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G = C42⋊D6order 192 = 26·3

The semidirect product of C42 and D6 acting faithfully

non-abelian, soluble, monomial, rational

Aliases: C42⋊D6, C23.3S4, C41D4⋊S3, C42⋊S3⋊C2, C42⋊C31C22, C22.3(C2×S4), C23.A42C2, SmallGroup(192,956)

Series: Derived Chief Lower central Upper central

C1C42C42⋊C3 — C42⋊D6
C1C22C42C42⋊C3C42⋊S3 — C42⋊D6
C42⋊C3 — C42⋊D6
C1

Generators and relations for C42⋊D6
 G = < a,b,c,d | a4=b4=c6=d2=1, dad=cbc-1=ab=ba, cac-1=dbd=b-1, dcd=c-1 >

Subgroups: 480 in 83 conjugacy classes, 10 normal (8 characteristic)
C1, C2 [×5], C3, C4 [×4], C22, C22 [×7], S3 [×2], C6, C8 [×2], C2×C4 [×5], D4 [×12], Q8 [×2], C23, C23 [×3], A4, D6, C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×6], C4○D4 [×4], S4 [×2], C2×A4, C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C42⋊C3, C2×S4, D44D4, C42⋊S3 [×2], C23.A4, C42⋊D6
Quotients: C1, C2 [×3], C22, S3, D6, S4, C2×S4, C42⋊D6

Character table of C42⋊D6

 class 12A2B2C2D2E34A4B4C4D68A8B
 size 13412121232661212322424
ρ111111111111111    trivial
ρ211-1-1-111111-1-11-1    linear of order 2
ρ311-11-1-1111-11-1-11    linear of order 2
ρ4111-11-1111-1-11-1-1    linear of order 2
ρ522-20-20-12200100    orthogonal lifted from D6
ρ6222020-12200-100    orthogonal lifted from S3
ρ733-3-1110-1-11-10-11    orthogonal lifted from C2×S4
ρ833-311-10-1-1-1101-1    orthogonal lifted from C2×S4
ρ9333-1-1-10-1-1-1-1011    orthogonal lifted from S4
ρ103331-110-1-1110-1-1    orthogonal lifted from S4
ρ116-20-2000-2202000    orthogonal faithful
ρ126-2000202-2-20000    orthogonal faithful
ρ136-202000-220-2000    orthogonal faithful
ρ146-2000-202-220000    orthogonal faithful

Permutation representations of C42⋊D6
On 12 points - transitive group 12T112
Generators in S12
(1 4 7 10)(2 5 8 11)
(1 10 7 4)(3 6 9 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 9)(2 8)(3 7)(4 12)(5 11)(6 10)

G:=sub<Sym(12)| (1,4,7,10)(2,5,8,11), (1,10,7,4)(3,6,9,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)>;

G:=Group( (1,4,7,10)(2,5,8,11), (1,10,7,4)(3,6,9,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10) );

G=PermutationGroup([(1,4,7,10),(2,5,8,11)], [(1,10,7,4),(3,6,9,12)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10)])

G:=TransitiveGroup(12,112);

On 12 points - transitive group 12T113
Generators in S12
(1 4 7 10)(2 5 8 11)
(1 10 7 4)(3 6 9 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 3)(4 6)(7 9)(10 12)

G:=sub<Sym(12)| (1,4,7,10)(2,5,8,11), (1,10,7,4)(3,6,9,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,3)(4,6)(7,9)(10,12)>;

G:=Group( (1,4,7,10)(2,5,8,11), (1,10,7,4)(3,6,9,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,3)(4,6)(7,9)(10,12) );

G=PermutationGroup([(1,4,7,10),(2,5,8,11)], [(1,10,7,4),(3,6,9,12)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,3),(4,6),(7,9),(10,12)])

G:=TransitiveGroup(12,113);

On 12 points - transitive group 12T114
Generators in S12
(1 7 6 10)(2 4)(3 9 5 12)(8 11)
(1 6)(2 11 4 8)(3 9 5 12)(7 10)
(1 2 3)(4 5 6)(7 8 9 10 11 12)
(2 3)(4 5)(7 10)(8 9)(11 12)

G:=sub<Sym(12)| (1,7,6,10)(2,4)(3,9,5,12)(8,11), (1,6)(2,11,4,8)(3,9,5,12)(7,10), (1,2,3)(4,5,6)(7,8,9,10,11,12), (2,3)(4,5)(7,10)(8,9)(11,12)>;

G:=Group( (1,7,6,10)(2,4)(3,9,5,12)(8,11), (1,6)(2,11,4,8)(3,9,5,12)(7,10), (1,2,3)(4,5,6)(7,8,9,10,11,12), (2,3)(4,5)(7,10)(8,9)(11,12) );

G=PermutationGroup([(1,7,6,10),(2,4),(3,9,5,12),(8,11)], [(1,6),(2,11,4,8),(3,9,5,12),(7,10)], [(1,2,3),(4,5,6),(7,8,9,10,11,12)], [(2,3),(4,5),(7,10),(8,9),(11,12)])

G:=TransitiveGroup(12,114);

On 12 points - transitive group 12T115
Generators in S12
(1 7 4 10)(3 12 6 9)
(2 8 5 11)(3 9 6 12)
(1 2 3)(4 5 6)(7 8 9 10 11 12)
(1 4)(2 6)(3 5)(7 10)(8 9)(11 12)

G:=sub<Sym(12)| (1,7,4,10)(3,12,6,9), (2,8,5,11)(3,9,6,12), (1,2,3)(4,5,6)(7,8,9,10,11,12), (1,4)(2,6)(3,5)(7,10)(8,9)(11,12)>;

G:=Group( (1,7,4,10)(3,12,6,9), (2,8,5,11)(3,9,6,12), (1,2,3)(4,5,6)(7,8,9,10,11,12), (1,4)(2,6)(3,5)(7,10)(8,9)(11,12) );

G=PermutationGroup([(1,7,4,10),(3,12,6,9)], [(2,8,5,11),(3,9,6,12)], [(1,2,3),(4,5,6),(7,8,9,10,11,12)], [(1,4),(2,6),(3,5),(7,10),(8,9),(11,12)])

G:=TransitiveGroup(12,115);

On 16 points - transitive group 16T431
Generators in S16
(1 10 4 7)(2 12 3 15)(5 14 16 6)(8 9 13 11)
(1 6 3 9)(2 11 4 14)(5 15 13 10)(7 16 12 8)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(2 4)(5 10)(6 9)(7 8)(12 16)(13 15)

G:=sub<Sym(16)| (1,10,4,7)(2,12,3,15)(5,14,16,6)(8,9,13,11), (1,6,3,9)(2,11,4,14)(5,15,13,10)(7,16,12,8), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16), (2,4)(5,10)(6,9)(7,8)(12,16)(13,15)>;

G:=Group( (1,10,4,7)(2,12,3,15)(5,14,16,6)(8,9,13,11), (1,6,3,9)(2,11,4,14)(5,15,13,10)(7,16,12,8), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16), (2,4)(5,10)(6,9)(7,8)(12,16)(13,15) );

G=PermutationGroup([(1,10,4,7),(2,12,3,15),(5,14,16,6),(8,9,13,11)], [(1,6,3,9),(2,11,4,14),(5,15,13,10),(7,16,12,8)], [(2,3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)], [(2,4),(5,10),(6,9),(7,8),(12,16),(13,15)])

G:=TransitiveGroup(16,431);

On 24 points - transitive group 24T530
Generators in S24
(1 18 8 24)(2 13 9 19)(4 21 11 15)(5 22 12 16)
(1 24 8 18)(3 14 10 20)(4 15 11 21)(6 23 7 17)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 18)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)

G:=sub<Sym(24)| (1,18,8,24)(2,13,9,19)(4,21,11,15)(5,22,12,16), (1,24,8,18)(3,14,10,20)(4,15,11,21)(6,23,7,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)>;

G:=Group( (1,18,8,24)(2,13,9,19)(4,21,11,15)(5,22,12,16), (1,24,8,18)(3,14,10,20)(4,15,11,21)(6,23,7,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19) );

G=PermutationGroup([(1,18,8,24),(2,13,9,19),(4,21,11,15),(5,22,12,16)], [(1,24,8,18),(3,14,10,20),(4,15,11,21),(6,23,7,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,18),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)])

G:=TransitiveGroup(24,530);

On 24 points - transitive group 24T531
Generators in S24
(2 5 16 13)(3 6 17 14)(7 19 22 10)(9 12 24 21)
(1 18 15 4)(2 13 16 5)(8 20 23 11)(9 21 24 12)
(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 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)

G:=sub<Sym(24)| (2,5,16,13)(3,6,17,14)(7,19,22,10)(9,12,24,21), (1,18,15,4)(2,13,16,5)(8,20,23,11)(9,21,24,12), (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)(5,8)(6,7)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24)>;

G:=Group( (2,5,16,13)(3,6,17,14)(7,19,22,10)(9,12,24,21), (1,18,15,4)(2,13,16,5)(8,20,23,11)(9,21,24,12), (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)(5,8)(6,7)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24) );

G=PermutationGroup([(2,5,16,13),(3,6,17,14),(7,19,22,10),(9,12,24,21)], [(1,18,15,4),(2,13,16,5),(8,20,23,11),(9,21,24,12)], [(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),(5,8),(6,7),(13,23),(14,22),(15,21),(16,20),(17,19),(18,24)])

G:=TransitiveGroup(24,531);

On 24 points - transitive group 24T532
Generators in S24
(1 8 20 18)(2 13 21 9)(3 22)(4 15 23 11)(5 12 24 16)(6 19)(7 17)(10 14)
(1 8 20 18)(2 21)(3 10 22 14)(4 15 23 11)(5 24)(6 17 19 7)(9 13)(12 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)
(1 19)(2 24)(3 23)(4 22)(5 21)(6 20)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

G:=sub<Sym(24)| (1,8,20,18)(2,13,21,9)(3,22)(4,15,23,11)(5,12,24,16)(6,19)(7,17)(10,14), (1,8,20,18)(2,21)(3,10,22,14)(4,15,23,11)(5,24)(6,17,19,7)(9,13)(12,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,19)(2,24)(3,23)(4,22)(5,21)(6,20)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;

G:=Group( (1,8,20,18)(2,13,21,9)(3,22)(4,15,23,11)(5,12,24,16)(6,19)(7,17)(10,14), (1,8,20,18)(2,21)(3,10,22,14)(4,15,23,11)(5,24)(6,17,19,7)(9,13)(12,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,19)(2,24)(3,23)(4,22)(5,21)(6,20)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );

G=PermutationGroup([(1,8,20,18),(2,13,21,9),(3,22),(4,15,23,11),(5,12,24,16),(6,19),(7,17),(10,14)], [(1,8,20,18),(2,21),(3,10,22,14),(4,15,23,11),(5,24),(6,17,19,7),(9,13),(12,16)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,19),(2,24),(3,23),(4,22),(5,21),(6,20),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)])

G:=TransitiveGroup(24,532);

On 24 points - transitive group 24T533
Generators in S24
(1 18 20 8)(2 13 21 9)(4 11 23 15)(5 12 24 16)
(1 8 20 18)(3 14 22 10)(4 15 23 11)(6 7 19 17)
(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 3)(4 6)(7 11)(8 10)(14 18)(15 17)(19 23)(20 22)

G:=sub<Sym(24)| (1,18,20,8)(2,13,21,9)(4,11,23,15)(5,12,24,16), (1,8,20,18)(3,14,22,10)(4,15,23,11)(6,7,19,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3)(4,6)(7,11)(8,10)(14,18)(15,17)(19,23)(20,22)>;

G:=Group( (1,18,20,8)(2,13,21,9)(4,11,23,15)(5,12,24,16), (1,8,20,18)(3,14,22,10)(4,15,23,11)(6,7,19,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3)(4,6)(7,11)(8,10)(14,18)(15,17)(19,23)(20,22) );

G=PermutationGroup([(1,18,20,8),(2,13,21,9),(4,11,23,15),(5,12,24,16)], [(1,8,20,18),(3,14,22,10),(4,15,23,11),(6,7,19,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,3),(4,6),(7,11),(8,10),(14,18),(15,17),(19,23),(20,22)])

G:=TransitiveGroup(24,533);

On 24 points - transitive group 24T534
Generators in S24
(1 15 9 18)(3 14 8 17)(4 22 11 19)(6 21 10 24)
(2 16 7 13)(3 17 8 14)(5 23 12 20)(6 24 10 21)
(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 11)(2 10)(3 12)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)

G:=sub<Sym(24)| (1,15,9,18)(3,14,8,17)(4,22,11,19)(6,21,10,24), (2,16,7,13)(3,17,8,14)(5,23,12,20)(6,24,10,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,11)(2,10)(3,12)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)>;

G:=Group( (1,15,9,18)(3,14,8,17)(4,22,11,19)(6,21,10,24), (2,16,7,13)(3,17,8,14)(5,23,12,20)(6,24,10,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,11)(2,10)(3,12)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22) );

G=PermutationGroup([(1,15,9,18),(3,14,8,17),(4,22,11,19),(6,21,10,24)], [(2,16,7,13),(3,17,8,14),(5,23,12,20),(6,24,10,21)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,11),(2,10),(3,12),(4,9),(5,8),(6,7),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)])

G:=TransitiveGroup(24,534);

On 24 points - transitive group 24T535
Generators in S24
(1 11)(2 23 6 20)(3 15 4 18)(5 9)(7 17 12 14)(8 21 10 24)(13 22)(16 19)
(1 19 5 22)(2 17 6 14)(3 10)(4 8)(7 23 12 20)(9 13 11 16)(15 24)(18 21)
(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 2)(5 6)(7 9)(11 12)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)

G:=sub<Sym(24)| (1,11)(2,23,6,20)(3,15,4,18)(5,9)(7,17,12,14)(8,21,10,24)(13,22)(16,19), (1,19,5,22)(2,17,6,14)(3,10)(4,8)(7,23,12,20)(9,13,11,16)(15,24)(18,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,2)(5,6)(7,9)(11,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)>;

G:=Group( (1,11)(2,23,6,20)(3,15,4,18)(5,9)(7,17,12,14)(8,21,10,24)(13,22)(16,19), (1,19,5,22)(2,17,6,14)(3,10)(4,8)(7,23,12,20)(9,13,11,16)(15,24)(18,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,2)(5,6)(7,9)(11,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21) );

G=PermutationGroup([(1,11),(2,23,6,20),(3,15,4,18),(5,9),(7,17,12,14),(8,21,10,24),(13,22),(16,19)], [(1,19,5,22),(2,17,6,14),(3,10),(4,8),(7,23,12,20),(9,13,11,16),(15,24),(18,21)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,2),(5,6),(7,9),(11,12),(13,20),(14,19),(15,24),(16,23),(17,22),(18,21)])

G:=TransitiveGroup(24,535);

On 24 points - transitive group 24T536
Generators in S24
(1 6)(2 17 12 14)(3 23 10 20)(4 22 7 19)(5 18 8 15)(9 11)(13 24)(16 21)
(1 13 11 16)(2 19 12 22)(3 5)(4 14 7 17)(6 24 9 21)(8 10)(15 20)(18 23)
(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 7)(2 9)(3 8)(4 11)(5 10)(6 12)(13 14)(15 18)(16 17)(19 24)(20 23)(21 22)

G:=sub<Sym(24)| (1,6)(2,17,12,14)(3,23,10,20)(4,22,7,19)(5,18,8,15)(9,11)(13,24)(16,21), (1,13,11,16)(2,19,12,22)(3,5)(4,14,7,17)(6,24,9,21)(8,10)(15,20)(18,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,7)(2,9)(3,8)(4,11)(5,10)(6,12)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22)>;

G:=Group( (1,6)(2,17,12,14)(3,23,10,20)(4,22,7,19)(5,18,8,15)(9,11)(13,24)(16,21), (1,13,11,16)(2,19,12,22)(3,5)(4,14,7,17)(6,24,9,21)(8,10)(15,20)(18,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,7)(2,9)(3,8)(4,11)(5,10)(6,12)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22) );

G=PermutationGroup([(1,6),(2,17,12,14),(3,23,10,20),(4,22,7,19),(5,18,8,15),(9,11),(13,24),(16,21)], [(1,13,11,16),(2,19,12,22),(3,5),(4,14,7,17),(6,24,9,21),(8,10),(15,20),(18,23)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,7),(2,9),(3,8),(4,11),(5,10),(6,12),(13,14),(15,18),(16,17),(19,24),(20,23),(21,22)])

G:=TransitiveGroup(24,536);

On 24 points - transitive group 24T537
Generators in S24
(1 18 8 24)(2 13 9 19)(4 21 11 15)(5 22 12 16)
(1 24 8 18)(3 14 10 20)(4 15 11 21)(6 23 7 17)
(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 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)

G:=sub<Sym(24)| (1,18,8,24)(2,13,9,19)(4,21,11,15)(5,22,12,16), (1,24,8,18)(3,14,10,20)(4,15,11,21)(6,23,7,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)>;

G:=Group( (1,18,8,24)(2,13,9,19)(4,21,11,15)(5,22,12,16), (1,24,8,18)(3,14,10,20)(4,15,11,21)(6,23,7,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23) );

G=PermutationGroup([(1,18,8,24),(2,13,9,19),(4,21,11,15),(5,22,12,16)], [(1,24,8,18),(3,14,10,20),(4,15,11,21),(6,23,7,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,22),(14,21),(15,20),(16,19),(17,24),(18,23)])

G:=TransitiveGroup(24,537);

On 24 points - transitive group 24T538
Generators in S24
(1 17 7 19)(2 15 8 23)(3 12)(4 22 10 14)(5 20 11 18)(6 9)(13 16)(21 24)
(1 22 7 14)(2 11)(3 13 9 21)(4 17 10 19)(5 8)(6 24 12 16)(15 18)(20 23)
(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 8)(6 7)(13 17)(14 16)(19 21)(22 24)

G:=sub<Sym(24)| (1,17,7,19)(2,15,8,23)(3,12)(4,22,10,14)(5,20,11,18)(6,9)(13,16)(21,24), (1,22,7,14)(2,11)(3,13,9,21)(4,17,10,19)(5,8)(6,24,12,16)(15,18)(20,23), (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)(5,8)(6,7)(13,17)(14,16)(19,21)(22,24)>;

G:=Group( (1,17,7,19)(2,15,8,23)(3,12)(4,22,10,14)(5,20,11,18)(6,9)(13,16)(21,24), (1,22,7,14)(2,11)(3,13,9,21)(4,17,10,19)(5,8)(6,24,12,16)(15,18)(20,23), (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)(5,8)(6,7)(13,17)(14,16)(19,21)(22,24) );

G=PermutationGroup([(1,17,7,19),(2,15,8,23),(3,12),(4,22,10,14),(5,20,11,18),(6,9),(13,16),(21,24)], [(1,22,7,14),(2,11),(3,13,9,21),(4,17,10,19),(5,8),(6,24,12,16),(15,18),(20,23)], [(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),(5,8),(6,7),(13,17),(14,16),(19,21),(22,24)])

G:=TransitiveGroup(24,538);

On 24 points - transitive group 24T539
Generators in S24
(1 8 16 22)(2 20 17 12)(3 6)(4 19 13 11)(5 9 14 23)(7 24)(10 21)(15 18)
(1 11 16 19)(2 5)(3 10 18 24)(4 22 13 8)(6 21 15 7)(9 20)(12 23)(14 17)
(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 6)(2 5)(3 4)(7 11)(8 10)(13 18)(14 17)(15 16)(19 21)(22 24)

G:=sub<Sym(24)| (1,8,16,22)(2,20,17,12)(3,6)(4,19,13,11)(5,9,14,23)(7,24)(10,21)(15,18), (1,11,16,19)(2,5)(3,10,18,24)(4,22,13,8)(6,21,15,7)(9,20)(12,23)(14,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,11)(8,10)(13,18)(14,17)(15,16)(19,21)(22,24)>;

G:=Group( (1,8,16,22)(2,20,17,12)(3,6)(4,19,13,11)(5,9,14,23)(7,24)(10,21)(15,18), (1,11,16,19)(2,5)(3,10,18,24)(4,22,13,8)(6,21,15,7)(9,20)(12,23)(14,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,11)(8,10)(13,18)(14,17)(15,16)(19,21)(22,24) );

G=PermutationGroup([(1,8,16,22),(2,20,17,12),(3,6),(4,19,13,11),(5,9,14,23),(7,24),(10,21),(15,18)], [(1,11,16,19),(2,5),(3,10,18,24),(4,22,13,8),(6,21,15,7),(9,20),(12,23),(14,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,11),(8,10),(13,18),(14,17),(15,16),(19,21),(22,24)])

G:=TransitiveGroup(24,539);

On 24 points - transitive group 24T540
Generators in S24
(1 14 6 17)(2 19 4 22)(3 9)(5 12)(7 24 10 21)(8 15 11 18)(13 23)(16 20)
(1 21 6 24)(2 8)(3 16 5 13)(4 11)(7 17 10 14)(9 20 12 23)(15 19)(18 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)
(1 5)(2 4)(3 6)(7 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

G:=sub<Sym(24)| (1,14,6,17)(2,19,4,22)(3,9)(5,12)(7,24,10,21)(8,15,11,18)(13,23)(16,20), (1,21,6,24)(2,8)(3,16,5,13)(4,11)(7,17,10,14)(9,20,12,23)(15,19)(18,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5)(2,4)(3,6)(7,12)(8,11)(9,10)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;

G:=Group( (1,14,6,17)(2,19,4,22)(3,9)(5,12)(7,24,10,21)(8,15,11,18)(13,23)(16,20), (1,21,6,24)(2,8)(3,16,5,13)(4,11)(7,17,10,14)(9,20,12,23)(15,19)(18,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5)(2,4)(3,6)(7,12)(8,11)(9,10)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );

G=PermutationGroup([(1,14,6,17),(2,19,4,22),(3,9),(5,12),(7,24,10,21),(8,15,11,18),(13,23),(16,20)], [(1,21,6,24),(2,8),(3,16,5,13),(4,11),(7,17,10,14),(9,20,12,23),(15,19),(18,22)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5),(2,4),(3,6),(7,12),(8,11),(9,10),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)])

G:=TransitiveGroup(24,540);

On 24 points - transitive group 24T541
Generators in S24
(1 11 16 22)(2 20 17 9)(3 6)(4 19 13 8)(5 12 14 23)(7 21)(10 24)(15 18)
(1 8 16 19)(2 5)(3 7 18 24)(4 22 13 11)(6 21 15 10)(9 23)(12 20)(14 17)
(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 15)(2 14)(3 13)(4 18)(5 17)(6 16)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)

G:=sub<Sym(24)| (1,11,16,22)(2,20,17,9)(3,6)(4,19,13,8)(5,12,14,23)(7,21)(10,24)(15,18), (1,8,16,19)(2,5)(3,7,18,24)(4,22,13,11)(6,21,15,10)(9,23)(12,20)(14,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23)>;

G:=Group( (1,11,16,22)(2,20,17,9)(3,6)(4,19,13,8)(5,12,14,23)(7,21)(10,24)(15,18), (1,8,16,19)(2,5)(3,7,18,24)(4,22,13,11)(6,21,15,10)(9,23)(12,20)(14,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23) );

G=PermutationGroup([(1,11,16,22),(2,20,17,9),(3,6),(4,19,13,8),(5,12,14,23),(7,21),(10,24),(15,18)], [(1,8,16,19),(2,5),(3,7,18,24),(4,22,13,11),(6,21,15,10),(9,23),(12,20),(14,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,15),(2,14),(3,13),(4,18),(5,17),(6,16),(7,22),(8,21),(9,20),(10,19),(11,24),(12,23)])

G:=TransitiveGroup(24,541);

Polynomial with Galois group C42⋊D6 over ℚ
actionf(x)Disc(f)
12T112x12-18x8-8x4+16272·298
12T113x12+4x10-3x8-6x6+13x4-12x2+4240·794
12T114x12-13x10+61x8-125x6+105x4-27x2+2225·7334
12T115x12-2x8+3x4-4-242·58

Matrix representation of C42⋊D6 in GL6(ℤ)

010000
-100000
000-100
001000
000010
000001
,
0-10000
100000
001000
000100
000001
0000-10
,
00000-1
0000-10
0-10000
-100000
000-100
00-1000
,
0000-10
00000-1
00-1000
000-100
-100000
0-10000

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,-1,0,0,0,0,-1,0,0,0,0,0],[0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0] >;

C42⋊D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes D_6
% in TeX

G:=Group("C4^2:D6");
// GroupNames label

G:=SmallGroup(192,956);
// by ID

G=gap.SmallGroup(192,956);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,170,675,2194,185,360,424,1271,1173,102,6053,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,d*a*d=c*b*c^-1=a*b=b*a,c*a*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Character table of C42⋊D6 in TeX

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