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## G = C24⋊4Dic3order 192 = 26·3

### 3rd semidirect product of C24 and Dic3 acting via Dic3/C2=S3

Aliases: C25.3S3, C23.13S4, C244Dic3, C22⋊(A4⋊C4), C22⋊A43C4, C2.1(C22⋊S4), (C2×C22⋊A4).2C2, SmallGroup(192,1495)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C22⋊A4 — C24⋊4Dic3
 Chief series C1 — C22 — C24 — C22⋊A4 — C2×C22⋊A4 — C24⋊4Dic3
 Lower central C22⋊A4 — C24⋊4Dic3
 Upper central C1 — C2

Generators and relations for C244Dic3
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=e3, eae-1=ab=ba, ac=ca, ad=da, faf-1=b, bc=cb, bd=db, ebe-1=fbf-1=a, ede-1=cd=dc, ece-1=fcf-1=d, fdf-1=c, fef-1=e-1 >

Subgroups: 702 in 150 conjugacy classes, 13 normal (7 characteristic)
C1, C2, C2 [×8], C3, C4 [×4], C22 [×3], C22 [×34], C6, C2×C4 [×12], C23 [×3], C23 [×34], Dic3, A4 [×4], C22⋊C4 [×12], C22×C4 [×4], C24, C24 [×8], C2×A4 [×4], C2×C22⋊C4 [×6], C25, A4⋊C4 [×3], C22⋊A4, C243C4, C2×C22⋊A4, C244Dic3
Quotients: C1, C2, C4, S3, Dic3, S4 [×3], A4⋊C4 [×3], C22⋊S4, C244Dic3

Character table of C244Dic3

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 6 size 1 1 3 3 3 3 3 3 6 6 32 12 12 12 12 12 12 12 12 32 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 1 -1 1 i -i -i -i i i i -i -1 linear of order 4 ρ4 1 -1 -1 1 1 -1 -1 1 1 -1 1 -i i i i -i -i -i i -1 linear of order 4 ρ5 2 2 2 2 2 2 2 2 2 2 -1 0 0 0 0 0 0 0 0 -1 orthogonal lifted from S3 ρ6 2 -2 -2 2 2 -2 -2 2 2 -2 -1 0 0 0 0 0 0 0 0 1 symplectic lifted from Dic3, Schur index 2 ρ7 3 3 -1 -1 -1 -1 3 3 -1 -1 0 -1 -1 1 -1 1 -1 1 1 0 orthogonal lifted from S4 ρ8 3 3 3 3 -1 -1 -1 -1 -1 -1 0 -1 1 1 -1 -1 1 1 -1 0 orthogonal lifted from S4 ρ9 3 3 -1 -1 3 3 -1 -1 -1 -1 0 -1 1 -1 -1 1 1 -1 1 0 orthogonal lifted from S4 ρ10 3 3 3 3 -1 -1 -1 -1 -1 -1 0 1 -1 -1 1 1 -1 -1 1 0 orthogonal lifted from S4 ρ11 3 3 -1 -1 -1 -1 3 3 -1 -1 0 1 1 -1 1 -1 1 -1 -1 0 orthogonal lifted from S4 ρ12 3 3 -1 -1 3 3 -1 -1 -1 -1 0 1 -1 1 1 -1 -1 1 -1 0 orthogonal lifted from S4 ρ13 3 -3 1 -1 3 -3 1 -1 -1 1 0 i i -i -i -i -i i i 0 complex lifted from A4⋊C4 ρ14 3 -3 1 -1 3 -3 1 -1 -1 1 0 -i -i i i i i -i -i 0 complex lifted from A4⋊C4 ρ15 3 -3 1 -1 -1 1 -3 3 -1 1 0 i -i i -i -i i -i i 0 complex lifted from A4⋊C4 ρ16 3 -3 -3 3 -1 1 1 -1 -1 1 0 -i -i -i i -i i i i 0 complex lifted from A4⋊C4 ρ17 3 -3 1 -1 -1 1 -3 3 -1 1 0 -i i -i i i -i i -i 0 complex lifted from A4⋊C4 ρ18 3 -3 -3 3 -1 1 1 -1 -1 1 0 i i i -i i -i -i -i 0 complex lifted from A4⋊C4 ρ19 6 6 -2 -2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C22⋊S4 ρ20 6 -6 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C244Dic3
On 12 points - transitive group 12T102
Generators in S12
(1 8)(2 5)(3 7)(4 11)(6 10)(9 12)
(1 11)(2 9)(3 6)(4 8)(5 12)(7 10)
(1 8)(2 12)(3 6)(4 11)(5 9)(7 10)
(1 11)(2 5)(3 10)(4 8)(6 7)(9 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 8 4 11)(2 7 5 10)(3 12 6 9)

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

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

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

G:=TransitiveGroup(12,102);

On 12 points - transitive group 12T107
Generators in S12
(2 5)(3 6)(7 10)(8 11)
(1 4)(3 6)(8 11)(9 12)
(2 5)(3 6)(7 10)(9 12)
(1 4)(2 5)(8 11)(9 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10 4 7)(2 9 5 12)(3 8 6 11)

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

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

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

G:=TransitiveGroup(12,107);

On 16 points - transitive group 16T434
Generators in S16
(1 6)(2 9)(3 12)(4 15)(5 7)(8 10)(11 13)(14 16)
(1 10)(2 7)(3 16)(4 13)(5 9)(6 8)(11 15)(12 14)
(1 8)(2 5)(3 12)(4 15)(6 10)(7 9)(11 13)(14 16)
(1 10)(2 7)(3 14)(4 11)(5 9)(6 8)(12 16)(13 15)
(1 2)(3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(1 3 2 4)(5 11 8 14)(6 16 9 13)(7 15 10 12)

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

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

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

G:=TransitiveGroup(16,434);

On 24 points - transitive group 24T386
Generators in S24
(2 15)(3 16)(5 18)(6 13)(8 20)(9 21)(11 23)(12 24)
(1 14)(3 16)(4 17)(6 13)(7 19)(9 21)(10 22)(12 24)
(1 4)(3 6)(7 10)(9 12)(13 16)(14 17)(19 22)(21 24)
(2 5)(3 6)(8 11)(9 12)(13 16)(15 18)(20 23)(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)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,386);

On 24 points - transitive group 24T433
Generators in S24
(2 16)(3 17)(5 13)(6 14)(7 23)(8 24)(10 20)(11 21)
(1 15)(3 17)(4 18)(6 14)(8 24)(9 19)(11 21)(12 22)
(2 13)(3 14)(5 16)(6 17)(7 20)(9 22)(10 23)(12 19)
(1 18)(2 13)(4 15)(5 16)(8 21)(9 22)(11 24)(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)
(1 10 4 7)(2 9 5 12)(3 8 6 11)(13 22 16 19)(14 21 17 24)(15 20 18 23)

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

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

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

G:=TransitiveGroup(24,433);

On 24 points - transitive group 24T494
Generators in S24
(1 4)(2 5)(7 10)(8 11)(13 16)(14 17)(19 22)(20 23)
(2 5)(3 6)(8 11)(9 12)(14 17)(15 18)(20 23)(21 24)
(1 4)(2 14)(3 18)(5 17)(6 15)(7 10)(8 23)(9 21)(11 20)(12 24)(13 16)(19 22)
(1 13)(2 17)(3 6)(4 16)(5 14)(7 22)(8 20)(9 12)(10 19)(11 23)(15 18)(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)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,494);

On 24 points - transitive group 24T495
Generators in S24
(1 13)(2 5)(3 18)(4 16)(6 15)(7 19)(8 11)(9 24)(10 22)(12 21)(14 17)(20 23)
(1 16)(2 14)(3 6)(4 13)(5 17)(7 22)(8 20)(9 12)(10 19)(11 23)(15 18)(21 24)
(1 22)(2 23)(4 19)(5 20)(7 16)(8 17)(10 13)(11 14)
(1 22)(3 24)(4 19)(6 21)(7 16)(9 18)(10 13)(12 15)
(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 16 4 13)(2 15 5 18)(3 14 6 17)(7 19 10 22)(8 24 11 21)(9 23 12 20)

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

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

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

G:=TransitiveGroup(24,495);

On 24 points - transitive group 24T512
Generators in S24
(1 14)(2 15)(4 17)(5 18)(7 19)(9 21)(10 22)(12 24)
(2 15)(3 16)(5 18)(6 13)(7 19)(8 20)(10 22)(11 23)
(1 14)(2 5)(3 13)(4 17)(6 16)(7 22)(8 11)(9 21)(10 19)(12 24)(15 18)(20 23)
(1 4)(2 18)(3 16)(5 15)(6 13)(7 10)(8 20)(9 24)(11 23)(12 21)(14 17)(19 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 4 23)(2 19 5 22)(3 24 6 21)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,512);

On 24 points - transitive group 24T513
Generators in S24
(2 15)(3 16)(5 18)(6 13)(8 20)(9 21)(11 23)(12 24)
(1 14)(3 16)(4 17)(6 13)(7 19)(9 21)(10 22)(12 24)
(2 5)(3 6)(7 10)(8 11)(13 16)(15 18)(19 22)(20 23)
(1 4)(2 5)(7 10)(9 12)(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)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,513);

On 24 points - transitive group 24T514
Generators in S24
(1 16)(2 5)(3 15)(4 13)(6 18)(7 10)(8 19)(9 23)(11 22)(12 20)(14 17)(21 24)
(1 13)(2 17)(3 6)(4 16)(5 14)(7 21)(8 11)(9 20)(10 24)(12 23)(15 18)(19 22)
(1 4)(2 14)(3 18)(5 17)(6 15)(7 10)(8 19)(9 23)(11 22)(12 20)(13 16)(21 24)
(1 13)(2 17)(3 6)(4 16)(5 14)(7 24)(8 22)(9 12)(10 21)(11 19)(15 18)(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)
(1 23 4 20)(2 22 5 19)(3 21 6 24)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,514);

On 24 points - transitive group 24T515
Generators in S24
(2 14)(3 15)(5 17)(6 18)(7 19)(9 21)(10 22)(12 24)
(1 13)(3 15)(4 16)(6 18)(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 13)(2 14)(4 16)(5 17)(7 19)(8 20)(10 22)(11 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)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,515);

Polynomial with Galois group C244Dic3 over ℚ
actionf(x)Disc(f)
12T102x12-5x10+20x8-70x6+145x4-280x2+208232·324·118·137
12T107x12-6x10+3x8+28x6-21x4-30x2+5244·316·57

Matrix representation of C244Dic3 in GL6(ℤ)

 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
,
 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 -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 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1 0 0 0

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

C244Dic3 in GAP, Magma, Sage, TeX

C_2^4\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("C2^4:4Dic3");
// GroupNames label

G:=SmallGroup(192,1495);
// by ID

G=gap.SmallGroup(192,1495);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,14,170,675,185,424,333,4037,1531,2358,608]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=e^3,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=f*b*f^-1=a,e*d*e^-1=c*d=d*c,e*c*e^-1=f*c*f^-1=d,f*d*f^-1=c,f*e*f^-1=e^-1>;
// generators/relations

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