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G = C24.5D6order 192 = 26·3

4th non-split extension by C24 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C24.5D6
 Chief series C1 — C22 — A4 — C2×A4 — C22×A4 — C22×S4 — C24.5D6
 Lower central A4 — C2×A4 — C24.5D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C24.5D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=a, f2=ba=ab, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=be5 >

Subgroups: 626 in 143 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, A4, D6, C2×C6, C22⋊C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×C12, S4, C2×A4, C22×S3, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, A4⋊C4, C4×A4, C2×S4, C2×S4, C22×A4, C23.23D4, C2×A4⋊C4, C2×C4×A4, C22×S4, C24.5D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, S4, D6⋊C4, C2×S4, C4×S4, C4⋊S4, A4⋊D4, C24.5D6

Character table of C24.5D6

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 3 3 3 3 12 12 8 2 2 6 6 12 12 12 12 12 12 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 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 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 1 -1 1 -1 1 -1 1 -i i i -i i 1 -i -i i -1 -1 -1 1 i -i -i i linear of order 4 ρ6 1 -1 -1 1 1 -1 1 -1 -1 1 1 -i i i -i -i -1 i i -i 1 -1 -1 1 i -i -i i linear of order 4 ρ7 1 -1 -1 1 1 -1 1 -1 1 -1 1 i -i -i i -i 1 i i -i -1 -1 -1 1 -i i i -i linear of order 4 ρ8 1 -1 -1 1 1 -1 1 -1 -1 1 1 i -i -i i i -1 -i -i i 1 -1 -1 1 -i i i -i linear of order 4 ρ9 2 2 2 2 2 2 2 2 0 0 -1 -2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 -2 -2 2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 -2 2 2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 -2 -2 -2 2 2 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 -1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ14 2 2 -2 -2 -2 -2 2 2 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 -1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ15 2 -2 -2 2 2 -2 2 -2 0 0 -1 2i -2i -2i 2i 0 0 0 0 0 0 1 1 -1 i -i -i i complex lifted from C4×S3 ρ16 2 -2 -2 2 2 -2 2 -2 0 0 -1 -2i 2i 2i -2i 0 0 0 0 0 0 1 1 -1 -i i i -i complex lifted from C4×S3 ρ17 2 -2 2 -2 -2 2 2 -2 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 -2 2 -2 -2 2 2 -2 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 3 3 3 3 -1 -1 -1 -1 1 1 0 3 3 -1 -1 -1 -1 1 -1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 3 3 -1 -1 -1 -1 -1 -1 0 -3 -3 1 1 -1 1 1 -1 1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 3 3 -1 -1 -1 -1 -1 -1 0 3 3 -1 -1 1 1 -1 1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ22 3 3 3 3 -1 -1 -1 -1 1 1 0 -3 -3 1 1 1 -1 -1 1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ23 3 -3 -3 3 -1 1 -1 1 1 -1 0 -3i 3i -i i -i -1 -i i i 1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ24 3 -3 -3 3 -1 1 -1 1 -1 1 0 -3i 3i -i i i 1 i -i -i -1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ25 3 -3 -3 3 -1 1 -1 1 -1 1 0 3i -3i i -i -i 1 -i i i -1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ26 3 -3 -3 3 -1 1 -1 1 1 -1 0 3i -3i i -i i -1 i -i -i 1 0 0 0 0 0 0 0 complex lifted from C4×S4 ρ27 6 6 -6 -6 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4 ρ28 6 -6 6 -6 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4

Permutation representations of C24.5D6
On 24 points - transitive group 24T414
Generators in S24
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(1 13)(2 14)(4 16)(5 17)(7 19)(8 20)(10 22)(11 23)
(1 13)(3 15)(4 16)(6 18)(7 19)(9 21)(10 22)(12 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 6 13 18)(2 17 14 5)(3 4 15 16)(7 12 19 24)(8 23 20 11)(9 10 21 22)```

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

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

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

`G:=TransitiveGroup(24,414);`

On 24 points - transitive group 24T415
Generators in S24
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)
(1 20)(2 21)(4 23)(5 24)(7 14)(8 15)(10 17)(11 18)
(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)
(1 12 14 13)(2 24 15 11)(3 10 16 23)(4 22 17 9)(5 8 18 21)(6 20 19 7)```

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

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

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

`G:=TransitiveGroup(24,415);`

On 24 points - transitive group 24T417
Generators in S24
```(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)
(1 7)(2 8)(4 10)(5 11)(13 19)(14 20)(16 22)(17 23)
(1 7)(3 9)(4 10)(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)
(1 6 16 21)(2 20 17 5)(3 4 18 19)(7 12 22 15)(8 14 23 11)(9 10 24 13)```

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

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

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

`G:=TransitiveGroup(24,417);`

Matrix representation of C24.5D6 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 1 12 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 1 0 0 0 0 12 1 0
,
 0 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 9 0 0 0 0 0 4 10 0 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 0 12 0 1
,
 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 3 0 0 0 0 0 3 9 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0 1

`G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,4,0,0,0,0,0,9,10,0,0,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,3,0,0,0,0,0,3,9,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;`

C24.5D6 in GAP, Magma, Sage, TeX

`C_2^4._5D_6`
`% in TeX`

`G:=Group("C2^4.5D6");`
`// GroupNames label`

`G:=SmallGroup(192,972);`
`// by ID`

`G=gap.SmallGroup(192,972);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,141,36,1124,4037,285,2358,475]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=a,f^2=b*a=a*b,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,f*c*f^-1=e*d*e^-1=c*d=d*c,e*c*e^-1=d,d*f=f*d,f*e*f^-1=b*e^5>;`
`// generators/relations`

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