Aliases: M4(2).A4, C8.A4⋊8C2, C8○D4⋊3C6, 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.A4).8C2, (C2×C4○D4).3C6, C4○D4.12(C2×C6), SmallGroup(192,1013)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — M4(2).A4 |
Subgroups: 197 in 73 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×4], Q8, Q8, C23, C12 [×2], C2×C6, C2×C8 [×4], M4(2), M4(2) [×5], C22×C4, C2×D4, C2×Q8, C4○D4 [×2], C4○D4 [×2], C24 [×2], SL2(𝔽3), C2×C12, C2×M4(2) [×2], C8○D4 [×2], C8○D4 [×2], C2×C4○D4, C3×M4(2), C2×SL2(𝔽3), C4.A4 [×2], Q8○M4(2), C8.A4 [×2], C2×C4.A4, M4(2).A4
Quotients:
C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C12 [×2], A4, C2×C6, C2×C12, C2×A4 [×3], C4×A4 [×2], C22×A4, C2×C4×A4, M4(2).A4
Generators and relations
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 >
►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)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)
(9 17 30)(10 18 31)(11 19 32)(12 20 25)(13 21 26)(14 22 27)(15 23 28)(16 24 29)
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)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25), (9,17,30)(10,18,31)(11,19,32)(12,20,25)(13,21,26)(14,22,27)(15,23,28)(16,24,29)>;
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)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25), (9,17,30)(10,18,31)(11,19,32)(12,20,25)(13,21,26)(14,22,27)(15,23,28)(16,24,29) );
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),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29)], [(1,23,5,19),(2,24,6,20),(3,17,7,21),(4,18,8,22),(9,26,13,30),(10,27,14,31),(11,28,15,32),(12,29,16,25)], [(9,17,30),(10,18,31),(11,19,32),(12,20,25),(13,21,26),(14,22,27),(15,23,28),(16,24,29)])
Matrix representation ►G ⊆ 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] >;
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 |
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