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G = C2×C24⋊C6order 192 = 26·3

Direct product of C2 and C24⋊C6

direct product, metabelian, soluble, monomial

Aliases: C2×C24⋊C6, C242A4, C251C6, C242(C2×C6), C231(C2×A4), C22≀C22C6, C22⋊A43C22, C22.2(C22×A4), (C2×C22≀C2)⋊C3, (C2×C22⋊A4)⋊1C2, SmallGroup(192,1000)

Series: Derived Chief Lower central Upper central

C1C24 — C2×C24⋊C6
C1C22C24C22⋊A4C24⋊C6 — C2×C24⋊C6
C24 — C2×C24⋊C6
C1C2

Generators and relations for C2×C24⋊C6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f6=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=ec=ce, cd=dc, fcf-1=bcde, fef-1=de=ed, fdf-1=e >

Subgroups: 706 in 142 conjugacy classes, 17 normal (11 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C22, C22 [×34], C6 [×3], C2×C4 [×4], D4 [×8], C23, C23 [×2], C23 [×32], A4 [×3], C2×C6, C22⋊C4 [×4], C22×C4, C2×D4 [×8], C24 [×2], C24 [×6], C2×A4 [×5], C2×C22⋊C4, C22≀C2 [×2], C22≀C2 [×2], C22×D4, C25, C22×A4, C22⋊A4, C2×C22≀C2, C24⋊C6 [×2], C2×C22⋊A4, C2×C24⋊C6
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C24⋊C6, C2×C24⋊C6

Character table of C2×C24⋊C6

 class 12A2B2C2D2E2F2G2H2I3A3B4A4B6A6B6C6D6E6F
 size 113344666616161212161616161616
ρ111111111111111111111    trivial
ρ21-1-11-111-1-11111-1-11-1-1-11    linear of order 2
ρ31111-1-1111111-1-1-1-111-1-1    linear of order 2
ρ41-1-111-11-1-1111-111-1-1-11-1    linear of order 2
ρ51111-1-11111ζ32ζ3-1-1ζ65ζ65ζ32ζ3ζ6ζ6    linear of order 6
ρ61111111111ζ3ζ3211ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ71-1-111-11-1-11ζ3ζ32-11ζ32ζ6ζ65ζ6ζ3ζ65    linear of order 6
ρ81111111111ζ32ζ311ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ91-1-11-111-1-11ζ3ζ321-1ζ6ζ32ζ65ζ6ζ65ζ3    linear of order 6
ρ101-1-11-111-1-11ζ32ζ31-1ζ65ζ3ζ6ζ65ζ6ζ32    linear of order 6
ρ111111-1-11111ζ3ζ32-1-1ζ6ζ6ζ3ζ32ζ65ζ65    linear of order 6
ρ121-1-111-11-1-11ζ32ζ3-11ζ3ζ65ζ6ζ65ζ32ζ6    linear of order 6
ρ133-3-33-33-111-100-11000000    orthogonal lifted from C2×A4
ρ143333-3-3-1-1-1-10011000000    orthogonal lifted from C2×A4
ρ153-3-333-3-111-1001-1000000    orthogonal lifted from C2×A4
ρ16333333-1-1-1-100-1-1000000    orthogonal lifted from A4
ρ176-62-200-2-2220000000000    orthogonal faithful
ρ186-62-20022-2-20000000000    orthogonal faithful
ρ1966-2-200-22-220000000000    orthogonal lifted from C24⋊C6
ρ2066-2-2002-22-20000000000    orthogonal lifted from C24⋊C6

Permutation representations of C2×C24⋊C6
On 12 points - transitive group 12T87
Generators in S12
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)
(3 12)(5 8)
(1 10)(5 8)
(2 11)(3 12)(5 8)(6 9)
(1 10)(2 11)(4 7)(5 8)
(1 2 3 4 5 6)(7 8 9 10 11 12)

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

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

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

G:=TransitiveGroup(12,87);

On 12 points - transitive group 12T88
Generators in S12
(1 6)(2 4)(3 5)(7 10)(8 11)(9 12)
(1 7)(3 9)(5 12)(6 10)
(2 11)(3 9)(4 8)(5 12)
(1 6)(3 5)(7 10)(9 12)
(2 4)(3 5)(8 11)(9 12)
(1 2 3)(4 5 6)(7 8 9 10 11 12)

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

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

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

G:=TransitiveGroup(12,88);

On 16 points - transitive group 16T416
Generators in S16
(1 4)(2 3)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 8)(4 13)(6 10)(11 15)
(1 10)(4 15)(6 8)(11 13)
(1 8)(2 5)(3 16)(4 13)(6 10)(7 9)(11 15)(12 14)
(1 10)(2 7)(3 12)(4 15)(5 9)(6 8)(11 13)(14 16)
(1 2)(3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,416);

On 24 points - transitive group 24T441
Generators in S24
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 23)(8 24)(9 19)(10 20)(11 21)(12 22)
(1 10)(2 11)(3 22)(4 23)(5 16)(6 17)(7 15)(8 24)(9 19)(12 14)(13 21)(18 20)
(1 18)(2 13)(3 12)(4 7)(5 24)(6 19)(8 16)(9 17)(10 20)(11 21)(14 22)(15 23)
(1 18)(2 21)(3 12)(4 15)(5 24)(6 9)(7 23)(8 16)(10 20)(11 13)(14 22)(17 19)
(1 20)(2 11)(3 14)(4 23)(5 8)(6 17)(7 15)(9 19)(10 18)(12 22)(13 21)(16 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)| (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,23)(8,24)(9,19)(10,20)(11,21)(12,22), (1,10)(2,11)(3,22)(4,23)(5,16)(6,17)(7,15)(8,24)(9,19)(12,14)(13,21)(18,20), (1,18)(2,13)(3,12)(4,7)(5,24)(6,19)(8,16)(9,17)(10,20)(11,21)(14,22)(15,23), (1,18)(2,21)(3,12)(4,15)(5,24)(6,9)(7,23)(8,16)(10,20)(11,13)(14,22)(17,19), (1,20)(2,11)(3,14)(4,23)(5,8)(6,17)(7,15)(9,19)(10,18)(12,22)(13,21)(16,24), (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( (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,23)(8,24)(9,19)(10,20)(11,21)(12,22), (1,10)(2,11)(3,22)(4,23)(5,16)(6,17)(7,15)(8,24)(9,19)(12,14)(13,21)(18,20), (1,18)(2,13)(3,12)(4,7)(5,24)(6,19)(8,16)(9,17)(10,20)(11,21)(14,22)(15,23), (1,18)(2,21)(3,12)(4,15)(5,24)(6,9)(7,23)(8,16)(10,20)(11,13)(14,22)(17,19), (1,20)(2,11)(3,14)(4,23)(5,8)(6,17)(7,15)(9,19)(10,18)(12,22)(13,21)(16,24), (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([(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,23),(8,24),(9,19),(10,20),(11,21),(12,22)], [(1,10),(2,11),(3,22),(4,23),(5,16),(6,17),(7,15),(8,24),(9,19),(12,14),(13,21),(18,20)], [(1,18),(2,13),(3,12),(4,7),(5,24),(6,19),(8,16),(9,17),(10,20),(11,21),(14,22),(15,23)], [(1,18),(2,21),(3,12),(4,15),(5,24),(6,9),(7,23),(8,16),(10,20),(11,13),(14,22),(17,19)], [(1,20),(2,11),(3,14),(4,23),(5,8),(6,17),(7,15),(9,19),(10,18),(12,22),(13,21),(16,24)], [(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(24,441);

On 24 points - transitive group 24T442
Generators in S24
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 14)(2 21)(3 22)(4 17)(5 10)(6 11)(7 15)(8 16)(9 23)(12 20)(13 19)(18 24)
(1 12)(2 7)(3 16)(4 23)(5 24)(6 13)(8 22)(9 17)(10 18)(11 19)(14 20)(15 21)
(2 15)(3 16)(5 18)(6 13)(7 21)(8 22)(10 24)(11 19)
(1 14)(2 15)(4 17)(5 18)(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)

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

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

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

G:=TransitiveGroup(24,442);

On 24 points - transitive group 24T443
Generators in S24
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 7)(3 16)(4 9)(5 18)(8 22)(10 24)(15 21)(17 23)
(1 14)(4 9)(5 18)(6 11)(10 24)(12 20)(13 19)(17 23)
(1 12)(2 21)(3 16)(4 9)(5 24)(6 13)(7 15)(8 22)(10 18)(11 19)(14 20)(17 23)
(1 20)(2 15)(3 8)(4 23)(5 18)(6 11)(7 21)(9 17)(10 24)(12 14)(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)| (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,7)(3,16)(4,9)(5,18)(8,22)(10,24)(15,21)(17,23), (1,14)(4,9)(5,18)(6,11)(10,24)(12,20)(13,19)(17,23), (1,12)(2,21)(3,16)(4,9)(5,24)(6,13)(7,15)(8,22)(10,18)(11,19)(14,20)(17,23), (1,20)(2,15)(3,8)(4,23)(5,18)(6,11)(7,21)(9,17)(10,24)(12,14)(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( (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,7)(3,16)(4,9)(5,18)(8,22)(10,24)(15,21)(17,23), (1,14)(4,9)(5,18)(6,11)(10,24)(12,20)(13,19)(17,23), (1,12)(2,21)(3,16)(4,9)(5,24)(6,13)(7,15)(8,22)(10,18)(11,19)(14,20)(17,23), (1,20)(2,15)(3,8)(4,23)(5,18)(6,11)(7,21)(9,17)(10,24)(12,14)(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([(1,12),(2,7),(3,8),(4,9),(5,10),(6,11),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(2,7),(3,16),(4,9),(5,18),(8,22),(10,24),(15,21),(17,23)], [(1,14),(4,9),(5,18),(6,11),(10,24),(12,20),(13,19),(17,23)], [(1,12),(2,21),(3,16),(4,9),(5,24),(6,13),(7,15),(8,22),(10,18),(11,19),(14,20),(17,23)], [(1,20),(2,15),(3,8),(4,23),(5,18),(6,11),(7,21),(9,17),(10,24),(12,14),(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(24,443);

On 24 points - transitive group 24T444
Generators in S24
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)
(3 19)(5 21)(10 18)(12 14)
(1 23)(5 21)(8 16)(12 14)
(2 24)(3 19)(5 21)(6 22)(7 15)(9 17)(10 18)(12 14)
(1 23)(2 24)(4 20)(5 21)(8 16)(9 17)(11 13)(12 14)
(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)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19), (3,19)(5,21)(10,18)(12,14), (1,23)(5,21)(8,16)(12,14), (2,24)(3,19)(5,21)(6,22)(7,15)(9,17)(10,18)(12,14), (1,23)(2,24)(4,20)(5,21)(8,16)(9,17)(11,13)(12,14), (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( (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19), (3,19)(5,21)(10,18)(12,14), (1,23)(5,21)(8,16)(12,14), (2,24)(3,19)(5,21)(6,22)(7,15)(9,17)(10,18)(12,14), (1,23)(2,24)(4,20)(5,21)(8,16)(9,17)(11,13)(12,14), (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([(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,20),(14,21),(15,22),(16,23),(17,24),(18,19)], [(3,19),(5,21),(10,18),(12,14)], [(1,23),(5,21),(8,16),(12,14)], [(2,24),(3,19),(5,21),(6,22),(7,15),(9,17),(10,18),(12,14)], [(1,23),(2,24),(4,20),(5,21),(8,16),(9,17),(11,13),(12,14)], [(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(24,444);

On 24 points - transitive group 24T445
Generators in S24
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 7)(3 22)(5 24)(6 11)(8 16)(10 18)(13 19)(15 21)
(1 20)(2 7)(4 9)(5 24)(10 18)(12 14)(15 21)(17 23)
(2 15)(3 16)(5 18)(6 13)(7 21)(8 22)(10 24)(11 19)
(1 14)(2 15)(4 17)(5 18)(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)

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

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

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

G:=TransitiveGroup(24,445);

On 24 points - transitive group 24T446
Generators in S24
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)
(2 11)(3 14)(4 7)(5 16)(8 24)(12 22)(13 21)(15 23)
(1 18)(4 7)(5 16)(6 9)(8 24)(10 20)(15 23)(17 19)
(1 10)(2 21)(3 14)(4 7)(5 24)(6 17)(8 16)(9 19)(11 13)(12 22)(15 23)(18 20)
(1 20)(2 13)(3 12)(4 23)(5 16)(6 9)(7 15)(8 24)(10 18)(11 21)(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)

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

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

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

G:=TransitiveGroup(24,446);

On 24 points - transitive group 24T447
Generators in S24
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 14)(2 7)(3 22)(5 10)(6 11)(8 16)(12 20)(13 19)(15 21)(18 24)
(1 12)(2 7)(3 16)(4 9)(5 24)(8 22)(10 18)(14 20)(15 21)(17 23)
(1 14)(3 16)(4 17)(6 13)(8 22)(9 23)(11 19)(12 20)
(2 15)(3 16)(5 18)(6 13)(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)

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

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

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

G:=TransitiveGroup(24,447);

On 24 points - transitive group 24T448
Generators in S24
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 18)(2 11)(3 12)(5 24)(6 9)(8 16)(10 20)(13 21)(14 22)(17 19)
(1 20)(2 11)(3 14)(4 7)(5 8)(10 18)(12 22)(13 21)(15 23)(16 24)
(1 18)(2 13)(4 15)(5 16)(7 23)(8 24)(10 20)(11 21)
(1 18)(3 14)(4 15)(6 17)(7 23)(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)| (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,18)(2,11)(3,12)(5,24)(6,9)(8,16)(10,20)(13,21)(14,22)(17,19), (1,20)(2,11)(3,14)(4,7)(5,8)(10,18)(12,22)(13,21)(15,23)(16,24), (1,18)(2,13)(4,15)(5,16)(7,23)(8,24)(10,20)(11,21), (1,18)(3,14)(4,15)(6,17)(7,23)(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( (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,18)(2,11)(3,12)(5,24)(6,9)(8,16)(10,20)(13,21)(14,22)(17,19), (1,20)(2,11)(3,14)(4,7)(5,8)(10,18)(12,22)(13,21)(15,23)(16,24), (1,18)(2,13)(4,15)(5,16)(7,23)(8,24)(10,20)(11,21), (1,18)(3,14)(4,15)(6,17)(7,23)(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([(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,18),(2,11),(3,12),(5,24),(6,9),(8,16),(10,20),(13,21),(14,22),(17,19)], [(1,20),(2,11),(3,14),(4,7),(5,8),(10,18),(12,22),(13,21),(15,23),(16,24)], [(1,18),(2,13),(4,15),(5,16),(7,23),(8,24),(10,20),(11,21)], [(1,18),(3,14),(4,15),(6,17),(7,23),(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(24,448);

On 24 points - transitive group 24T449
Generators in S24
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 24)(2 6)(3 14)(4 20)(5 18)(7 23)(8 15)(9 11)(10 21)(12 17)(13 19)(16 22)
(1 5)(2 16)(3 20)(4 14)(6 22)(7 17)(8 10)(9 19)(11 13)(12 23)(15 21)(18 24)
(1 10)(3 12)(4 7)(5 8)(14 17)(15 18)(20 23)(21 24)
(2 11)(3 12)(4 7)(6 9)(13 16)(14 17)(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)

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

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

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

G:=TransitiveGroup(24,449);

On 24 points - transitive group 24T450
Generators in S24
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)
(1 4)(2 13)(3 17)(5 21)(6 19)(7 22)(8 11)(9 16)(10 14)(12 24)(15 23)(18 20)
(1 23)(2 21)(3 6)(4 15)(5 13)(7 10)(8 20)(9 24)(11 18)(12 16)(14 22)(17 19)
(2 12)(3 7)(5 9)(6 10)(13 24)(14 19)(16 21)(17 22)
(1 11)(2 12)(4 8)(5 9)(13 24)(15 20)(16 21)(18 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)| (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,4)(2,13)(3,17)(5,21)(6,19)(7,22)(8,11)(9,16)(10,14)(12,24)(15,23)(18,20), (1,23)(2,21)(3,6)(4,15)(5,13)(7,10)(8,20)(9,24)(11,18)(12,16)(14,22)(17,19), (2,12)(3,7)(5,9)(6,10)(13,24)(14,19)(16,21)(17,22), (1,11)(2,12)(4,8)(5,9)(13,24)(15,20)(16,21)(18,23), (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( (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,4)(2,13)(3,17)(5,21)(6,19)(7,22)(8,11)(9,16)(10,14)(12,24)(15,23)(18,20), (1,23)(2,21)(3,6)(4,15)(5,13)(7,10)(8,20)(9,24)(11,18)(12,16)(14,22)(17,19), (2,12)(3,7)(5,9)(6,10)(13,24)(14,19)(16,21)(17,22), (1,11)(2,12)(4,8)(5,9)(13,24)(15,20)(16,21)(18,23), (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([(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23)], [(1,4),(2,13),(3,17),(5,21),(6,19),(7,22),(8,11),(9,16),(10,14),(12,24),(15,23),(18,20)], [(1,23),(2,21),(3,6),(4,15),(5,13),(7,10),(8,20),(9,24),(11,18),(12,16),(14,22),(17,19)], [(2,12),(3,7),(5,9),(6,10),(13,24),(14,19),(16,21),(17,22)], [(1,11),(2,12),(4,8),(5,9),(13,24),(15,20),(16,21),(18,23)], [(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(24,450);

On 24 points - transitive group 24T451
Generators in S24
(1 4)(2 5)(3 6)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 16)(14 17)(15 18)
(2 21)(3 7)(5 9)(6 19)(10 13)(12 15)(16 22)(18 24)
(1 11)(2 21)(4 23)(5 9)(8 17)(12 15)(14 20)(18 24)
(2 15)(3 16)(5 18)(6 13)(7 22)(9 24)(10 19)(12 21)
(1 14)(2 15)(4 17)(5 18)(8 23)(9 24)(11 20)(12 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)

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

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

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

G:=TransitiveGroup(24,451);

On 24 points - transitive group 24T452
Generators in S24
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)
(1 13)(2 6)(3 22)(4 15)(5 20)(7 18)(8 23)(9 12)(10 19)(11 16)(14 21)(17 24)
(1 5)(2 24)(3 15)(4 22)(6 17)(7 19)(8 11)(9 14)(10 18)(12 21)(13 20)(16 23)
(1 11)(3 10)(4 7)(5 8)(13 16)(15 18)(19 22)(20 23)
(2 12)(3 10)(4 7)(6 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)

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

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

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

G:=TransitiveGroup(24,452);

Polynomial with Galois group C2×C24⋊C6 over ℚ
actionf(x)Disc(f)
12T87x12-36x10+393x8-1821x6+3662x4-2535x2+169212·710·116·1310·414·5034
12T88x12-27x10+276x8-1313x6+2862x4-2376x2+216227·331·1974

Matrix representation of C2×C24⋊C6 in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
100000
010000
000-100
00-1000
00000-1
0000-10
,
0-10000
-100000
001000
000100
00000-1
0000-10
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
0000-10
00000-1
,
001000
000-100
000010
00000-1
100000
0-10000

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,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-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,-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],[0,0,0,0,1,0,0,0,0,0,0,-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] >;

C2×C24⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes C_6
% in TeX

G:=Group("C2xC2^4:C6");
// GroupNames label

G:=SmallGroup(192,1000);
// by ID

G=gap.SmallGroup(192,1000);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,1683,185,4204,333,1027,1784]);
// Polycyclic

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

Export

Character table of C2×C24⋊C6 in TeX

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