Aliases: M4(2).A4, C8○D4⋊3C6, C8.A4⋊8C2, C8.5(C2×A4), C4.3(C4×A4), C4○D4.C12, Q8○M4(2)⋊C3, (C2×Q8).C12, C4.A4.2C4, C22.7(C4×A4), Q8.5(C2×C12), C4.15(C22×A4), C4.A4.19C22, (C2×SL2(𝔽3)).1C4, SL2(𝔽3).10(C2×C4), C2.11(C2×C4×A4), (C2×C4).10(C2×A4), (C2×C4○D4).3C6, (C2×C4.A4).8C2, C4○D4.12(C2×C6), SmallGroup(192,1013)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — M4(2).A4 |
Generators and relations for M4(2).A4
G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >
Subgroups: 197 in 73 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C2×C12, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C3×M4(2), C2×SL2(𝔽3), C4.A4, Q8○M4(2), C8.A4, C2×C4.A4, M4(2).A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C22×A4, C2×C4×A4, M4(2).A4
►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)(25 29)(27 31)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)
(1 16 5 12)(2 9 6 13)(3 10 7 14)(4 11 8 15)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(9 29 20)(10 30 21)(11 31 22)(12 32 23)(13 25 24)(14 26 17)(15 27 18)(16 28 19)
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)(25,29)(27,31), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,16,5,12)(2,9,6,13)(3,10,7,14)(4,11,8,15)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (9,29,20)(10,30,21)(11,31,22)(12,32,23)(13,25,24)(14,26,17)(15,27,18)(16,28,19)>;
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)(25,29)(27,31), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,16,5,12)(2,9,6,13)(3,10,7,14)(4,11,8,15)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (9,29,20)(10,30,21)(11,31,22)(12,32,23)(13,25,24)(14,26,17)(15,27,18)(16,28,19) );
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),(25,29),(27,31)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,29,13,25),(10,30,14,26),(11,31,15,27),(12,32,16,28)], [(1,16,5,12),(2,9,6,13),(3,10,7,14),(4,11,8,15),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29)], [(9,29,20),(10,30,21),(11,31,22),(12,32,23),(13,25,24),(14,26,17),(15,27,18),(16,28,19)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 6 | 6 | 4 | 4 | 1 | 1 | 2 | 6 | 6 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 4 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | A4 | C2×A4 | C2×A4 | C4×A4 | C4×A4 | M4(2).A4 |
| kernel | M4(2).A4 | C8.A4 | C2×C4.A4 | Q8○M4(2) | C2×SL2(𝔽3) | C4.A4 | C8○D4 | C2×C4○D4 | C2×Q8 | C4○D4 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 6 |
Matrix representation of M4(2).A4 ►in GL4(𝔽5) generated by
| 0 | 1 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 2 | 0 |
| 0 | 3 | 0 | 2 |
| 2 | 0 | 4 | 0 |
| 0 | 2 | 0 | 1 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 2 | 0 |
| 0 | 4 | 0 | 3 |
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,2,0,0,1,0,0,0,0,0,0,3,0,0,4,0],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[3,0,2,0,0,3,0,3,0,0,2,0,0,0,0,2],[2,0,0,0,0,2,0,0,4,0,3,0,0,1,0,3],[4,0,2,0,0,4,0,3,2,0,0,0,0,3,0,0] >;
M4(2).A4 in GAP, Magma, Sage, TeX
M_4(2).A_4
% in TeX
G:=Group("M4(2).A4"); // GroupNames label
G:=SmallGroup(192,1013);
// by ID
G=gap.SmallGroup(192,1013);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,1373,92,248,438,172,775,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=e^3=1,c^2=d^2=a^4,b*a*b=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^4*c,e*c*e^-1=a^4*c*d,e*d*e^-1=c>;
// generators/relations