non-abelian, soluble, monomial
Aliases: A4≀C2, C22⋊A4⋊C6, C22⋊S4⋊C3, (A42)⋊1C2, C22⋊A4⋊1S3, C24⋊2(C3×S3), SmallGroup(288,1025)
Series: Derived ►Chief ►Lower central ►Upper central
| C22⋊A4 — A4≀C2 |
Subgroups: 442 in 46 conjugacy classes, 7 normal (all characteristic)
C1, C2 [×3], C3 [×3], C4, C22 [×6], S3, C6 [×2], C2×C4, D4 [×2], C23 [×3], C32, A4 [×5], C2×C6, C22⋊C4, C2×D4, C24, C3×S3, S4, C2×A4 [×2], C22≀C2, C3×A4, C22×A4, C22⋊A4 [×2], C24⋊C6, C22⋊S4, A42, A4≀C2
Quotients:
C1, C2, C3, S3, C6, C3×S3, A4≀C2
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f6=1, eae-1=ab=ba, ac=ca, faf-1=ad=da, fdf-1=bc=cb, bd=db, ebe-1=a, fbf-1=abc, ece-1=cd=dc, fcf-1=abd, ede-1=c, fef-1=e-1 >
►On 8 points - transitive group 8T42
Generators in S8
(1 4)(2 3)(5 7)(6 8)
(1 8)(2 5)(3 7)(4 6)
(1 8)(4 6)
(1 6)(4 8)
(3 5 7)(4 8 6)
(1 2)(3 4 5 6 7 8)
G:=sub<Sym(8)| (1,4)(2,3)(5,7)(6,8), (1,8)(2,5)(3,7)(4,6), (1,8)(4,6), (1,6)(4,8), (3,5,7)(4,8,6), (1,2)(3,4,5,6,7,8)>;
G:=Group( (1,4)(2,3)(5,7)(6,8), (1,8)(2,5)(3,7)(4,6), (1,8)(4,6), (1,6)(4,8), (3,5,7)(4,8,6), (1,2)(3,4,5,6,7,8) );
G=PermutationGroup([(1,4),(2,3),(5,7),(6,8)], [(1,8),(2,5),(3,7),(4,6)], [(1,8),(4,6)], [(1,6),(4,8)], [(3,5,7),(4,8,6)], [(1,2),(3,4,5,6,7,8)])
G:=TransitiveGroup(8,42);
►On 12 points - transitive group 12T126
Generators in S12
(1 10)(2 11)(3 12)(6 9)
(1 10)(2 11)(4 7)(5 8)
(1 10)(5 8)
(3 12)(5 8)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
G:=sub<Sym(12)| (1,10)(2,11)(3,12)(6,9), (1,10)(2,11)(4,7)(5,8), (1,10)(5,8), (3,12)(5,8), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12)>;
G:=Group( (1,10)(2,11)(3,12)(6,9), (1,10)(2,11)(4,7)(5,8), (1,10)(5,8), (3,12)(5,8), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12) );
G=PermutationGroup([(1,10),(2,11),(3,12),(6,9)], [(1,10),(2,11),(4,7),(5,8)], [(1,10),(5,8)], [(3,12),(5,8)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)])
G:=TransitiveGroup(12,126);
►On 12 points - transitive group 12T128
Generators in S12
(1 4)(2 7)(5 6)(9 11)
(2 7)(3 10)(8 12)(9 11)
(1 4)(2 9)(3 8)(5 6)(7 11)(10 12)
(1 5)(2 7)(3 12)(4 6)(8 10)(9 11)
(1 2 3)(4 7 10)(5 11 8)(6 9 12)
(2 3)(4 5 6)(7 8 9 10 11 12)
G:=sub<Sym(12)| (1,4)(2,7)(5,6)(9,11), (2,7)(3,10)(8,12)(9,11), (1,4)(2,9)(3,8)(5,6)(7,11)(10,12), (1,5)(2,7)(3,12)(4,6)(8,10)(9,11), (1,2,3)(4,7,10)(5,11,8)(6,9,12), (2,3)(4,5,6)(7,8,9,10,11,12)>;
G:=Group( (1,4)(2,7)(5,6)(9,11), (2,7)(3,10)(8,12)(9,11), (1,4)(2,9)(3,8)(5,6)(7,11)(10,12), (1,5)(2,7)(3,12)(4,6)(8,10)(9,11), (1,2,3)(4,7,10)(5,11,8)(6,9,12), (2,3)(4,5,6)(7,8,9,10,11,12) );
G=PermutationGroup([(1,4),(2,7),(5,6),(9,11)], [(2,7),(3,10),(8,12),(9,11)], [(1,4),(2,9),(3,8),(5,6),(7,11),(10,12)], [(1,5),(2,7),(3,12),(4,6),(8,10),(9,11)], [(1,2,3),(4,7,10),(5,11,8),(6,9,12)], [(2,3),(4,5,6),(7,8,9,10,11,12)])
G:=TransitiveGroup(12,128);
►On 12 points - transitive group 12T129
Generators in S12
(1 10)(3 12)(5 9)(6 7)
(1 6)(3 5)(7 10)(9 12)
(1 7)(2 4)(3 12)(5 9)(6 10)(8 11)
(1 10)(2 11)(3 5)(4 8)(6 7)(9 12)
(1 10 7)(2 8 11)(3 12 9)
(1 2 3)(4 5 6)(7 8 9 10 11 12)
G:=sub<Sym(12)| (1,10)(3,12)(5,9)(6,7), (1,6)(3,5)(7,10)(9,12), (1,7)(2,4)(3,12)(5,9)(6,10)(8,11), (1,10)(2,11)(3,5)(4,8)(6,7)(9,12), (1,10,7)(2,8,11)(3,12,9), (1,2,3)(4,5,6)(7,8,9,10,11,12)>;
G:=Group( (1,10)(3,12)(5,9)(6,7), (1,6)(3,5)(7,10)(9,12), (1,7)(2,4)(3,12)(5,9)(6,10)(8,11), (1,10)(2,11)(3,5)(4,8)(6,7)(9,12), (1,10,7)(2,8,11)(3,12,9), (1,2,3)(4,5,6)(7,8,9,10,11,12) );
G=PermutationGroup([(1,10),(3,12),(5,9),(6,7)], [(1,6),(3,5),(7,10),(9,12)], [(1,7),(2,4),(3,12),(5,9),(6,10),(8,11)], [(1,10),(2,11),(3,5),(4,8),(6,7),(9,12)], [(1,10,7),(2,8,11),(3,12,9)], [(1,2,3),(4,5,6),(7,8,9,10,11,12)])
G:=TransitiveGroup(12,129);
►On 16 points: primitive - transitive group 16T708
Generators in S16
(1 8)(2 5)(3 14)(4 11)(6 10)(7 12)(9 13)(15 16)
(1 2)(3 4)(5 8)(6 15)(7 13)(9 12)(10 16)(11 14)
(1 14)(2 11)(3 8)(4 5)(6 13)(7 15)(9 10)(12 16)
(1 12)(2 9)(3 15)(4 6)(5 13)(7 8)(10 11)(14 16)
(2 5 8)(3 9 6)(4 7 10)(11 13 15)(12 16 14)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
G:=sub<Sym(16)| (1,8)(2,5)(3,14)(4,11)(6,10)(7,12)(9,13)(15,16), (1,2)(3,4)(5,8)(6,15)(7,13)(9,12)(10,16)(11,14), (1,14)(2,11)(3,8)(4,5)(6,13)(7,15)(9,10)(12,16), (1,12)(2,9)(3,15)(4,6)(5,13)(7,8)(10,11)(14,16), (2,5,8)(3,9,6)(4,7,10)(11,13,15)(12,16,14), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16)>;
G:=Group( (1,8)(2,5)(3,14)(4,11)(6,10)(7,12)(9,13)(15,16), (1,2)(3,4)(5,8)(6,15)(7,13)(9,12)(10,16)(11,14), (1,14)(2,11)(3,8)(4,5)(6,13)(7,15)(9,10)(12,16), (1,12)(2,9)(3,15)(4,6)(5,13)(7,8)(10,11)(14,16), (2,5,8)(3,9,6)(4,7,10)(11,13,15)(12,16,14), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16) );
G=PermutationGroup([(1,8),(2,5),(3,14),(4,11),(6,10),(7,12),(9,13),(15,16)], [(1,2),(3,4),(5,8),(6,15),(7,13),(9,12),(10,16),(11,14)], [(1,14),(2,11),(3,8),(4,5),(6,13),(7,15),(9,10),(12,16)], [(1,12),(2,9),(3,15),(4,6),(5,13),(7,8),(10,11),(14,16)], [(2,5,8),(3,9,6),(4,7,10),(11,13,15),(12,16,14)], [(2,3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)])
G:=TransitiveGroup(16,708);
►On 18 points - transitive group 18T112
Generators in S18
(1 4)(2 5)(10 16)(12 18)
(7 13)(9 15)(10 16)(12 18)
(1 4)(2 5)(7 13)(8 14)(11 17)(12 18)
(2 5)(3 6)(9 15)(10 16)(11 17)(12 18)
(1 18 9)(2 10 13)(3 14 11)(4 12 15)(5 16 7)(6 8 17)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (1,4)(2,5)(10,16)(12,18), (7,13)(9,15)(10,16)(12,18), (1,4)(2,5)(7,13)(8,14)(11,17)(12,18), (2,5)(3,6)(9,15)(10,16)(11,17)(12,18), (1,18,9)(2,10,13)(3,14,11)(4,12,15)(5,16,7)(6,8,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (1,4)(2,5)(10,16)(12,18), (7,13)(9,15)(10,16)(12,18), (1,4)(2,5)(7,13)(8,14)(11,17)(12,18), (2,5)(3,6)(9,15)(10,16)(11,17)(12,18), (1,18,9)(2,10,13)(3,14,11)(4,12,15)(5,16,7)(6,8,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([(1,4),(2,5),(10,16),(12,18)], [(7,13),(9,15),(10,16),(12,18)], [(1,4),(2,5),(7,13),(8,14),(11,17),(12,18)], [(2,5),(3,6),(9,15),(10,16),(11,17),(12,18)], [(1,18,9),(2,10,13),(3,14,11),(4,12,15),(5,16,7),(6,8,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)])
G:=TransitiveGroup(18,112);
►On 18 points - transitive group 18T113
Generators in S18
(1 6)(2 4)(8 13)(12 17)
(8 13)(9 14)(11 16)(12 17)
(1 6)(2 4)(7 18)(8 13)(9 14)(10 15)
(2 4)(3 5)(7 18)(8 13)(11 16)(12 17)
(1 13 16)(2 17 14)(3 15 18)(4 12 9)(5 10 7)(6 8 11)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (1,6)(2,4)(8,13)(12,17), (8,13)(9,14)(11,16)(12,17), (1,6)(2,4)(7,18)(8,13)(9,14)(10,15), (2,4)(3,5)(7,18)(8,13)(11,16)(12,17), (1,13,16)(2,17,14)(3,15,18)(4,12,9)(5,10,7)(6,8,11), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (1,6)(2,4)(8,13)(12,17), (8,13)(9,14)(11,16)(12,17), (1,6)(2,4)(7,18)(8,13)(9,14)(10,15), (2,4)(3,5)(7,18)(8,13)(11,16)(12,17), (1,13,16)(2,17,14)(3,15,18)(4,12,9)(5,10,7)(6,8,11), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([(1,6),(2,4),(8,13),(12,17)], [(8,13),(9,14),(11,16),(12,17)], [(1,6),(2,4),(7,18),(8,13),(9,14),(10,15)], [(2,4),(3,5),(7,18),(8,13),(11,16),(12,17)], [(1,13,16),(2,17,14),(3,15,18),(4,12,9),(5,10,7),(6,8,11)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)])
G:=TransitiveGroup(18,113);
►On 24 points - transitive group 24T692
Generators in S24
(1 14)(2 21)(3 22)(4 9)(5 10)(6 13)(7 15)(8 16)(11 19)(12 20)(17 23)(18 24)
(1 20)(2 15)(3 8)(4 23)(5 18)(6 11)(7 21)(9 17)(10 24)(12 14)(13 19)(16 22)
(1 20)(3 8)(5 18)(10 24)(12 14)(16 22)
(1 12)(3 16)(5 24)(8 22)(10 18)(14 20)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)(13 15 17)(14 18 16)(19 21 23)(20 24 22)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
G:=sub<Sym(24)| (1,14)(2,21)(3,22)(4,9)(5,10)(6,13)(7,15)(8,16)(11,19)(12,20)(17,23)(18,24), (1,20)(2,15)(3,8)(4,23)(5,18)(6,11)(7,21)(9,17)(10,24)(12,14)(13,19)(16,22), (1,20)(3,8)(5,18)(10,24)(12,14)(16,22), (1,12)(3,16)(5,24)(8,22)(10,18)(14,20), (1,5,3)(2,4,6)(7,9,11)(8,12,10)(13,15,17)(14,18,16)(19,21,23)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,14)(2,21)(3,22)(4,9)(5,10)(6,13)(7,15)(8,16)(11,19)(12,20)(17,23)(18,24), (1,20)(2,15)(3,8)(4,23)(5,18)(6,11)(7,21)(9,17)(10,24)(12,14)(13,19)(16,22), (1,20)(3,8)(5,18)(10,24)(12,14)(16,22), (1,12)(3,16)(5,24)(8,22)(10,18)(14,20), (1,5,3)(2,4,6)(7,9,11)(8,12,10)(13,15,17)(14,18,16)(19,21,23)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([(1,14),(2,21),(3,22),(4,9),(5,10),(6,13),(7,15),(8,16),(11,19),(12,20),(17,23),(18,24)], [(1,20),(2,15),(3,8),(4,23),(5,18),(6,11),(7,21),(9,17),(10,24),(12,14),(13,19),(16,22)], [(1,20),(3,8),(5,18),(10,24),(12,14),(16,22)], [(1,12),(3,16),(5,24),(8,22),(10,18),(14,20)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10),(13,15,17),(14,18,16),(19,21,23),(20,24,22)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])
G:=TransitiveGroup(24,692);
►On 24 points - transitive group 24T694
Generators in S24
(2 21)(3 10)(5 16)(6 13)(8 12)(14 18)(15 17)(19 23)
(1 24)(2 21)(3 10)(4 7)(8 12)(9 11)(19 23)(20 22)
(1 22)(2 23)(3 12)(4 11)(5 16)(6 13)(7 9)(8 10)(14 18)(15 17)(19 21)(20 24)
(1 20)(2 21)(3 10)(4 9)(5 14)(6 17)(7 11)(8 12)(13 15)(16 18)(19 23)(22 24)
(1 6 3)(2 4 5)(7 16 21)(8 22 17)(9 18 23)(10 24 13)(11 14 19)(12 20 15)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
G:=sub<Sym(24)| (2,21)(3,10)(5,16)(6,13)(8,12)(14,18)(15,17)(19,23), (1,24)(2,21)(3,10)(4,7)(8,12)(9,11)(19,23)(20,22), (1,22)(2,23)(3,12)(4,11)(5,16)(6,13)(7,9)(8,10)(14,18)(15,17)(19,21)(20,24), (1,20)(2,21)(3,10)(4,9)(5,14)(6,17)(7,11)(8,12)(13,15)(16,18)(19,23)(22,24), (1,6,3)(2,4,5)(7,16,21)(8,22,17)(9,18,23)(10,24,13)(11,14,19)(12,20,15), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (2,21)(3,10)(5,16)(6,13)(8,12)(14,18)(15,17)(19,23), (1,24)(2,21)(3,10)(4,7)(8,12)(9,11)(19,23)(20,22), (1,22)(2,23)(3,12)(4,11)(5,16)(6,13)(7,9)(8,10)(14,18)(15,17)(19,21)(20,24), (1,20)(2,21)(3,10)(4,9)(5,14)(6,17)(7,11)(8,12)(13,15)(16,18)(19,23)(22,24), (1,6,3)(2,4,5)(7,16,21)(8,22,17)(9,18,23)(10,24,13)(11,14,19)(12,20,15), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([(2,21),(3,10),(5,16),(6,13),(8,12),(14,18),(15,17),(19,23)], [(1,24),(2,21),(3,10),(4,7),(8,12),(9,11),(19,23),(20,22)], [(1,22),(2,23),(3,12),(4,11),(5,16),(6,13),(7,9),(8,10),(14,18),(15,17),(19,21),(20,24)], [(1,20),(2,21),(3,10),(4,9),(5,14),(6,17),(7,11),(8,12),(13,15),(16,18),(19,23),(22,24)], [(1,6,3),(2,4,5),(7,16,21),(8,22,17),(9,18,23),(10,24,13),(11,14,19),(12,20,15)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])
G:=TransitiveGroup(24,694);
►On 24 points - transitive group 24T695
Generators in S24
(1 8)(2 13)(4 15)(5 12)(9 21)(11 23)(16 24)(18 20)
(1 20)(2 21)(4 23)(5 24)(8 18)(9 13)(11 15)(12 16)
(1 8)(2 9)(3 22)(4 15)(5 16)(6 19)(7 17)(10 14)(11 23)(12 24)(13 21)(18 20)
(1 20)(2 13)(3 14)(4 23)(5 12)(6 7)(8 18)(9 21)(10 22)(11 15)(16 24)(17 19)
(7 17 19)(8 20 18)(9 13 21)(10 22 14)(11 15 23)(12 24 16)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
G:=sub<Sym(24)| (1,8)(2,13)(4,15)(5,12)(9,21)(11,23)(16,24)(18,20), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (7,17,19)(8,20,18)(9,13,21)(10,22,14)(11,15,23)(12,24,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,8)(2,13)(4,15)(5,12)(9,21)(11,23)(16,24)(18,20), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (7,17,19)(8,20,18)(9,13,21)(10,22,14)(11,15,23)(12,24,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([(1,8),(2,13),(4,15),(5,12),(9,21),(11,23),(16,24),(18,20)], [(1,20),(2,21),(4,23),(5,24),(8,18),(9,13),(11,15),(12,16)], [(1,8),(2,9),(3,22),(4,15),(5,16),(6,19),(7,17),(10,14),(11,23),(12,24),(13,21),(18,20)], [(1,20),(2,13),(3,14),(4,23),(5,12),(6,7),(8,18),(9,21),(10,22),(11,15),(16,24),(17,19)], [(7,17,19),(8,20,18),(9,13,21),(10,22,14),(11,15,23),(12,24,16)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])
G:=TransitiveGroup(24,695);
►On 24 points - transitive group 24T702
Generators in S24
(1 20)(2 21)(3 18)(4 13)(5 10)(6 11)(7 17)(8 22)(9 23)(12 16)(14 24)(15 19)
(1 12)(2 17)(3 22)(4 9)(5 14)(6 19)(7 21)(8 18)(10 24)(11 15)(13 23)(16 20)
(2 17)(4 9)(6 19)(7 21)(11 15)(13 23)
(2 7)(4 23)(6 15)(9 13)(11 19)(17 21)
(1 5 3)(2 4 6)(7 13 19)(8 20 14)(9 15 21)(10 22 16)(11 17 23)(12 24 18)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
G:=sub<Sym(24)| (1,20)(2,21)(3,18)(4,13)(5,10)(6,11)(7,17)(8,22)(9,23)(12,16)(14,24)(15,19), (1,12)(2,17)(3,22)(4,9)(5,14)(6,19)(7,21)(8,18)(10,24)(11,15)(13,23)(16,20), (2,17)(4,9)(6,19)(7,21)(11,15)(13,23), (2,7)(4,23)(6,15)(9,13)(11,19)(17,21), (1,5,3)(2,4,6)(7,13,19)(8,20,14)(9,15,21)(10,22,16)(11,17,23)(12,24,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,20)(2,21)(3,18)(4,13)(5,10)(6,11)(7,17)(8,22)(9,23)(12,16)(14,24)(15,19), (1,12)(2,17)(3,22)(4,9)(5,14)(6,19)(7,21)(8,18)(10,24)(11,15)(13,23)(16,20), (2,17)(4,9)(6,19)(7,21)(11,15)(13,23), (2,7)(4,23)(6,15)(9,13)(11,19)(17,21), (1,5,3)(2,4,6)(7,13,19)(8,20,14)(9,15,21)(10,22,16)(11,17,23)(12,24,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([(1,20),(2,21),(3,18),(4,13),(5,10),(6,11),(7,17),(8,22),(9,23),(12,16),(14,24),(15,19)], [(1,12),(2,17),(3,22),(4,9),(5,14),(6,19),(7,21),(8,18),(10,24),(11,15),(13,23),(16,20)], [(2,17),(4,9),(6,19),(7,21),(11,15),(13,23)], [(2,7),(4,23),(6,15),(9,13),(11,19),(17,21)], [(1,5,3),(2,4,6),(7,13,19),(8,20,14),(9,15,21),(10,22,16),(11,17,23),(12,24,18)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])
G:=TransitiveGroup(24,702);
►On 24 points - transitive group 24T703
Generators in S24
(1 20)(2 21)(3 10)(4 11)(5 16)(6 17)(7 19)(8 18)(9 13)(12 24)(14 22)(15 23)
(1 18)(2 9)(3 22)(4 15)(5 12)(6 19)(7 17)(8 20)(10 14)(11 23)(13 21)(16 24)
(2 9)(4 15)(6 19)(7 17)(11 23)(13 21)
(2 13)(4 23)(6 7)(9 21)(11 15)(17 19)
(7 17 19)(8 20 18)(9 13 21)(10 22 14)(11 15 23)(12 24 16)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
G:=sub<Sym(24)| (1,20)(2,21)(3,10)(4,11)(5,16)(6,17)(7,19)(8,18)(9,13)(12,24)(14,22)(15,23), (1,18)(2,9)(3,22)(4,15)(5,12)(6,19)(7,17)(8,20)(10,14)(11,23)(13,21)(16,24), (2,9)(4,15)(6,19)(7,17)(11,23)(13,21), (2,13)(4,23)(6,7)(9,21)(11,15)(17,19), (7,17,19)(8,20,18)(9,13,21)(10,22,14)(11,15,23)(12,24,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,20)(2,21)(3,10)(4,11)(5,16)(6,17)(7,19)(8,18)(9,13)(12,24)(14,22)(15,23), (1,18)(2,9)(3,22)(4,15)(5,12)(6,19)(7,17)(8,20)(10,14)(11,23)(13,21)(16,24), (2,9)(4,15)(6,19)(7,17)(11,23)(13,21), (2,13)(4,23)(6,7)(9,21)(11,15)(17,19), (7,17,19)(8,20,18)(9,13,21)(10,22,14)(11,15,23)(12,24,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([(1,20),(2,21),(3,10),(4,11),(5,16),(6,17),(7,19),(8,18),(9,13),(12,24),(14,22),(15,23)], [(1,18),(2,9),(3,22),(4,15),(5,12),(6,19),(7,17),(8,20),(10,14),(11,23),(13,21),(16,24)], [(2,9),(4,15),(6,19),(7,17),(11,23),(13,21)], [(2,13),(4,23),(6,7),(9,21),(11,15),(17,19)], [(7,17,19),(8,20,18),(9,13,21),(10,22,14),(11,15,23),(12,24,16)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])
G:=TransitiveGroup(24,703);
►On 24 points - transitive group 24T704
Generators in S24
(1 18)(2 17)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 15)(14 16)(19 23)(20 24)
(1 16)(2 13)(3 23)(4 20)(5 11)(6 8)(7 9)(10 12)(14 18)(15 17)(19 21)(22 24)
(1 16)(4 20)(6 8)(10 12)(14 18)(22 24)
(1 14)(4 24)(6 12)(8 10)(16 18)(20 22)
(1 4 6)(2 5 3)(7 21 13)(8 14 22)(9 23 15)(10 16 24)(11 19 17)(12 18 20)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
G:=sub<Sym(24)| (1,18)(2,17)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(19,23)(20,24), (1,16)(2,13)(3,23)(4,20)(5,11)(6,8)(7,9)(10,12)(14,18)(15,17)(19,21)(22,24), (1,16)(4,20)(6,8)(10,12)(14,18)(22,24), (1,14)(4,24)(6,12)(8,10)(16,18)(20,22), (1,4,6)(2,5,3)(7,21,13)(8,14,22)(9,23,15)(10,16,24)(11,19,17)(12,18,20), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,18)(2,17)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(19,23)(20,24), (1,16)(2,13)(3,23)(4,20)(5,11)(6,8)(7,9)(10,12)(14,18)(15,17)(19,21)(22,24), (1,16)(4,20)(6,8)(10,12)(14,18)(22,24), (1,14)(4,24)(6,12)(8,10)(16,18)(20,22), (1,4,6)(2,5,3)(7,21,13)(8,14,22)(9,23,15)(10,16,24)(11,19,17)(12,18,20), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([(1,18),(2,17),(3,21),(4,22),(5,9),(6,10),(7,11),(8,12),(13,15),(14,16),(19,23),(20,24)], [(1,16),(2,13),(3,23),(4,20),(5,11),(6,8),(7,9),(10,12),(14,18),(15,17),(19,21),(22,24)], [(1,16),(4,20),(6,8),(10,12),(14,18),(22,24)], [(1,14),(4,24),(6,12),(8,10),(16,18),(20,22)], [(1,4,6),(2,5,3),(7,21,13),(8,14,22),(9,23,15),(10,16,24),(11,19,17),(12,18,20)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])
G:=TransitiveGroup(24,704);
Matrix representation ►G ⊆ GL6(ℤ)
| 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 |
| -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | -1 | -1 | 0 | 0 | 0 |
G:=sub<GL(6,Integers())| [0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0],[1,-1,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1,1,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0] >;
Character table of A4≀C2
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | |
| size | 1 | 6 | 9 | 12 | 8 | 8 | 16 | 16 | 32 | 36 | 24 | 24 | 48 | 48 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | 2 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 0 | ζ65 | ζ6 | -1+√-3 | -1-√-3 | -1 | 0 | ζ6 | ζ65 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 2 | 0 | ζ6 | ζ65 | -1-√-3 | -1+√-3 | -1 | 0 | ζ65 | ζ6 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 6 | 2 | -2 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal faithful |
| ρ11 | 6 | 2 | -2 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | complex faithful |
| ρ12 | 6 | 2 | -2 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | complex faithful |
| ρ13 | 9 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ14 | 9 | -3 | 1 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal faithful |
In GAP, Magma, Sage, TeX
A_4\wr C_2
% in TeX
G:=Group("A4wrC2"); // GroupNames label
G:=SmallGroup(288,1025);
// by ID
G=gap.SmallGroup(288,1025);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,2,2,254,2019,766,185,5044,326,333,761,4548,1531,1770,1777,608]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^3=f^6=1,e*a*e^-1=a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,f*d*f^-1=b*c=c*b,b*d=d*b,e*b*e^-1=a,f*b*f^-1=a*b*c,e*c*e^-1=c*d=d*c,f*c*f^-1=a*b*d,e*d*e^-1=c,f*e*f^-1=e^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀