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G = C24⋊D6order 192 = 26·3

1st semidirect product of C24 and D6 acting faithfully

non-abelian, soluble, monomial, rational

Aliases: C231S4, C241D6, C22⋊S4⋊C2, C22≀C2⋊S3, C22.2(C2×S4), C24⋊C62C2, C22⋊A42C22, Aut(C2×Q8), SmallGroup(192,955)

Series: Derived Chief Lower central Upper central

C1C24C22⋊A4 — C24⋊D6
C1C22C24C22⋊A4C22⋊S4 — C24⋊D6
C22⋊A4 — C24⋊D6
C1

Generators and relations for C24⋊D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, ad=da, eae-1=cb=fbf=bc, faf=ebe-1=abd, bd=db, ede-1=fdf=cd=dc, ece-1=d, cf=fc, fef=e-1 >

Subgroups: 600 in 99 conjugacy classes, 10 normal (8 characteristic)
C1, C2 [×6], C3, C4 [×5], C22, C22 [×13], S3 [×2], C6, C2×C4 [×8], D4 [×13], Q8, C23, C23 [×6], A4 [×3], D6, C22⋊C4 [×5], C2×D4 [×8], C4○D4 [×3], C24, S4 [×4], C2×A4, C23⋊C4 [×3], C22≀C2, C22≀C2 [×2], 2+ 1+4, C2×S4, C22⋊A4, C2≀C22, C24⋊C6, C22⋊S4 [×2], C24⋊D6
Quotients: C1, C2 [×3], C22, S3, D6, S4, C2×S4, C24⋊D6

Character table of C24⋊D6

 class 12A2B2C2D2E2F34A4B4C4D4E6
 size 13466121232121212242432
ρ111111111111111    trivial
ρ211-1111-111-1-1-11-1    linear of order 2
ρ311-111-111-1-111-1-1    linear of order 2
ρ411111-1-11-11-1-1-11    linear of order 2
ρ522-22200-10-20001    orthogonal lifted from D6
ρ62222200-102000-1    orthogonal lifted from S3
ρ7333-1-1-1-10-1-1-1110    orthogonal lifted from S4
ρ833-3-1-11-1011-11-10    orthogonal lifted from C2×S4
ρ933-3-1-1-110-111-110    orthogonal lifted from C2×S4
ρ10333-1-11101-11-1-10    orthogonal lifted from S4
ρ116-20-22200-200000    orthogonal faithful
ρ126-202-20-20002000    orthogonal faithful
ρ136-202-202000-2000    orthogonal faithful
ρ146-20-22-200200000    orthogonal faithful

Permutation representations of C24⋊D6
On 8 points - transitive group 8T41
Generators in S8
(1 4)(6 8)
(1 6)(4 8)
(1 6)(2 3)(4 8)(5 7)
(1 8)(2 5)(3 7)(4 6)
(1 2)(3 4 5 6 7 8)
(1 2)(3 6)(4 5)(7 8)

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

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

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

G:=TransitiveGroup(8,41);

On 12 points - transitive group 12T108
Generators in S12
(2 11)(4 7)
(4 7)(6 9)
(1 10)(3 12)(4 7)(6 9)
(2 11)(3 12)(5 8)(6 9)
(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)| (2,11)(4,7), (4,7)(6,9), (1,10)(3,12)(4,7)(6,9), (2,11)(3,12)(5,8)(6,9), (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( (2,11)(4,7), (4,7)(6,9), (1,10)(3,12)(4,7)(6,9), (2,11)(3,12)(5,8)(6,9), (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([(2,11),(4,7)], [(4,7),(6,9)], [(1,10),(3,12),(4,7),(6,9)], [(2,11),(3,12),(5,8),(6,9)], [(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(12,108);

On 12 points - transitive group 12T109
Generators in S12
(2 11)(4 7)
(4 7)(6 9)
(1 10)(3 12)(4 7)(6 9)
(2 11)(3 12)(5 8)(6 9)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)

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

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

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

G:=TransitiveGroup(12,109);

On 12 points - transitive group 12T110
Generators in S12
(1 11)(2 4)(3 10)(5 7)(6 8)(9 12)
(1 6)(2 12)(3 7)(4 9)(5 10)(8 11)
(2 4)(3 5)(7 10)(9 12)
(1 6)(2 4)(8 11)(9 12)
(1 2 3)(4 5 6)(7 8 9 10 11 12)
(2 3)(4 5)(7 9)(10 12)

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

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

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

G:=TransitiveGroup(12,110);

On 12 points - transitive group 12T111
Generators in S12
(1 11)(3 7)(5 10)(6 8)
(2 9)(3 7)(4 12)(5 10)
(2 4)(3 5)(7 10)(9 12)
(1 6)(2 4)(8 11)(9 12)
(1 2 3)(4 5 6)(7 8 9 10 11 12)
(1 6)(2 5)(3 4)(7 9)(10 12)

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

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

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

G:=TransitiveGroup(12,111);

On 16 points - transitive group 16T435
Generators in S16
(1 10)(2 5)(3 15)(4 14)(6 8)(7 9)(11 13)(12 16)
(1 6)(2 7)(3 11)(4 16)(5 9)(8 10)(12 14)(13 15)
(1 8)(2 5)(3 15)(4 12)(6 10)(7 9)(11 13)(14 16)
(1 10)(2 7)(3 11)(4 14)(5 9)(6 8)(12 16)(13 15)
(1 2)(3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(1 4)(2 3)(5 15)(6 14)(7 13)(8 12)(9 11)(10 16)

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

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

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

G:=TransitiveGroup(16,435);

On 16 points - transitive group 16T436
Generators in S16
(1 5)(2 11)(3 14)(4 8)(6 16)(7 9)(10 15)(12 13)
(1 7)(2 16)(3 10)(4 13)(5 9)(6 11)(8 12)(14 15)
(1 3)(2 4)(5 14)(6 12)(7 10)(8 11)(9 15)(13 16)
(1 2)(3 4)(5 11)(6 9)(7 16)(8 14)(10 13)(12 15)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(2 4)(5 6)(7 10)(8 9)(11 15)(12 14)

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

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

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

G:=TransitiveGroup(16,436);

On 24 points - transitive group 24T516
Generators in S24
(4 14)(6 16)(7 21)(9 23)
(2 18)(6 16)(9 23)(11 19)
(2 18)(3 13)(5 15)(6 16)(8 22)(9 23)(11 19)(12 20)
(1 17)(2 18)(4 14)(5 15)(7 21)(8 22)(10 24)(11 19)
(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 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)

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

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

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

G:=TransitiveGroup(24,516);

On 24 points - transitive group 24T517
Generators in S24
(1 20)(2 13)(4 15)(5 24)(8 18)(9 21)(11 23)(12 16)
(1 20)(3 22)(4 15)(6 17)(7 19)(8 18)(10 14)(11 23)
(1 8)(3 10)(4 11)(6 7)(14 22)(15 23)(17 19)(18 20)
(2 9)(3 10)(5 12)(6 7)(13 21)(14 22)(16 24)(17 19)
(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 12)(3 11)(4 10)(5 9)(6 8)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

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

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

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

G:=TransitiveGroup(24,517);

On 24 points - transitive group 24T518
Generators in S24
(1 24)(2 13)(4 15)(5 22)(8 18)(9 19)(11 21)(12 16)
(1 24)(3 20)(4 15)(6 17)(7 23)(8 18)(10 14)(11 21)
(1 8)(3 10)(4 11)(6 7)(14 20)(15 21)(17 23)(18 24)
(2 9)(3 10)(5 12)(6 7)(13 19)(14 20)(16 22)(17 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 23)(2 22)(3 21)(4 20)(5 19)(6 24)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(24,518);

On 24 points - transitive group 24T519
Generators in S24
(2 9)(4 15)(6 19)(7 17)(11 23)(13 21)
(2 21)(4 11)(6 17)(7 19)(9 13)(15 23)
(1 8)(2 21)(3 14)(4 11)(5 24)(6 17)(7 19)(9 13)(10 22)(12 16)(15 23)(18 20)
(1 20)(2 13)(3 10)(4 23)(5 16)(6 7)(8 18)(9 21)(11 15)(12 24)(14 22)(17 19)
(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 18)(8 17)(9 16)(10 15)(11 14)(12 13)(19 20)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,519);

On 24 points - transitive group 24T520
Generators in S24
(1 18)(2 13)(3 10)(4 21)(5 22)(6 7)(8 24)(9 19)(11 15)(12 16)(14 20)(17 23)
(1 24)(2 9)(3 14)(4 15)(5 12)(6 23)(7 17)(8 18)(10 20)(11 21)(13 19)(16 22)
(1 8)(3 10)(4 11)(6 7)(14 20)(15 21)(17 23)(18 24)
(2 9)(3 10)(5 12)(6 7)(13 19)(14 20)(16 22)(17 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 3)(4 6)(7 11)(8 10)(14 18)(15 17)(20 24)(21 23)

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

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

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

G:=TransitiveGroup(24,520);

On 24 points - transitive group 24T521
Generators in S24
(1 18)(2 9)(3 14)(6 7)(8 20)(10 22)(13 21)(17 19)
(2 9)(3 14)(4 11)(5 16)(10 22)(12 24)(13 21)(15 23)
(1 8)(2 21)(3 14)(4 11)(5 24)(6 17)(7 19)(9 13)(10 22)(12 16)(15 23)(18 20)
(1 20)(2 13)(3 10)(4 23)(5 16)(6 7)(8 18)(9 21)(11 15)(12 24)(14 22)(17 19)
(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 15)(8 14)(9 13)(10 18)(11 17)(12 16)(19 23)(20 22)

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

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

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

G:=TransitiveGroup(24,521);

On 24 points - transitive group 24T522
Generators in S24
(2 23)(3 21)(4 17)(5 15)(7 20)(8 24)(11 14)(12 18)
(1 19)(2 23)(4 17)(6 13)(7 20)(9 22)(10 16)(11 14)
(1 9)(2 7)(4 11)(6 10)(13 16)(14 17)(19 22)(20 23)
(1 9)(3 8)(5 12)(6 10)(13 16)(15 18)(19 22)(21 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 11)(2 10)(3 12)(4 9)(5 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)

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

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

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

G:=TransitiveGroup(24,522);

On 24 points - transitive group 24T523
Generators in S24
(1 16)(2 21)(3 12)(4 17)(5 9)(6 20)(7 13)(8 24)(10 23)(11 14)(15 22)(18 19)
(1 23)(2 11)(3 18)(4 8)(5 22)(6 13)(7 20)(9 15)(10 16)(12 19)(14 21)(17 24)
(1 7)(3 9)(5 12)(6 10)(13 16)(15 18)(19 22)(20 23)
(2 8)(3 9)(4 11)(5 12)(14 17)(15 18)(19 22)(21 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 3)(5 6)(7 9)(10 12)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(24,523);

On 24 points - transitive group 24T524
Generators in S24
(1 23)(2 7)(3 16)(4 20)(5 12)(6 13)(8 19)(9 14)(10 22)(11 17)(15 24)(18 21)
(1 9)(2 18)(3 19)(4 11)(5 15)(6 22)(7 21)(8 16)(10 13)(12 24)(14 23)(17 20)
(2 5)(3 6)(7 12)(8 10)(13 16)(15 18)(19 22)(21 24)
(1 4)(2 5)(7 12)(9 11)(14 17)(15 18)(20 23)(21 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 9)(2 8)(3 7)(4 11)(5 10)(6 12)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)

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

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

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

G:=TransitiveGroup(24,524);

On 24 points - transitive group 24T525
Generators in S24
(1 24)(2 13)(4 15)(5 22)(8 18)(9 19)(11 21)(12 16)
(1 24)(3 20)(4 15)(6 17)(7 23)(8 18)(10 14)(11 21)
(1 8)(3 10)(4 11)(6 7)(14 20)(15 21)(17 23)(18 24)
(2 9)(3 10)(5 12)(6 7)(13 19)(14 20)(16 22)(17 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 3)(4 6)(7 11)(8 10)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(24,525);

On 24 points - transitive group 24T526
Generators in S24
(1 22)(2 20)(3 6)(4 11)(5 9)(7 24)(8 16)(10 21)(12 17)(13 19)(14 23)(15 18)
(1 11)(2 5)(3 24)(4 22)(6 7)(8 13)(9 20)(10 18)(12 23)(14 17)(15 21)(16 19)
(1 16)(3 18)(4 13)(6 15)(7 21)(8 22)(10 24)(11 19)
(2 17)(3 18)(5 14)(6 15)(7 21)(9 23)(10 24)(12 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)
(1 6)(2 5)(3 4)(7 19)(8 24)(9 23)(10 22)(11 21)(12 20)(13 18)(14 17)(15 16)

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

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

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

G:=TransitiveGroup(24,526);

On 24 points - transitive group 24T527
Generators in S24
(1 17)(2 21)(3 11)(4 20)(5 18)(6 8)(7 15)(9 14)(10 24)(12 23)(13 16)(19 22)
(1 14)(2 10)(3 13)(4 23)(5 7)(6 22)(8 19)(9 17)(11 16)(12 20)(15 18)(21 24)
(1 12)(3 8)(4 9)(6 11)(13 19)(14 20)(16 22)(17 23)
(2 7)(3 8)(5 10)(6 11)(13 19)(15 21)(16 22)(18 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 6)(2 5)(3 4)(7 10)(8 9)(11 12)(13 17)(14 16)(19 23)(20 22)

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

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

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

G:=TransitiveGroup(24,527);

On 24 points - transitive group 24T528
Generators in S24
(1 19)(2 18)(3 8)(4 11)(5 17)(6 20)(7 23)(9 14)(10 15)(12 22)(13 24)(16 21)
(1 14)(2 7)(3 21)(4 13)(5 22)(6 10)(8 16)(9 19)(11 24)(12 17)(15 20)(18 23)
(1 12)(3 11)(4 8)(5 9)(13 16)(14 17)(19 22)(21 24)
(2 10)(3 11)(4 8)(6 7)(13 16)(15 18)(20 23)(21 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 11)(2 10)(3 12)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)

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

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

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

G:=TransitiveGroup(24,528);

On 24 points - transitive group 24T529
Generators in S24
(1 17)(2 21)(3 11)(4 20)(5 18)(6 8)(7 15)(9 14)(10 24)(12 23)(13 16)(19 22)
(1 14)(2 10)(3 13)(4 23)(5 7)(6 22)(8 19)(9 17)(11 16)(12 20)(15 18)(21 24)
(1 12)(3 8)(4 9)(6 11)(13 19)(14 20)(16 22)(17 23)
(2 7)(3 8)(5 10)(6 11)(13 19)(15 21)(16 22)(18 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 11)(2 10)(3 9)(4 8)(5 7)(6 12)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)

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

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

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

G:=TransitiveGroup(24,529);

Polynomial with Galois group C24⋊D6 over ℚ
actionf(x)Disc(f)
8T41x8-x7-7x6+7x5+12x4-12x3-x2+524·54·72·3132
12T108x12-8x10+13x8+8x6-25x4+6x2+4230·1736
12T109x12-15x10+86x8-235x6+309x4-170x2+25212·512·1974
12T110x12-19x10+141x8-509x6+891x4-611x2+27212·37·234·1074
12T111x12-34x8-28x6-46x4+6x2-14-225·58·75·690294

Matrix representation of C24⋊D6 in GL6(ℤ)

100000
010000
001000
000-1-1-1
000001
000010
,
100000
010000
001000
000001
000-1-1-1
000100
,
001000
-1-1-1000
100000
000001
000-1-1-1
000100
,
010000
100000
-1-1-1000
000010
000100
000-1-1-1
,
000100
000-1-1-1
000010
100000
-1-1-1000
010000
,
000100
000-1-1-1
000001
100000
-1-1-1000
001000

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

C24⋊D6 in GAP, Magma, Sage, TeX

C_2^4\rtimes D_6
% in TeX

G:=Group("C2^4:D6");
// GroupNames label

G:=SmallGroup(192,955);
// by ID

G=gap.SmallGroup(192,955);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,170,675,2194,185,424,1271,333,6053,1027,1784]);
// Polycyclic

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

Export

Character table of C24⋊D6 in TeX

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