G = SL2(𝔽3)⋊5D4 order 192 = 26·3
1st semidirect product of SL2(𝔽3) and D4 acting through Inn(SL2(𝔽3))
Aliases: SL2(𝔽3)⋊5D4, Q8⋊5D4⋊C3, (C4×Q8)⋊2C6, C2.3(D4×A4), Q8.1(C3×D4), C22⋊C4.2A4, C22⋊2(C4.A4), C2.3(D4.A4), C23.26(C2×A4), (C22×Q8).3C6, (C4×SL2(𝔽3))⋊5C2, C22.22(C22×A4), (C22×SL2(𝔽3))⋊1C2, (C2×SL2(𝔽3)).27C22, (C2×C4○D4)⋊1C6, (C2×C4.A4)⋊2C2, (C2×C4).4(C2×A4), C2.3(C2×C4.A4), (C2×Q8).38(C2×C6), SmallGroup(192,1003)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C22×SL2(𝔽3) — SL2(𝔽3)⋊5D4 |
Generators and relations for SL2(𝔽3)⋊5D4
G = < a,b,c,d,e | a4=c3=d4=e2=1, b2=a2, bab-1=dad-1=a-1, cac-1=b, ae=ea, cbc-1=ab, bd=db, be=eb, dcd-1=a-1c, ce=ec, ede=d-1 >
Subgroups: 315 in 90 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C12, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, SL2(𝔽3), C2×C12, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C3×C22⋊C4, C2×SL2(𝔽3), C2×SL2(𝔽3), C4.A4, Q8⋊5D4, C4×SL2(𝔽3), C22×SL2(𝔽3), C2×C4.A4, SL2(𝔽3)⋊5D4
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, C3×D4, C2×A4, C4.A4, C22×A4, D4×A4, C2×C4.A4, D4.A4, SL2(𝔽3)⋊5D4
►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 7 3 5)(2 6 4 8)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)
(2 6 7)(4 8 5)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)
(1 23 9 31)(2 22 10 30)(3 21 11 29)(4 24 12 32)(5 17 13 25)(6 20 14 28)(7 19 15 27)(8 18 16 26)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32), (2,6,7)(4,8,5)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29), (1,23,9,31)(2,22,10,30)(3,21,11,29)(4,24,12,32)(5,17,13,25)(6,20,14,28)(7,19,15,27)(8,18,16,26), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)>;
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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32), (2,6,7)(4,8,5)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29), (1,23,9,31)(2,22,10,30)(3,21,11,29)(4,24,12,32)(5,17,13,25)(6,20,14,28)(7,19,15,27)(8,18,16,26), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32) );
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)], [(1,7,3,5),(2,6,4,8),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32)], [(2,6,7),(4,8,5),(10,14,15),(12,16,13),(18,22,23),(20,24,21),(26,30,31),(28,32,29)], [(1,23,9,31),(2,22,10,30),(3,21,11,29),(4,24,12,32),(5,17,13,25),(6,20,14,28),(7,19,15,27),(8,18,16,26)], [(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 |
| type | + | + | + | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 | C4.A4 | A4 | C2×A4 | C2×A4 | D4.A4 | D4.A4 | D4×A4 |
| kernel | SL2(𝔽3)⋊5D4 | C4×SL2(𝔽3) | C22×SL2(𝔽3) | C2×C4.A4 | Q8⋊5D4 | C4×Q8 | C22×Q8 | C2×C4○D4 | SL2(𝔽3) | Q8 | C22 | C22⋊C4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 12 | 1 | 2 | 1 | 1 | 2 | 1 |
Matrix representation of SL2(𝔽3)⋊5D4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 4 |
| 0 | 0 | 4 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 10 | 9 |
| 1 | 11 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 11 | 6 |
| 0 | 0 | 6 | 2 |
| 1 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,3,4,0,0,4,10],[1,0,0,0,0,1,0,0,0,0,1,10,0,0,0,9],[1,1,0,0,11,12,0,0,0,0,11,6,0,0,6,2],[1,1,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
SL2(𝔽3)⋊5D4 in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_3)\rtimes_5D_4 % in TeX
G:=Group("SL(2,3):5D4"); // GroupNames label
G:=SmallGroup(192,1003);
// by ID
G=gap.SmallGroup(192,1003);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,680,438,172,775,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^4=e^2=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=b,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations