C2^4:C12 - GroupNames

G = C₂⁴⋊C₁₂ order 192 = 2⁶·3

1^st semidirect product of C₂⁴ and C₁₂ acting via C₁₂/C₂=C₆

Aliases: C₂⁴⋊₁C₁₂, C₂⁵.₁C₆, C₂⁴⋊₃C₄⋊C₃, C₂²⋊A₄⋊₂C₄, (C₂²×C₄)⋊₁A₄, C₂².₂(C₄×A₄), C₂³.₁₁(C₂×A₄), C₂.₁(C₂⁴⋊C₆), (C₂×C₂²⋊A₄).₁C₂, SmallGroup(192,191)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂⁴⋊C₁₂

Chief series

C₁ — C₂² — C₂⁴ — C₂⁵ — C₂×C₂²⋊A₄ — C₂⁴⋊C₁₂

Lower central

C₂⁴ — C₂⁴⋊C₁₂

Upper central

C₁ — C₂

Generators and relations for C₂⁴⋊C₁₂
G = < a,b,c,d,e | a²=b²=c²=d²=e¹²=1, ab=ba, ebe^-1=ac=ca, ad=da, eae^-1=abcd, bc=cb, bd=db, ece^-1=cd=dc, ede^-1=c >

Subgroups: 544 in 106 conjugacy classes, 11 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₂×C₄, C₂³, C₂³, C₁₂, A₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂×A₄, C₂×C₂²⋊C₄, C₂⁵, C₄×A₄, C₂²⋊A₄, C₂⁴⋊₃C₄, C₂×C₂²⋊A₄, C₂⁴⋊C₁₂
Quotients: C₁, C₂, C₃, C₄, C₆, C₁₂, A₄, C₂×A₄, C₄×A₄, C₂⁴⋊C₆, C₂⁴⋊C₁₂

Character table of C₂⁴⋊C₁₂

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	6A	6B	12A	12B	12C	12D
size	1	1	3	3	6	6	6	6	16	16	4	4	12	12	16	16	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₄	1	1	1	1	1	1	1	1	ζ₃²	ζ₃	-1	-1	-1	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	linear of order 6
ρ₅	1	1	1	1	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₆	1	1	1	1	1	1	1	1	ζ₃	ζ₃²	-1	-1	-1	-1	ζ₃²	ζ₃	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₇	1	-1	-1	1	1	1	-1	-1	1	1	i	-i	i	-i	-1	-1	i	-i	i	-i	linear of order 4
ρ₈	1	-1	-1	1	1	1	-1	-1	1	1	-i	i	-i	i	-1	-1	-i	i	-i	i	linear of order 4
ρ₉	1	-1	-1	1	1	1	-1	-1	ζ₃²	ζ₃	i	-i	i	-i	ζ₆⁵	ζ₆	ζ₄ζ₃	ζ₄³ζ₃	ζ₄ζ₃²	ζ₄³ζ₃²	linear of order 12
ρ₁₀	1	-1	-1	1	1	1	-1	-1	ζ₃	ζ₃²	-i	i	-i	i	ζ₆	ζ₆⁵	ζ₄³ζ₃²	ζ₄ζ₃²	ζ₄³ζ₃	ζ₄ζ₃	linear of order 12
ρ₁₁	1	-1	-1	1	1	1	-1	-1	ζ₃	ζ₃²	i	-i	i	-i	ζ₆	ζ₆⁵	ζ₄ζ₃²	ζ₄³ζ₃²	ζ₄ζ₃	ζ₄³ζ₃	linear of order 12
ρ₁₂	1	-1	-1	1	1	1	-1	-1	ζ₃²	ζ₃	-i	i	-i	i	ζ₆⁵	ζ₆	ζ₄³ζ₃	ζ₄ζ₃	ζ₄³ζ₃²	ζ₄ζ₃²	linear of order 12
ρ₁₃	3	3	3	3	-1	-1	-1	-1	0	0	3	3	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₄	3	3	3	3	-1	-1	-1	-1	0	0	-3	-3	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₅	3	-3	-3	3	-1	-1	1	1	0	0	-3i	3i	i	-i	0	0	0	0	0	0	complex lifted from C₄×A₄
ρ₁₆	3	-3	-3	3	-1	-1	1	1	0	0	3i	-3i	-i	i	0	0	0	0	0	0	complex lifted from C₄×A₄
ρ₁₇	6	-6	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₈	6	6	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₆
ρ₁₉	6	-6	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₀	6	6	-2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₆

Permutation representations of C₂⁴⋊C₁₂
►On 12 points - transitive group 12T99

Generators in S₁₂

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

(1 10)(3 12)(4 7)(6 9)

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

(1 7)(3 9)(4 10)(6 12)

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

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

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

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

G:=TransitiveGroup(12,99);

►On 12 points - transitive group 12T105

Generators in S₁₂

(1 7)(4 10)(5 11)(6 12)

(2 8)(3 9)(4 10)(5 11)

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

(1 7)(3 9)(4 10)(6 12)

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

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

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

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

G:=TransitiveGroup(12,105);

►On 16 points - transitive group 16T418

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,418);

►On 24 points - transitive group 24T483

Generators in S₂₄

(2 16)(3 23)(5 13)(6 20)(8 22)(9 17)(11 19)(12 14)

(1 21)(3 23)(4 18)(6 20)(7 15)(9 17)(10 24)(12 14)

(2 8)(3 9)(5 11)(6 12)(13 19)(14 20)(16 22)(17 23)

(1 7)(3 9)(4 10)(6 12)(14 20)(15 21)(17 23)(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)

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

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

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

G:=TransitiveGroup(24,483);

►On 24 points - transitive group 24T484

Generators in S₂₄

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

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

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

(2 8)(3 9)(5 11)(6 12)(14 20)(15 21)(17 23)(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)

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

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

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

G:=TransitiveGroup(24,484);

►On 24 points - transitive group 24T501

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,501);

►On 24 points - transitive group 24T502

Generators in S₂₄

(1 7)(2 17)(5 11)(6 21)(8 23)(12 15)(14 20)(16 22)

(3 9)(4 19)(5 11)(6 21)(10 13)(12 15)(14 20)(18 24)

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

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

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

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

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

G:=TransitiveGroup(24,502);

►On 24 points - transitive group 24T503

Generators in S₂₄

(2 18)(6 22)(8 24)(12 16)

(4 20)(6 22)(10 14)(12 16)

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

(1 17)(3 19)(4 20)(6 22)(7 23)(9 13)(10 14)(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)

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

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

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

G:=TransitiveGroup(24,503);

►On 24 points - transitive group 24T504

Generators in S₂₄

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

(1 7)(3 9)(4 20)(6 22)(10 14)(12 16)(13 19)(17 23)

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

(1 23)(3 13)(4 14)(6 16)(7 17)(9 19)(10 20)(12 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)| (2,18)(3,9)(5,11)(6,22)(8,24)(12,16)(13,19)(15,21), (1,7)(3,9)(4,20)(6,22)(10,14)(12,16)(13,19)(17,23), (2,24)(3,13)(5,15)(6,16)(8,18)(9,19)(11,21)(12,22), (1,23)(3,13)(4,14)(6,16)(7,17)(9,19)(10,20)(12,22), (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( (2,18)(3,9)(5,11)(6,22)(8,24)(12,16)(13,19)(15,21), (1,7)(3,9)(4,20)(6,22)(10,14)(12,16)(13,19)(17,23), (2,24)(3,13)(5,15)(6,16)(8,18)(9,19)(11,21)(12,22), (1,23)(3,13)(4,14)(6,16)(7,17)(9,19)(10,20)(12,22), (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([[(2,18),(3,9),(5,11),(6,22),(8,24),(12,16),(13,19),(15,21)], [(1,7),(3,9),(4,20),(6,22),(10,14),(12,16),(13,19),(17,23)], [(2,24),(3,13),(5,15),(6,16),(8,18),(9,19),(11,21),(12,22)], [(1,23),(3,13),(4,14),(6,16),(7,17),(9,19),(10,20),(12,22)], [(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(24,504);

►On 24 points - transitive group 24T505

Generators in S₂₄

(1 14)(2 21)(5 18)(6 13)(7 20)(8 15)(11 24)(12 19)

(3 16)(4 23)(5 18)(6 13)(9 22)(10 17)(11 24)(12 19)

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

(1 20)(2 15)(3 9)(4 23)(5 18)(6 12)(7 14)(8 21)(10 17)(11 24)(13 19)(16 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)(5,18)(6,13)(7,20)(8,15)(11,24)(12,19), (3,16)(4,23)(5,18)(6,13)(9,22)(10,17)(11,24)(12,19), (1,14)(2,8)(3,22)(4,17)(5,11)(6,13)(7,20)(9,16)(10,23)(12,19)(15,21)(18,24), (1,20)(2,15)(3,9)(4,23)(5,18)(6,12)(7,14)(8,21)(10,17)(11,24)(13,19)(16,22), (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)(5,18)(6,13)(7,20)(8,15)(11,24)(12,19), (3,16)(4,23)(5,18)(6,13)(9,22)(10,17)(11,24)(12,19), (1,14)(2,8)(3,22)(4,17)(5,11)(6,13)(7,20)(9,16)(10,23)(12,19)(15,21)(18,24), (1,20)(2,15)(3,9)(4,23)(5,18)(6,12)(7,14)(8,21)(10,17)(11,24)(13,19)(16,22), (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),(5,18),(6,13),(7,20),(8,15),(11,24),(12,19)], [(3,16),(4,23),(5,18),(6,13),(9,22),(10,17),(11,24),(12,19)], [(1,14),(2,8),(3,22),(4,17),(5,11),(6,13),(7,20),(9,16),(10,23),(12,19),(15,21),(18,24)], [(1,20),(2,15),(3,9),(4,23),(5,18),(6,12),(7,14),(8,21),(10,17),(11,24),(13,19),(16,22)], [(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(24,505);

►On 24 points - transitive group 24T506

Generators in S₂₄

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

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

(1 7)(2 8)(4 10)(5 11)(13 19)(15 21)(16 22)(18 24)

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

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

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

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

G:=TransitiveGroup(24,506);

►On 24 points - transitive group 24T507

Generators in S₂₄

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

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

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

(2 14)(3 15)(5 17)(6 18)(8 20)(9 21)(11 23)(12 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)

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

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

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

G:=TransitiveGroup(24,507);

Polynomial with Galois group C₂⁴⋊C₁₂ over ℚ

action	f(x)	Disc(f)
12T99	x¹²-12x¹⁰+60x⁸-160x⁶+228x⁴-144x²+8	2⁵⁷·3¹⁶
12T105	x¹²-6x¹¹+2x¹⁰+45x⁹-58x⁸-104x⁷+174x⁶+97x⁵-186x⁴-34x³+69x²-5	5⁷·13⁸·31⁶

Matrix representation of C₂⁴⋊C₁₂ ►in GL₆(ℤ)

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

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	0	-1	0
0	0	0	0	0	-1

-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	0	1	0
0	0	0	0	0	1

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

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

C₂⁴⋊C₁₂ in GAP, Magma, Sage, TeX

C_2^4\rtimes C_{12}

% in TeX

G:=Group("C2^4:C12");

// GroupNames label

G:=SmallGroup(192,191);

// by ID

G=gap.SmallGroup(192,191);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,1683,346,4204,641,2028,3541]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^12=1,a*b=b*a,e*b*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c*d,b*c=c*b,b*d=d*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;

// generators/relations

Export

Character table of C₂⁴⋊C₁₂ in TeX

G = C24⋊C12 order 192 = 26·3

1st semidirect product of C24 and C12 acting via C12/C2=C6

G = C₂⁴⋊C₁₂ order 192 = 2⁶·3

1^st semidirect product of C₂⁴ and C₁₂ acting via C₁₂/C₂=C₆