direct product, non-abelian, soluble
Aliases: D4×SL2(𝔽3), (D4×Q8)⋊C3, Q8⋊(C3×D4), (C4×Q8)⋊3C6, C2.4(D4×A4), (C2×D4).1A4, C4⋊(C2×SL2(𝔽3)), (C22×Q8)⋊2C6, C23.27(C2×A4), C2.4(D4.A4), (C4×SL2(𝔽3))⋊8C2, C22.23(C22×A4), C22⋊2(C2×SL2(𝔽3)), (C22×SL2(𝔽3))⋊2C2, C2.2(C22×SL2(𝔽3)), (C2×SL2(𝔽3)).28C22, (C2×C4).6(C2×A4), (C2×Q8).39(C2×C6), SmallGroup(192,1004)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C22×SL2(𝔽3) — D4×SL2(𝔽3) |
Generators and relations for D4×SL2(𝔽3)
G = < a,b,c,d,e | a4=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 339 in 105 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, SL2(𝔽3), C2×C12, C3×D4, C22×C6, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C2×SL2(𝔽3), C2×SL2(𝔽3), C6×D4, D4×Q8, C4×SL2(𝔽3), C22×SL2(𝔽3), D4×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, SL2(𝔽3), C3×D4, C2×A4, C2×SL2(𝔽3), C22×A4, D4×A4, C22×SL2(𝔽3), D4.A4, D4×SL2(𝔽3)
►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 4)(6 8)(9 11)(14 16)(17 19)(22 24)(25 27)(30 32)
(1 13 5 18)(2 14 6 19)(3 15 7 20)(4 16 8 17)(9 25 30 24)(10 26 31 21)(11 27 32 22)(12 28 29 23)
(1 21 5 26)(2 22 6 27)(3 23 7 28)(4 24 8 25)(9 16 30 17)(10 13 31 18)(11 14 32 19)(12 15 29 20)
(9 25 17)(10 26 18)(11 27 19)(12 28 20)(13 31 21)(14 32 22)(15 29 23)(16 30 24)
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), (2,4)(6,8)(9,11)(14,16)(17,19)(22,24)(25,27)(30,32), (1,13,5,18)(2,14,6,19)(3,15,7,20)(4,16,8,17)(9,25,30,24)(10,26,31,21)(11,27,32,22)(12,28,29,23), (1,21,5,26)(2,22,6,27)(3,23,7,28)(4,24,8,25)(9,16,30,17)(10,13,31,18)(11,14,32,19)(12,15,29,20), (9,25,17)(10,26,18)(11,27,19)(12,28,20)(13,31,21)(14,32,22)(15,29,23)(16,30,24)>;
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), (2,4)(6,8)(9,11)(14,16)(17,19)(22,24)(25,27)(30,32), (1,13,5,18)(2,14,6,19)(3,15,7,20)(4,16,8,17)(9,25,30,24)(10,26,31,21)(11,27,32,22)(12,28,29,23), (1,21,5,26)(2,22,6,27)(3,23,7,28)(4,24,8,25)(9,16,30,17)(10,13,31,18)(11,14,32,19)(12,15,29,20), (9,25,17)(10,26,18)(11,27,19)(12,28,20)(13,31,21)(14,32,22)(15,29,23)(16,30,24) );
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)], [(2,4),(6,8),(9,11),(14,16),(17,19),(22,24),(25,27),(30,32)], [(1,13,5,18),(2,14,6,19),(3,15,7,20),(4,16,8,17),(9,25,30,24),(10,26,31,21),(11,27,32,22),(12,28,29,23)], [(1,21,5,26),(2,22,6,27),(3,23,7,28),(4,24,8,25),(9,16,30,17),(10,13,31,18),(11,14,32,19),(12,15,29,20)], [(9,25,17),(10,26,18),(11,27,19),(12,28,20),(13,31,21),(14,32,22),(15,29,23),(16,30,24)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | ··· | 6N | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 |
| type | + | + | + | + | - | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | SL2(𝔽3) | SL2(𝔽3) | C3×D4 | A4 | C2×A4 | C2×A4 | D4.A4 | D4.A4 | D4×A4 |
| kernel | D4×SL2(𝔽3) | C4×SL2(𝔽3) | C22×SL2(𝔽3) | D4×Q8 | C4×Q8 | C22×Q8 | SL2(𝔽3) | D4 | D4 | Q8 | C2×D4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 4 | 8 | 2 | 1 | 1 | 2 | 1 | 2 | 1 |
Matrix representation of D4×SL2(𝔽3) ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 7 |
| 0 | 0 | 9 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 12 |
| 3 | 9 | 0 | 0 |
| 9 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 10 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,9,0,0,7,1],[1,0,0,0,0,1,0,0,0,0,1,4,0,0,0,12],[3,9,0,0,9,10,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,10,0,0,1,4,0,0,0,0,1,0,0,0,0,1] >;
D4×SL2(𝔽3) in GAP, Magma, Sage, TeX
D_4\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("D4xSL(2,3)"); // GroupNames label
G:=SmallGroup(192,1004);
// by ID
G=gap.SmallGroup(192,1004);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,438,172,775,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,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=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations