C2^4:D6 - GroupNames

G = C₂⁴⋊D₆ order 192 = 2⁶·3

1^st semidirect product of C₂⁴ and D₆ acting faithfully

non-abelian, soluble, monomial, rational

Aliases: C₂³⋊₁S₄, C₂⁴⋊₁D₆, C₂²⋊S₄⋊C₂, C₂²≀C₂⋊S₃, C₂².₂(C₂×S₄), C₂⁴⋊C₆⋊₂C₂, C₂²⋊A₄⋊₂C₂², Aut(C₂×Q₈), SmallGroup(192,955)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²⋊A₄ — C₂⁴⋊D₆

Chief series

C₁ — C₂² — C₂⁴ — C₂²⋊A₄ — C₂²⋊S₄ — C₂⁴⋊D₆

Lower central

C₂²⋊A₄ — C₂⁴⋊D₆

Upper central

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, C₃, C₄, C₂², C₂², S₃, C₆, C₂×C₄, D₄, Q₈, C₂³, C₂³, A₄, D₆, C₂²⋊C₄, C₂×D₄, C₄○D₄, C₂⁴, S₄, C₂×A₄, C₂³⋊C₄, C₂²≀C₂, C₂²≀C₂, 2₊¹⁺⁴, C₂×S₄, C₂²⋊A₄, C₂≀C₂², C₂⁴⋊C₆, C₂²⋊S₄, C₂⁴⋊D₆
Quotients: C₁, C₂, C₂², S₃, D₆, S₄, C₂×S₄, C₂⁴⋊D₆

Character table of C₂⁴⋊D₆

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	4D	4E	6
size	1	3	4	6	6	12	12	32	12	12	12	24	24	32
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	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	linear of order 2
ρ₅	2	2	-2	2	2	0	0	-1	0	-2	0	0	0	1	orthogonal lifted from D₆
ρ₆	2	2	2	2	2	0	0	-1	0	2	0	0	0	-1	orthogonal lifted from S₃
ρ₇	3	3	3	-1	-1	-1	-1	0	-1	-1	-1	1	1	0	orthogonal lifted from S₄
ρ₈	3	3	-3	-1	-1	1	-1	0	1	1	-1	1	-1	0	orthogonal lifted from C₂×S₄
ρ₉	3	3	-3	-1	-1	-1	1	0	-1	1	1	-1	1	0	orthogonal lifted from C₂×S₄
ρ₁₀	3	3	3	-1	-1	1	1	0	1	-1	1	-1	-1	0	orthogonal lifted from S₄
ρ₁₁	6	-2	0	-2	2	2	0	0	-2	0	0	0	0	0	orthogonal faithful
ρ₁₂	6	-2	0	2	-2	0	-2	0	0	0	2	0	0	0	orthogonal faithful
ρ₁₃	6	-2	0	2	-2	0	2	0	0	0	-2	0	0	0	orthogonal faithful
ρ₁₄	6	-2	0	-2	2	-2	0	0	2	0	0	0	0	0	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 9)(2 7)(5 12)(6 10)

(1 9)(3 11)(4 8)(5 12)

(1 5)(3 4)(8 11)(9 12)

(2 6)(3 4)(7 10)(8 11)

(1 2 3)(4 5 6)(7 8 9 10 11 12)

(1 4)(2 6)(3 5)(8 12)(9 11)

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

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

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

G:=TransitiveGroup(12,111);

►On 16 points - transitive group 16T435

Generators in S₁₆

(1 13)(2 12)(3 9)(4 10)(5 7)(6 8)(11 15)(14 16)

(1 15)(2 14)(3 5)(4 6)(7 9)(8 10)(11 13)(12 16)

(1 13)(2 16)(3 7)(4 10)(5 9)(6 8)(11 15)(12 14)

(1 15)(2 12)(3 9)(4 6)(5 7)(8 10)(11 13)(14 16)

(1 2)(3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)

(1 4)(2 3)(5 12)(6 11)(7 16)(8 15)(9 14)(10 13)

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

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

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

G:=TransitiveGroup(16,435);

►On 16 points - transitive group 16T436

Generators in S₁₆

(1 9)(2 11)(3 14)(4 6)(5 7)(8 15)(10 16)(12 13)

(1 5)(2 16)(3 8)(4 13)(6 12)(7 9)(10 11)(14 15)

(1 3)(2 4)(5 8)(6 11)(7 15)(9 14)(10 12)(13 16)

(1 2)(3 4)(5 16)(6 14)(7 10)(8 13)(9 11)(12 15)

(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)

(2 4)(5 8)(6 7)(9 10)(11 15)(12 14)

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

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

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

G:=TransitiveGroup(16,436);

►On 24 points - transitive group 24T516

Generators in S₂₄

(4 14)(6 16)(7 23)(9 19)

(2 18)(6 16)(9 19)(11 21)

(2 18)(3 13)(5 15)(6 16)(8 24)(9 19)(11 21)(12 22)

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

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

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

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

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

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

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

G:=TransitiveGroup(24,517);

►On 24 points - transitive group 24T518

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

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

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

G:=TransitiveGroup(24,518);

►On 24 points - transitive group 24T519

Generators in S₂₄

(2 21)(4 15)(6 7)(9 13)(11 23)(17 19)

(2 9)(4 23)(6 17)(7 19)(11 15)(13 21)

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

(1 8)(2 13)(3 22)(4 11)(5 16)(6 19)(7 17)(9 21)(10 14)(12 24)(15 23)(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)

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

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

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

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

G:=TransitiveGroup(24,519);

►On 24 points - transitive group 24T520

Generators in S₂₄

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

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

(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 3)(4 6)(7 11)(8 10)(14 18)(15 17)(19 23)(20 22)

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

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

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

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 14)(3 18)(4 21)(5 19)(7 24)(8 22)(10 17)(11 15)

(1 16)(2 14)(4 21)(6 23)(7 24)(9 20)(10 17)(12 13)

(1 12)(2 10)(4 7)(6 9)(13 16)(14 17)(20 23)(21 24)

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

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

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

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

G:=TransitiveGroup(24,522);

►On 24 points - transitive group 24T523

Generators in S₂₄

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

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

(1 10)(3 12)(5 9)(6 7)(13 16)(15 18)(19 22)(20 23)

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

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

(1 9)(2 7)(4 11)(6 10)(13 16)(15 18)(19 22)(21 24)

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

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

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

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

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), (1,3)(4,6)(7,11)(8,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)>;

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), (1,3)(4,6)(7,11)(8,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22) );

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)], [(1,3),(4,6),(7,11),(8,10),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)]])

G:=TransitiveGroup(24,525);

►On 24 points - transitive group 24T526

Generators in S₂₄

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

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

(1 16)(3 18)(4 13)(6 15)(7 20)(8 21)(10 23)(11 24)

(2 17)(3 18)(5 14)(6 15)(7 20)(9 22)(10 23)(12 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 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)

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

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

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

G:=TransitiveGroup(24,526);

►On 24 points - transitive group 24T527

Generators in S₂₄

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

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

(1 22)(3 24)(4 19)(6 21)(7 13)(8 14)(10 16)(11 17)

(2 23)(3 24)(5 20)(6 21)(7 13)(9 15)(10 16)(12 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)

(1 6)(2 5)(3 4)(7 11)(8 10)(13 17)(14 16)(19 24)(20 23)(21 22)

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

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

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

G:=TransitiveGroup(24,527);

►On 24 points - transitive group 24T528

Generators in S₂₄

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

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

(1 10)(3 12)(4 8)(6 7)(14 17)(15 18)(19 22)(21 24)

(2 11)(3 12)(5 9)(6 7)(13 16)(14 17)(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 12)(2 11)(3 10)(4 7)(5 9)(6 8)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)

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

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

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

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

action	f(x)	Disc(f)
8T41	x⁸-x⁷-7x⁶+7x⁵+12x⁴-12x³-x²+5	2⁴·5⁴·7²·313²
12T108	x¹²-8x¹⁰+13x⁸+8x⁶-25x⁴+6x²+4	2³⁰·173⁶
12T109	x¹²-15x¹⁰+86x⁸-235x⁶+309x⁴-170x²+25	2¹²·5¹²·197⁴
12T110	x¹²-19x¹⁰+141x⁸-509x⁶+891x⁴-611x²+27	2¹²·3⁷·23⁴·107⁴
12T111	x¹²-34x⁸-28x⁶-46x⁴+6x²-14	-2²⁵·5⁸·7⁵·69029⁴

Matrix representation of C₂⁴⋊D₆ ►in GL₆(ℤ)

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

0	0	1	0	0	0
-1	-1	-1	0	0	0
1	0	0	0	0	0
0	0	0	0	0	1
0	0	0	-1	-1	-1
0	0	0	1	0	0

0	1	0	0	0	0
1	0	0	0	0	0
-1	-1	-1	0	0	0
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	-1	-1	-1

0	0	0	1	0	0
0	0	0	-1	-1	-1
0	0	0	0	1	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	-1	-1	-1
0	0	0	0	0	1
1	0	0	0	0	0
-1	-1	-1	0	0	0
0	0	1	0	0	0

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

G = C24⋊D6 order 192 = 26·3

1st semidirect product of C24 and D6 acting faithfully

G = C₂⁴⋊D₆ order 192 = 2⁶·3

1^st semidirect product of C₂⁴ and D₆ acting faithfully