p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8○M5(2), D4○M5(2), C8.25C24, M5(2)○M5(2), M4(2)○M5(2), C16.15C23, M5(2)⋊17C22, D4○C16⋊8C2, C8○D4.5C4, C4○D4.4C8, D4.9(C2×C8), C4○D4○M5(2), (C2×D4).11C8, (C2×Q8).10C8, Q8.10(C2×C8), (C2×C16)⋊13C22, M5(2)○(C8○D4), C23.12(C2×C8), C4.23(C22×C8), C2.12(C23×C8), C4.64(C23×C4), C8.64(C22×C4), (C2×M5(2))⋊21C2, (C2×C8).618C23, C8○D4.20C22, C22.5(C22×C8), M4(2).36(C2×C4), (C2×M4(2)).35C4, (C22×C8).463C22, (C2×C4).33(C2×C8), (C2×C8).153(C2×C4), C4○D4.39(C2×C4), (C2×C8○D4).22C2, (C2×C4○D4).32C4, (C2×C4).477(C22×C4), (C22×C4).371(C2×C4), SmallGroup(128,2139)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8○M5(2)
G = < a,b,c,d | a4=b2=d2=1, c8=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >
Subgroups: 196 in 180 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, M5(2), C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×M5(2), D4○C16, C2×C8○D4, Q8○M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C23×C8, Q8○M5(2)
►On 32 points
(1 27 9 19)(2 28 10 20)(3 29 11 21)(4 30 12 22)(5 31 13 23)(6 32 14 24)(7 17 15 25)(8 18 16 26)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
G:=sub<Sym(32)| (1,27,9,19)(2,28,10,20)(3,29,11,21)(4,30,12,22)(5,31,13,23)(6,32,14,24)(7,17,15,25)(8,18,16,26), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)>;
G:=Group( (1,27,9,19)(2,28,10,20)(3,29,11,21)(4,30,12,22)(5,31,13,23)(6,32,14,24)(7,17,15,25)(8,18,16,26), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32) );
G=PermutationGroup([[(1,27,9,19),(2,28,10,20),(3,29,11,21),(4,30,12,22),(5,31,13,23),(6,32,14,24),(7,17,15,25),(8,18,16,26)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)]])
68 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 4A | 4B | 4C | ··· | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 16A | ··· | 16AF |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | Q8○M5(2) |
| kernel | Q8○M5(2) | C2×M5(2) | D4○C16 | C2×C8○D4 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C1 |
| # reps | 1 | 6 | 8 | 1 | 6 | 8 | 2 | 12 | 4 | 16 | 4 |
Matrix representation of Q8○M5(2) ►in GL4(𝔽17) generated by
| 13 | 0 | 15 | 9 |
| 2 | 4 | 15 | 12 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 16 | 13 | 0 | 5 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 13 | 5 | 5 |
| 0 | 0 | 0 | 1 |
| 4 | 8 | 13 | 7 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 15 | 12 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [13,2,0,0,0,4,0,0,15,15,0,1,9,12,16,0],[16,0,0,0,13,1,0,0,0,0,16,0,5,0,0,1],[4,0,4,0,13,0,8,2,5,0,13,0,5,1,7,0],[1,0,0,0,0,1,0,0,15,0,16,0,12,0,0,16] >;
Q8○M5(2) in GAP, Magma, Sage, TeX
Q_8\circ M_5(2)
% in TeX
G:=Group("Q8oM5(2)"); // GroupNames label
G:=SmallGroup(128,2139);
// by ID
G=gap.SmallGroup(128,2139);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,723,2019,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^8=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c>;
// generators/relations