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G = C24⋊C12order 192 = 26·3

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

metabelian, soluble, monomial

Aliases: C241C12, C25.1C6, C243C4⋊C3, C22⋊A42C4, (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

C1C24 — C24⋊C12
C1C22C24C25C2×C22⋊A4 — C24⋊C12
C24 — C24⋊C12
C1C2

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 [×6], C3, C4 [×2], C22, C22 [×30], C6, C2×C4 [×4], C23, C23 [×30], C12, A4 [×3], C22⋊C4 [×4], C22×C4, C22×C4, C24, C24 [×6], C2×A4 [×3], C2×C22⋊C4 [×2], C25, C4×A4, C22⋊A4, C243C4, 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 12A2B2C2D2E2F2G3A3B4A4B4C4D6A6B12A12B12C12D
 size 113366661616441212161616161616
ρ111111111111111111111    trivial
ρ21111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ311111111ζ32ζ31111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ411111111ζ32ζ3-1-1-1-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ511111111ζ3ζ321111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ611111111ζ3ζ32-1-1-1-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ71-1-1111-1-111i-ii-i-1-1i-ii-i    linear of order 4
ρ81-1-1111-1-111-ii-ii-1-1-ii-ii    linear of order 4
ρ91-1-1111-1-1ζ32ζ3i-ii-iζ65ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ101-1-1111-1-1ζ3ζ32-ii-iiζ6ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ111-1-1111-1-1ζ3ζ32i-ii-iζ6ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ121-1-1111-1-1ζ32ζ3-ii-iiζ65ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ133333-1-1-1-10033-1-1000000    orthogonal lifted from A4
ρ143333-1-1-1-100-3-311000000    orthogonal lifted from C2×A4
ρ153-3-33-1-11100-3i3ii-i000000    complex lifted from C4×A4
ρ163-3-33-1-111003i-3i-ii000000    complex lifted from C4×A4
ρ176-62-2-222-2000000000000    orthogonal faithful
ρ1866-2-2-22-22000000000000    orthogonal lifted from C24⋊C6
ρ196-62-22-2-22000000000000    orthogonal faithful
ρ2066-2-22-22-2000000000000    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 11)(2 16)(3 5)(4 10)(6 14)(7 15)(8 12)(9 13)
(1 15)(2 8)(3 9)(4 14)(5 13)(6 10)(7 11)(12 16)
(1 15)(2 12)(3 9)(4 6)(5 13)(7 11)(8 16)(10 14)
(1 7)(2 16)(3 13)(4 10)(5 9)(6 14)(8 12)(11 15)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

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

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

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

G:=TransitiveGroup(24,484);

On 24 points - transitive group 24T501
Generators in S24
(1 21)(2 8)(3 9)(4 18)(5 19)(6 14)(7 15)(10 24)(11 13)(12 20)(16 22)(17 23)
(1 7)(2 16)(3 17)(4 24)(5 13)(6 12)(8 22)(9 23)(10 18)(11 19)(14 20)(15 21)
(1 7)(2 22)(3 17)(4 10)(5 13)(6 20)(8 16)(9 23)(11 19)(12 14)(15 21)(18 24)
(1 15)(2 8)(3 23)(4 18)(5 11)(6 14)(7 21)(9 17)(10 24)(12 20)(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)| (1,21)(2,8)(3,9)(4,18)(5,19)(6,14)(7,15)(10,24)(11,13)(12,20)(16,22)(17,23), (1,7)(2,16)(3,17)(4,24)(5,13)(6,12)(8,22)(9,23)(10,18)(11,19)(14,20)(15,21), (1,7)(2,22)(3,17)(4,10)(5,13)(6,20)(8,16)(9,23)(11,19)(12,14)(15,21)(18,24), (1,15)(2,8)(3,23)(4,18)(5,11)(6,14)(7,21)(9,17)(10,24)(12,20)(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( (1,21)(2,8)(3,9)(4,18)(5,19)(6,14)(7,15)(10,24)(11,13)(12,20)(16,22)(17,23), (1,7)(2,16)(3,17)(4,24)(5,13)(6,12)(8,22)(9,23)(10,18)(11,19)(14,20)(15,21), (1,7)(2,22)(3,17)(4,10)(5,13)(6,20)(8,16)(9,23)(11,19)(12,14)(15,21)(18,24), (1,15)(2,8)(3,23)(4,18)(5,11)(6,14)(7,21)(9,17)(10,24)(12,20)(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([(1,21),(2,8),(3,9),(4,18),(5,19),(6,14),(7,15),(10,24),(11,13),(12,20),(16,22),(17,23)], [(1,7),(2,16),(3,17),(4,24),(5,13),(6,12),(8,22),(9,23),(10,18),(11,19),(14,20),(15,21)], [(1,7),(2,22),(3,17),(4,10),(5,13),(6,20),(8,16),(9,23),(11,19),(12,14),(15,21),(18,24)], [(1,15),(2,8),(3,23),(4,18),(5,11),(6,14),(7,21),(9,17),(10,24),(12,20),(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(24,501);

On 24 points - transitive group 24T502
Generators in S24
(1 7)(2 24)(5 11)(6 16)(8 18)(12 22)(15 21)(17 23)
(3 9)(4 14)(5 11)(6 16)(10 20)(12 22)(13 19)(15 21)
(1 7)(2 18)(3 13)(4 10)(5 21)(6 16)(8 24)(9 19)(11 15)(12 22)(14 20)(17 23)
(1 23)(2 8)(3 19)(4 14)(5 11)(6 22)(7 17)(9 13)(10 20)(12 16)(15 21)(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)| (1,7)(2,24)(5,11)(6,16)(8,18)(12,22)(15,21)(17,23), (3,9)(4,14)(5,11)(6,16)(10,20)(12,22)(13,19)(15,21), (1,7)(2,18)(3,13)(4,10)(5,21)(6,16)(8,24)(9,19)(11,15)(12,22)(14,20)(17,23), (1,23)(2,8)(3,19)(4,14)(5,11)(6,22)(7,17)(9,13)(10,20)(12,16)(15,21)(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( (1,7)(2,24)(5,11)(6,16)(8,18)(12,22)(15,21)(17,23), (3,9)(4,14)(5,11)(6,16)(10,20)(12,22)(13,19)(15,21), (1,7)(2,18)(3,13)(4,10)(5,21)(6,16)(8,24)(9,19)(11,15)(12,22)(14,20)(17,23), (1,23)(2,8)(3,19)(4,14)(5,11)(6,22)(7,17)(9,13)(10,20)(12,16)(15,21)(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([(1,7),(2,24),(5,11),(6,16),(8,18),(12,22),(15,21),(17,23)], [(3,9),(4,14),(5,11),(6,16),(10,20),(12,22),(13,19),(15,21)], [(1,7),(2,18),(3,13),(4,10),(5,21),(6,16),(8,24),(9,19),(11,15),(12,22),(14,20),(17,23)], [(1,23),(2,8),(3,19),(4,14),(5,11),(6,22),(7,17),(9,13),(10,20),(12,16),(15,21),(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(24,502);

On 24 points - transitive group 24T503
Generators in S24
(2 24)(6 16)(8 18)(12 22)
(4 14)(6 16)(10 20)(12 22)
(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)| (2,24)(6,16)(8,18)(12,22), (4,14)(6,16)(10,20)(12,22), (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( (2,24)(6,16)(8,18)(12,22), (4,14)(6,16)(10,20)(12,22), (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([(2,24),(6,16),(8,18),(12,22)], [(4,14),(6,16),(10,20),(12,22)], [(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(24,503);

On 24 points - transitive group 24T504
Generators in S24
(2 15)(3 9)(5 11)(6 19)(8 21)(12 13)(16 22)(18 24)
(1 7)(3 9)(4 17)(6 19)(10 23)(12 13)(14 20)(16 22)
(2 21)(3 22)(5 24)(6 13)(8 15)(9 16)(11 18)(12 19)
(1 20)(3 22)(4 23)(6 13)(7 14)(9 16)(10 17)(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)

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

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

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

G:=TransitiveGroup(24,504);

On 24 points - transitive group 24T505
Generators in S24
(1 24)(2 19)(5 16)(6 23)(7 18)(8 13)(11 22)(12 17)
(3 14)(4 21)(5 16)(6 23)(9 20)(10 15)(11 22)(12 17)
(1 24)(2 8)(3 20)(4 15)(5 11)(6 23)(7 18)(9 14)(10 21)(12 17)(13 19)(16 22)
(1 18)(2 13)(3 9)(4 21)(5 16)(6 12)(7 24)(8 19)(10 15)(11 22)(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)| (1,24)(2,19)(5,16)(6,23)(7,18)(8,13)(11,22)(12,17), (3,14)(4,21)(5,16)(6,23)(9,20)(10,15)(11,22)(12,17), (1,24)(2,8)(3,20)(4,15)(5,11)(6,23)(7,18)(9,14)(10,21)(12,17)(13,19)(16,22), (1,18)(2,13)(3,9)(4,21)(5,16)(6,12)(7,24)(8,19)(10,15)(11,22)(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( (1,24)(2,19)(5,16)(6,23)(7,18)(8,13)(11,22)(12,17), (3,14)(4,21)(5,16)(6,23)(9,20)(10,15)(11,22)(12,17), (1,24)(2,8)(3,20)(4,15)(5,11)(6,23)(7,18)(9,14)(10,21)(12,17)(13,19)(16,22), (1,18)(2,13)(3,9)(4,21)(5,16)(6,12)(7,24)(8,19)(10,15)(11,22)(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([(1,24),(2,19),(5,16),(6,23),(7,18),(8,13),(11,22),(12,17)], [(3,14),(4,21),(5,16),(6,23),(9,20),(10,15),(11,22),(12,17)], [(1,24),(2,8),(3,20),(4,15),(5,11),(6,23),(7,18),(9,14),(10,21),(12,17),(13,19),(16,22)], [(1,18),(2,13),(3,9),(4,21),(5,16),(6,12),(7,24),(8,19),(10,15),(11,22),(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(24,505);

On 24 points - transitive group 24T506
Generators in S24
(2 24)(3 19)(4 10)(5 21)(6 22)(8 18)(9 13)(11 15)(12 16)(14 20)
(1 17)(2 8)(3 19)(4 20)(6 16)(7 23)(9 13)(10 14)(12 22)(18 24)
(1 7)(2 8)(4 10)(5 11)(14 20)(15 21)(17 23)(18 24)
(2 8)(3 9)(5 11)(6 12)(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)

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

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

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

G:=TransitiveGroup(24,506);

On 24 points - transitive group 24T507
Generators in S24
(2 21)(3 9)(4 17)(5 11)(6 12)(8 15)(10 23)(13 19)(16 22)(18 24)
(1 7)(2 15)(3 9)(4 10)(6 13)(8 21)(12 19)(14 20)(16 22)(17 23)
(1 14)(2 15)(4 17)(5 18)(7 20)(8 21)(10 23)(11 24)
(2 15)(3 16)(5 18)(6 19)(8 21)(9 22)(11 24)(12 13)
(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)| (2,21)(3,9)(4,17)(5,11)(6,12)(8,15)(10,23)(13,19)(16,22)(18,24), (1,7)(2,15)(3,9)(4,10)(6,13)(8,21)(12,19)(14,20)(16,22)(17,23), (1,14)(2,15)(4,17)(5,18)(7,20)(8,21)(10,23)(11,24), (2,15)(3,16)(5,18)(6,19)(8,21)(9,22)(11,24)(12,13), (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( (2,21)(3,9)(4,17)(5,11)(6,12)(8,15)(10,23)(13,19)(16,22)(18,24), (1,7)(2,15)(3,9)(4,10)(6,13)(8,21)(12,19)(14,20)(16,22)(17,23), (1,14)(2,15)(4,17)(5,18)(7,20)(8,21)(10,23)(11,24), (2,15)(3,16)(5,18)(6,19)(8,21)(9,22)(11,24)(12,13), (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([(2,21),(3,9),(4,17),(5,11),(6,12),(8,15),(10,23),(13,19),(16,22),(18,24)], [(1,7),(2,15),(3,9),(4,10),(6,13),(8,21),(12,19),(14,20),(16,22),(17,23)], [(1,14),(2,15),(4,17),(5,18),(7,20),(8,21),(10,23),(11,24)], [(2,15),(3,16),(5,18),(6,19),(8,21),(9,22),(11,24),(12,13)], [(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
actionf(x)Disc(f)
12T99x12-12x10+60x8-160x6+228x4-144x2+8257·316
12T105x12-6x11+2x10+45x9-58x8-104x7+174x6+97x5-186x4-34x3+69x2-557·138·316

Matrix representation of C24⋊C12 in GL6(ℤ)

-100000
0-10000
000100
001000
00000-1
0000-10
,
010000
100000
000-100
00-1000
0000-10
00000-1
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
00-1000
000-100
000010
000001
,
000-100
001000
00000-1
000010
0-10000
100000

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

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