C2^2:S4 - GroupNames

G = C₂²⋊S₄ order 96 = 2⁵·3

The semidirect product of C₂² and S₄ acting via S₄/C₂²=S₃

non-abelian, soluble, monomial, rational

Aliases: C₂²⋊S₄, C₂⁴⋊₃S₃, C₂²⋊A₄⋊₂C₂, SmallGroup(96,227)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²⋊A₄ — C₂²⋊S₄

Chief series

C₁ — C₂² — C₂⁴ — C₂²⋊A₄ — C₂²⋊S₄

Lower central

C₂²⋊A₄ — C₂²⋊S₄

Upper central

C₁

Generators and relations for C₂²⋊S₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e³=f²=1, ebe^-1=ab=ba, ac=ca, ad=da, eae^-1=faf=b, bc=cb, bd=db, fbf=a, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 250 in 56 conjugacy classes, 7 normal (4 characteristic)
C₁, C₂, C₃, C₄, C₂², C₂², S₃, C₂×C₄, D₄, C₂³, A₄, C₂²⋊C₄, C₂×D₄, C₂⁴, S₄, C₂²≀C₂, C₂²⋊A₄, C₂²⋊S₄
Quotients: C₁, C₂, S₃, S₄, C₂²⋊S₄

Character table of C₂²⋊S₄

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C
size	1	3	3	3	6	12	32	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	1	-1	-1	-1	linear of order 2
ρ₃	2	2	2	2	2	0	-1	0	0	0	orthogonal lifted from S₃
ρ₄	3	-1	3	-1	-1	-1	0	1	-1	1	orthogonal lifted from S₄
ρ₅	3	-1	-1	3	-1	1	0	-1	-1	1	orthogonal lifted from S₄
ρ₆	3	3	-1	-1	-1	-1	0	-1	1	1	orthogonal lifted from S₄
ρ₇	3	-1	3	-1	-1	1	0	-1	1	-1	orthogonal lifted from S₄
ρ₈	3	-1	-1	3	-1	-1	0	1	1	-1	orthogonal lifted from S₄
ρ₉	3	3	-1	-1	-1	1	0	1	-1	-1	orthogonal lifted from S₄
ρ₁₀	6	-2	-2	-2	2	0	0	0	0	0	orthogonal faithful

Permutation representations of C₂²⋊S₄
►On 8 points - transitive group 8T34

Generators in S₈

(1 6)(2 5)(3 4)(7 8)

(1 8)(2 4)(3 5)(6 7)

(1 7)(2 4)(3 5)(6 8)

(1 8)(2 5)(3 4)(6 7)

(3 4 5)(6 7 8)

(1 2)(3 7)(4 6)(5 8)

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

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

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

G:=TransitiveGroup(8,34);

►On 12 points - transitive group 12T66

Generators in S₁₂

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(12,66);

►On 12 points - transitive group 12T67

Generators in S₁₂

(1 11)(2 12)(4 8)(6 7)

(1 11)(3 10)(5 9)(6 7)

(1 7)(2 8)(4 12)(6 11)

(2 8)(3 9)(4 12)(5 10)

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

(2 3)(4 5)(8 9)(10 12)

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

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

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

G:=TransitiveGroup(12,67);

►On 12 points - transitive group 12T68

Generators in S₁₂

(1 11)(2 12)(4 8)(6 7)

(1 11)(3 10)(5 9)(6 7)

(1 7)(2 8)(4 12)(6 11)

(2 8)(3 9)(4 12)(5 10)

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

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

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

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

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

G:=TransitiveGroup(12,68);

►On 12 points - transitive group 12T69

Generators in S₁₂

(2 7)(3 8)(4 10)(6 12)

(1 9)(2 7)(5 11)(6 12)

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

(1 9)(2 7)(4 10)(6 12)

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

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

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

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

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

G:=TransitiveGroup(12,69);

►On 16 points - transitive group 16T194

Generators in S₁₆

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,194);

►On 24 points - transitive group 24T195

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,195);

►On 24 points - transitive group 24T196

Generators in S₂₄

(1 18)(2 16)(4 7)(5 8)(10 22)(11 23)(13 21)(15 20)

(1 18)(3 17)(4 7)(6 9)(10 22)(12 24)(14 19)(15 20)

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

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

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

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

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

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

G:=TransitiveGroup(24,196);

►On 24 points - transitive group 24T197

Generators in S₂₄

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

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

(1 22)(2 23)(4 15)(5 13)(7 20)(8 21)(10 18)(11 16)

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

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

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

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

G:=TransitiveGroup(24,197);

►On 24 points - transitive group 24T198

Generators in S₂₄

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

(1 11)(2 12)(5 23)(6 24)(7 19)(9 21)(14 17)(15 18)

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

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

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

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

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

G:=TransitiveGroup(24,198);

►On 24 points - transitive group 24T199

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,199);

►On 24 points - transitive group 24T200

Generators in S₂₄

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

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

(1 14)(3 13)(5 21)(6 19)(7 24)(9 23)(10 16)(11 17)

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

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

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

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

G:=TransitiveGroup(24,200);

C₂²⋊S₄ is a maximal subgroup of
C₂⁴⋊D₆ A₄≀C₂ PSO₄⁺ (𝔽₃) (C₂×C₆)⋊S₄ C₂⁴⋊₄D₁₅
C₂²⋊S₄ is a maximal quotient of
Q₈.₁S₄ Q₈⋊S₄ C₂³.S₄ Q₈.S₄ C₂³⋊S₄ Q₈⋊₂S₄ C₂⁴⋊₄Dic₃ C₂⁴⋊D₉ (C₂×C₆)⋊S₄ C₂⁴⋊₄D₁₅

Polynomial with Galois group C₂²⋊S₄ over ℚ

action	f(x)	Disc(f)
8T34	x⁸+4x⁶+4x⁵+35x⁴+8x³+66x²+62x+63	2⁸·7²·19²·59²·709⁴
12T66	x¹²-3x¹⁰-3x⁸+6x⁶-9x⁴-27x²+27	2³⁶·3³¹·5⁸
12T67	x¹²-x⁸-x⁶-x⁴+1	2¹²·229⁴
12T68	x¹²+x¹⁰+6x⁸+3x⁶+6x⁴+x²+1	2²⁰·3⁴·19⁶
12T69	x¹²-3x¹⁰-2x⁸+9x⁶-5x²+1	2²⁰·37⁶

Matrix representation of C₂²⋊S₄ ►in GL₆(ℤ)

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

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

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

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

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

C₂²⋊S₄ in GAP, Magma, Sage, TeX

C_2^2\rtimes S_4

% in TeX

G:=Group("C2^2:S4");

// GroupNames label

G:=SmallGroup(96,227);

// by ID

G=gap.SmallGroup(96,227);

# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,49,218,116,147,225,1444,1090,869,443]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^3=f^2=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f=b,b*c=c*b,b*d=d*b,f*b*f=a,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;

// generators/relations

Export

Character table of C₂²⋊S₄ in TeX

G = C22⋊S4 order 96 = 25·3

The semidirect product of C22 and S4 acting via S4/C22=S3

G = C₂²⋊S₄ order 96 = 2⁵·3

The semidirect product of C₂² and S₄ acting via S₄/C₂²=S₃