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) |
Subgroups: 339 in 105 conjugacy classes, 31 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C6 [×7], C2×C4, C2×C4 [×8], D4 [×4], Q8, Q8 [×5], C23 [×2], C12 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4 [×4], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×6], SL2(𝔽3), C2×C12, C3×D4 [×4], C22×C6 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C4⋊Q8, C22×Q8 [×2], C2×SL2(𝔽3), C2×SL2(𝔽3) [×2], C6×D4, D4×Q8, C4×SL2(𝔽3), C22×SL2(𝔽3) [×2], D4×SL2(𝔽3)
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, SL2(𝔽3) [×4], C3×D4, C2×A4 [×3], C2×SL2(𝔽3) [×6], C22×A4, D4×A4, C22×SL2(𝔽3), D4.A4, D4×SL2(𝔽3)
Generators and relations
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 >
►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 21 26 18)(2 22 27 19)(3 23 28 20)(4 24 25 17)(5 13 31 10)(6 14 32 11)(7 15 29 12)(8 16 30 9)
(1 13 26 10)(2 14 27 11)(3 15 28 12)(4 16 25 9)(5 18 31 21)(6 19 32 22)(7 20 29 23)(8 17 30 24)
(5 13 21)(6 14 22)(7 15 23)(8 16 24)(9 17 30)(10 18 31)(11 19 32)(12 20 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), (2,4)(6,8)(9,11)(14,16)(17,19)(22,24)(25,27)(30,32), (1,21,26,18)(2,22,27,19)(3,23,28,20)(4,24,25,17)(5,13,31,10)(6,14,32,11)(7,15,29,12)(8,16,30,9), (1,13,26,10)(2,14,27,11)(3,15,28,12)(4,16,25,9)(5,18,31,21)(6,19,32,22)(7,20,29,23)(8,17,30,24), (5,13,21)(6,14,22)(7,15,23)(8,16,24)(9,17,30)(10,18,31)(11,19,32)(12,20,29)>;
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), (2,4)(6,8)(9,11)(14,16)(17,19)(22,24)(25,27)(30,32), (1,21,26,18)(2,22,27,19)(3,23,28,20)(4,24,25,17)(5,13,31,10)(6,14,32,11)(7,15,29,12)(8,16,30,9), (1,13,26,10)(2,14,27,11)(3,15,28,12)(4,16,25,9)(5,18,31,21)(6,19,32,22)(7,20,29,23)(8,17,30,24), (5,13,21)(6,14,22)(7,15,23)(8,16,24)(9,17,30)(10,18,31)(11,19,32)(12,20,29) );
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)], [(2,4),(6,8),(9,11),(14,16),(17,19),(22,24),(25,27),(30,32)], [(1,21,26,18),(2,22,27,19),(3,23,28,20),(4,24,25,17),(5,13,31,10),(6,14,32,11),(7,15,29,12),(8,16,30,9)], [(1,13,26,10),(2,14,27,11),(3,15,28,12),(4,16,25,9),(5,18,31,21),(6,19,32,22),(7,20,29,23),(8,17,30,24)], [(5,13,21),(6,14,22),(7,15,23),(8,16,24),(9,17,30),(10,18,31),(11,19,32),(12,20,29)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
D_4\times 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