(12 16)
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), (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,11,18,25,5,15,22,29)(2,12,19,26,6,16,23,30)(3,13,20,27,7,9,24,31)(4,14,21,28,8,10,17,32), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,13)(10,14)(11,15)(12,16)>;
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), (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,11,18,25,5,15,22,29)(2,12,19,26,6,16,23,30)(3,13,20,27,7,9,24,31)(4,14,21,28,8,10,17,32), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,13)(10,14)(11,15)(12,16) );
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)], [(1,5),(3,7),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31)], [(1,11,18,25,5,15,22,29),(2,12,19,26,6,16,23,30),(3,13,20,27,7,9,24,31),(4,14,21,28,8,10,17,32)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,13),(10,14),(11,15),(12,16)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 8A | ··· | 8L | 8M | ··· | 8V |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 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 | C4 | C4 | D4 | C4○D4 | C4○D4 | M4(2)○D8 |
| kernel | M4(2)○D8 | C8○2M4(2) | C42⋊C22 | C2×C8.C4 | C8○D8 | C8.26D4 | Q8○M4(2) | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C4○D8 | M4(2) | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 2 | 4 |
Matrix representation of M4(2)○D8 ►in GL4(𝔽17) generated by
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 14 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 14 | 14 |
| 0 | 0 | 3 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,0,4,0,0,0,0,4,16,0,0,0,0,16,0,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[14,3,0,0,14,14,0,0,0,0,14,3,0,0,14,14],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
M4(2)○D8 in GAP, Magma, Sage, TeX
M_4(2)\circ D_8
% in TeX
G:=Group("M4(2)oD8"); // GroupNames label
G:=SmallGroup(128,1689);
// by ID
G=gap.SmallGroup(128,1689);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,521,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^4*c^3>;
// generators/relations