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G = C22×S4order 96 = 25·3

Direct product of C22 and S4

direct product, non-abelian, soluble, monomial, rational

Aliases: C22×S4, C23⋊D6, A4⋊C23, C242S3, (C2×A4)⋊C22, C22⋊(C22×S3), (C22×A4)⋊3C2, SmallGroup(96,226)

Series: Derived Chief Lower central Upper central

C1C22A4 — C22×S4
C1C22A4S4C2×S4 — C22×S4
A4 — C22×S4
C1C22

Generators and relations for C22×S4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 420 in 131 conjugacy classes, 26 normal (7 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22 [×2], C22 [×26], S3 [×4], C6 [×3], C2×C4 [×6], D4 [×16], C23 [×3], C23 [×14], A4, D6 [×6], C2×C6, C22×C4, C2×D4 [×12], C24, C24, S4 [×4], C2×A4 [×3], C22×S3, C22×D4, C2×S4 [×6], C22×A4, C22×S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], C22×S4

Character table of C22×S4

 class 12A2B2C2D2E2F2G2H2I2J2K34A4B4C4D6A6B6C
 size 11113333666686666888
ρ111111111111111111111    trivial
ρ21-1-111-1-1111-1-1111-1-11-1-1    linear of order 2
ρ31-11-1-11-111-11-111-1-11-1-11    linear of order 2
ρ411-1-1-1-1111-1-1111-11-1-11-1    linear of order 2
ρ511111111-1-1-1-11-1-1-1-1111    linear of order 2
ρ61-1-111-1-11-1-1111-1-1111-1-1    linear of order 2
ρ71-11-1-11-11-11-111-111-1-1-11    linear of order 2
ρ811-1-1-1-111-111-11-11-11-11-1    linear of order 2
ρ92-22-2-22-220000-1000011-1    orthogonal lifted from D6
ρ1022-2-2-2-2220000-100001-11    orthogonal lifted from D6
ρ11222222220000-10000-1-1-1    orthogonal lifted from S3
ρ122-2-222-2-220000-10000-111    orthogonal lifted from D6
ρ133-33-31-11-11-11-10-111-1000    orthogonal lifted from C2×S4
ρ143-3-33-111-111-1-10-1-111000    orthogonal lifted from C2×S4
ρ153333-1-1-1-1-1-1-1-101111000    orthogonal lifted from S4
ρ1633-3-311-1-1-111-101-11-1000    orthogonal lifted from C2×S4
ρ173333-1-1-1-111110-1-1-1-1000    orthogonal lifted from S4
ρ1833-3-311-1-11-1-110-11-11000    orthogonal lifted from C2×S4
ρ193-33-31-11-1-11-1101-1-11000    orthogonal lifted from C2×S4
ρ203-3-33-111-1-1-111011-1-1000    orthogonal lifted from C2×S4

Permutation representations of C22×S4
On 12 points - transitive group 12T48
Generators in S12
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)
(1 9)(2 7)(4 12)(6 11)
(2 7)(3 8)(4 12)(5 10)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 9)(2 8)(3 7)(4 10)(5 12)(6 11)

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

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

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

G:=TransitiveGroup(12,48);

On 16 points - transitive group 16T182
Generators in S16
(1 3)(2 4)(5 10)(6 8)(7 9)(11 14)(12 15)(13 16)
(1 4)(2 3)(5 15)(6 16)(7 14)(8 13)(9 11)(10 12)
(1 10)(2 15)(3 5)(4 12)(6 7)(8 9)(11 13)(14 16)
(1 8)(2 16)(3 6)(4 13)(5 7)(9 10)(11 12)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 4)(2 3)(5 14)(6 16)(7 15)(8 13)(9 12)(10 11)

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

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

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

G:=TransitiveGroup(16,182);

On 24 points - transitive group 24T125
Generators in S24
(1 14)(2 15)(3 13)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 16)(11 17)(12 18)
(1 11)(2 12)(3 10)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 11)(2 15)(3 16)(4 20)(5 23)(6 7)(8 22)(9 21)(10 13)(12 18)(14 17)(19 24)
(1 17)(2 12)(3 13)(4 8)(5 21)(6 24)(7 19)(9 23)(10 16)(11 14)(15 18)(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 20)(2 19)(3 21)(4 14)(5 13)(6 15)(7 12)(8 11)(9 10)(16 23)(17 22)(18 24)

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

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

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

G:=TransitiveGroup(24,125);

On 24 points - transitive group 24T126
Generators in S24
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)
(1 14)(2 12)(3 20)(4 17)(5 9)(6 23)(7 16)(8 24)(10 13)(11 21)(15 19)(18 22)
(1 21)(2 15)(3 10)(4 24)(5 18)(6 7)(8 17)(9 22)(11 14)(12 19)(13 20)(16 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)
(2 3)(4 5)(8 9)(10 15)(11 14)(12 13)(16 23)(17 22)(18 24)(19 20)

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

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

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

G:=TransitiveGroup(24,126);

On 24 points - transitive group 24T150
Generators in S24
(1 9)(2 7)(3 8)(4 22)(5 23)(6 24)(10 14)(11 15)(12 13)(16 19)(17 20)(18 21)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 7)(3 8)(4 22)(6 24)(10 14)(12 13)(16 19)(18 21)
(1 9)(3 8)(4 22)(5 23)(10 14)(11 15)(16 19)(17 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 22)(2 24)(3 23)(4 9)(5 8)(6 7)(10 17)(11 16)(12 18)(13 21)(14 20)(15 19)

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

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

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

G:=TransitiveGroup(24,150);

On 24 points - transitive group 24T151
Generators in S24
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(1 9)(2 7)(4 24)(6 23)(11 15)(12 13)(17 21)(18 19)
(2 7)(3 8)(4 24)(5 22)(10 14)(12 13)(16 20)(18 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 11)(2 10)(3 12)(4 20)(5 19)(6 21)(7 14)(8 13)(9 15)(16 24)(17 23)(18 22)

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

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

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

G:=TransitiveGroup(24,151);

On 24 points - transitive group 24T152
Generators in S24
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(1 9)(2 7)(4 24)(6 23)(11 15)(12 13)(17 21)(18 19)
(2 7)(3 8)(4 24)(5 22)(10 14)(12 13)(16 20)(18 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 15)(2 14)(3 13)(4 16)(5 18)(6 17)(7 10)(8 12)(9 11)(19 22)(20 24)(21 23)

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

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

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

G:=TransitiveGroup(24,152);

C22×S4 is a maximal subgroup of
C24.5D6
C22×S4 is a maximal quotient of
C24.10D6  D42S4  Q84S4  GL2(𝔽3)⋊C22  Q8.6S4  Q8.7S4  D4.4S4  D4.5S4

Polynomial with Galois group C22×S4 over ℚ
actionf(x)Disc(f)
12T48x12-15x10+85x8-224x6+270x4-120x2+4230·56·474

Matrix representation of C22×S4 in GL5(ℤ)

10000
01000
00-100
000-10
0000-1
,
-10000
0-1000
00-100
000-10
0000-1
,
10000
01000
00-100
000-10
00-101
,
10000
01000
00100
001-10
0010-1
,
0-1000
1-1000
0010-2
0000-1
0001-1
,
01000
10000
00100
00001
00010

G:=sub<GL(5,Integers())| [1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,-1,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-2,-1,-1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C22×S4 in GAP, Magma, Sage, TeX

C_2^2\times S_4
% in TeX

G:=Group("C2^2xS4");
// GroupNames label

G:=SmallGroup(96,226);
// by ID

G=gap.SmallGroup(96,226);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,387,1444,202,869,347]);
// Polycyclic

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

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