non-abelian, soluble, monomial
Aliases: D4⋊2S4, C24.D6, (C4×S4)⋊2C2, (D4×A4)⋊3C2, C4.9(C2×S4), A4⋊Q8⋊3C2, A4⋊2(C4○D4), A4⋊D4⋊2C2, (C22×D4)⋊3S3, C22.4(C2×S4), C2.7(C22×S4), (C22×C4).5D6, C22⋊(D4⋊2S3), A4⋊C4.3C22, (C4×A4).5C22, (C2×S4).2C22, (C2×A4).6C23, (C22×A4).C22, C23.6(C22×S3), (C2×A4⋊C4)⋊3C2, SmallGroup(192,1473)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2S4
G = < a,b,c,d,e,f | a4=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a2b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 654 in 173 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, A4, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, S4, C2×A4, C2×A4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, A4⋊C4, A4⋊C4, C4×A4, D4⋊2S3, C2×S4, C22×A4, D4⋊5D4, A4⋊Q8, C4×S4, C2×A4⋊C4, A4⋊D4, D4×A4, D4⋊2S4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, S4, C22×S3, D4⋊2S3, C2×S4, C22×S4, D4⋊2S4
Character table of D4⋊2S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 12 | |
| size | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 12 | 8 | 2 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 8 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 3 | 3 | -3 | 3 | -1 | -1 | 1 | -1 | -1 | 0 | -3 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | -3 | -1 | -1 | -1 | 1 | -1 | 0 | -3 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | -3 | -3 | -1 | -1 | 1 | 1 | -1 | 0 | 3 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 3 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 0 | 3 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -3 | -3 | -1 | -1 | 1 | 1 | 1 | 0 | 3 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 3 | 3 | 3 | -3 | -1 | -1 | -1 | 1 | 1 | 0 | -3 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ22 | 3 | 3 | -3 | 3 | -1 | -1 | 1 | -1 | 1 | 0 | -3 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 4 | -4 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ24 | 6 | -6 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 6 | -6 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T295
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 22)(23 24)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 11 13)(2 12 14)(3 9 15)(4 10 16)(5 18 21)(6 19 22)(7 20 23)(8 17 24)
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(9 18)(10 19)(11 20)(12 17)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,11,13)(2,12,14)(3,9,15)(4,10,16)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,23)(2,24)(3,21)(4,22)(5,15)(6,16)(7,13)(8,14)(9,18)(10,19)(11,20)(12,17)>;
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), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,11,13)(2,12,14)(3,9,15)(4,10,16)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,23)(2,24)(3,21)(4,22)(5,15)(6,16)(7,13)(8,14)(9,18)(10,19)(11,20)(12,17) );
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)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,22),(23,24)], [(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(13,15),(14,16),(21,23),(22,24)], [(1,11,13),(2,12,14),(3,9,15),(4,10,16),(5,18,21),(6,19,22),(7,20,23),(8,17,24)], [(1,23),(2,24),(3,21),(4,22),(5,15),(6,16),(7,13),(8,14),(9,18),(10,19),(11,20),(12,17)]])
G:=TransitiveGroup(24,295);
►On 24 points - transitive group 24T318
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 22)(2 21)(3 24)(4 23)(5 13)(6 16)(7 15)(8 14)(9 20)(10 19)(11 18)(12 17)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 6)(2 12 7)(3 9 8)(4 10 5)(13 23 19)(14 24 20)(15 21 17)(16 22 18)
(5 10)(6 11)(7 12)(8 9)(13 17)(14 18)(15 19)(16 20)(21 23)(22 24)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22)(2,21)(3,24)(4,23)(5,13)(6,16)(7,15)(8,14)(9,20)(10,19)(11,18)(12,17), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,6)(2,12,7)(3,9,8)(4,10,5)(13,23,19)(14,24,20)(15,21,17)(16,22,18), (5,10)(6,11)(7,12)(8,9)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24)>;
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), (1,22)(2,21)(3,24)(4,23)(5,13)(6,16)(7,15)(8,14)(9,20)(10,19)(11,18)(12,17), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,6)(2,12,7)(3,9,8)(4,10,5)(13,23,19)(14,24,20)(15,21,17)(16,22,18), (5,10)(6,11)(7,12)(8,9)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24) );
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)], [(1,22),(2,21),(3,24),(4,23),(5,13),(6,16),(7,15),(8,14),(9,20),(10,19),(11,18),(12,17)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,11,6),(2,12,7),(3,9,8),(4,10,5),(13,23,19),(14,24,20),(15,21,17),(16,22,18)], [(5,10),(6,11),(7,12),(8,9),(13,17),(14,18),(15,19),(16,20),(21,23),(22,24)]])
G:=TransitiveGroup(24,318);
►On 24 points - transitive group 24T359
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(9 11)(13 15)(18 20)(22 24)
(1 17)(2 18)(3 19)(4 20)(5 7)(6 8)(9 22)(10 23)(11 24)(12 21)(13 15)(14 16)
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 24)(10 21)(11 22)(12 23)(17 19)(18 20)
(1 21 14)(2 22 15)(3 23 16)(4 24 13)(5 18 11)(6 19 12)(7 20 9)(8 17 10)
(1 18)(2 19)(3 20)(4 17)(5 21)(6 22)(7 23)(8 24)(9 16)(10 13)(11 14)(12 15)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(9,11)(13,15)(18,20)(22,24), (1,17)(2,18)(3,19)(4,20)(5,7)(6,8)(9,22)(10,23)(11,24)(12,21)(13,15)(14,16), (1,3)(2,4)(5,13)(6,14)(7,15)(8,16)(9,24)(10,21)(11,22)(12,23)(17,19)(18,20), (1,21,14)(2,22,15)(3,23,16)(4,24,13)(5,18,11)(6,19,12)(7,20,9)(8,17,10), (1,18)(2,19)(3,20)(4,17)(5,21)(6,22)(7,23)(8,24)(9,16)(10,13)(11,14)(12,15)>;
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), (2,4)(5,7)(9,11)(13,15)(18,20)(22,24), (1,17)(2,18)(3,19)(4,20)(5,7)(6,8)(9,22)(10,23)(11,24)(12,21)(13,15)(14,16), (1,3)(2,4)(5,13)(6,14)(7,15)(8,16)(9,24)(10,21)(11,22)(12,23)(17,19)(18,20), (1,21,14)(2,22,15)(3,23,16)(4,24,13)(5,18,11)(6,19,12)(7,20,9)(8,17,10), (1,18)(2,19)(3,20)(4,17)(5,21)(6,22)(7,23)(8,24)(9,16)(10,13)(11,14)(12,15) );
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)], [(2,4),(5,7),(9,11),(13,15),(18,20),(22,24)], [(1,17),(2,18),(3,19),(4,20),(5,7),(6,8),(9,22),(10,23),(11,24),(12,21),(13,15),(14,16)], [(1,3),(2,4),(5,13),(6,14),(7,15),(8,16),(9,24),(10,21),(11,22),(12,23),(17,19),(18,20)], [(1,21,14),(2,22,15),(3,23,16),(4,24,13),(5,18,11),(6,19,12),(7,20,9),(8,17,10)], [(1,18),(2,19),(3,20),(4,17),(5,21),(6,22),(7,23),(8,24),(9,16),(10,13),(11,14),(12,15)]])
G:=TransitiveGroup(24,359);
►On 24 points - transitive group 24T396
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 12)(2 11)(3 10)(4 9)(5 21)(6 24)(7 23)(8 22)(13 17)(14 20)(15 19)(16 18)
(1 12)(2 9)(3 10)(4 11)(5 23)(6 24)(7 21)(8 22)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 21)(6 22)(7 23)(8 24)(9 11)(10 12)(13 17)(14 18)(15 19)(16 20)
(1 5 18)(2 6 19)(3 7 20)(4 8 17)(9 22 13)(10 23 14)(11 24 15)(12 21 16)
(5 18)(6 19)(7 20)(8 17)(9 11)(10 12)(13 24)(14 21)(15 22)(16 23)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,21)(6,24)(7,23)(8,22)(13,17)(14,20)(15,19)(16,18), (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,21)(6,22)(7,23)(8,24)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20), (1,5,18)(2,6,19)(3,7,20)(4,8,17)(9,22,13)(10,23,14)(11,24,15)(12,21,16), (5,18)(6,19)(7,20)(8,17)(9,11)(10,12)(13,24)(14,21)(15,22)(16,23)>;
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), (1,12)(2,11)(3,10)(4,9)(5,21)(6,24)(7,23)(8,22)(13,17)(14,20)(15,19)(16,18), (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,21)(6,22)(7,23)(8,24)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20), (1,5,18)(2,6,19)(3,7,20)(4,8,17)(9,22,13)(10,23,14)(11,24,15)(12,21,16), (5,18)(6,19)(7,20)(8,17)(9,11)(10,12)(13,24)(14,21)(15,22)(16,23) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,21),(6,24),(7,23),(8,22),(13,17),(14,20),(15,19),(16,18)], [(1,12),(2,9),(3,10),(4,11),(5,23),(6,24),(7,21),(8,22),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(5,21),(6,22),(7,23),(8,24),(9,11),(10,12),(13,17),(14,18),(15,19),(16,20)], [(1,5,18),(2,6,19),(3,7,20),(4,8,17),(9,22,13),(10,23,14),(11,24,15),(12,21,16)], [(5,18),(6,19),(7,20),(8,17),(9,11),(10,12),(13,24),(14,21),(15,22),(16,23)]])
G:=TransitiveGroup(24,396);
Matrix representation of D4⋊2S4 ►in GL5(𝔽13)
| 8 | 0 | 0 | 0 | 0 |
| 11 | 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 |
| 0 | 0 | 5 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 11 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 2 | 1 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 | 1 |
| 0 | 0 | 5 | 1 | 0 |
G:=sub<GL(5,GF(13))| [8,11,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,5,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,8,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,8,5,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,2,0,0,0,0,1,0,0,11,5,8],[1,3,0,0,0,0,12,0,0,0,0,0,1,8,5,0,0,0,0,1,0,0,0,1,0] >;
D4⋊2S4 in GAP, Magma, Sage, TeX
D_4\rtimes_2S_4
% in TeX
G:=Group("D4:2S4"); // GroupNames label
G:=SmallGroup(192,1473);
// by ID
G=gap.SmallGroup(192,1473);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,254,135,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^2*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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