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## G = C24.10D6order 192 = 26·3

### 9th 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 — C24.10D6
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C4×S4 — C24.10D6
 Lower central A4 — C2×A4 — C24.10D6
 Upper central C1 — C4 — C2×C4

Generators and relations for C24.10D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=e5 >

Subgroups: 626 in 171 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×8], C22 [×2], C22 [×14], S3 [×2], C6 [×2], C2×C4, C2×C4 [×18], D4 [×14], Q8 [×2], C23, C23 [×6], Dic3 [×2], C12 [×2], A4, D6 [×2], C2×C6, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, S4 [×2], C2×A4, C2×A4, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, A4⋊C4 [×2], C4×A4 [×2], C4○D12, C2×S4 [×2], C22×A4, C22.19C24, A4⋊Q8, C4×S4 [×2], C4⋊S4, A4⋊D4 [×2], C2×C4×A4, C24.10D6
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, S4, C22×S3, C4○D12, C2×S4 [×3], C22×S4, C24.10D6

Character table of C24.10D6

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

Permutation representations of C24.10D6
On 24 points - transitive group 24T292
Generators in S24
```(13 19)(14 20)(15 21)(16 22)(17 23)(18 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)
(2 8)(3 9)(5 11)(6 12)(13 19)(15 21)(16 22)(18 24)
(1 7)(2 8)(4 10)(5 11)(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)
(1 21 7 15)(2 14 8 20)(3 19 9 13)(4 24 10 18)(5 17 11 23)(6 22 12 16)```

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

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

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

`G:=TransitiveGroup(24,292);`

On 24 points - transitive group 24T319
Generators in S24
```(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)
(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 7)(2 8)(4 10)(5 11)(14 20)(15 21)(17 23)(18 24)
(1 7)(3 9)(4 10)(6 12)(13 19)(14 20)(16 22)(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)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)```

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

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

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

`G:=TransitiveGroup(24,319);`

On 24 points - transitive group 24T395
Generators in S24
```(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 13)(2 20)(3 9)(4 16)(5 23)(6 12)(7 19)(8 14)(10 22)(11 17)(15 21)(18 24)
(1 19)(2 8)(3 15)(4 22)(5 11)(6 18)(7 13)(9 21)(10 16)(12 24)(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)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)```

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

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

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

`G:=TransitiveGroup(24,395);`

Matrix representation of C24.10D6 in GL5(𝔽13)

 0 5 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 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 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 8 0 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 12 0 0 0 0 0 12 0
,
 0 8 0 0 0 8 0 0 0 0 0 0 0 0 12 0 0 0 12 0 0 0 12 0 0

`G:=sub<GL(5,GF(13))| [0,8,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[8,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0,0],[0,8,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,12,0,0] >;`

C24.10D6 in GAP, Magma, Sage, TeX

`C_2^4._{10}D_6`
`% in TeX`

`G:=Group("C2^4.10D6");`
`// GroupNames label`

`G:=SmallGroup(192,1471);`
`// by ID`

`G=gap.SmallGroup(192,1471);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,254,1124,4037,285,2358,475]);`
`// Polycyclic`

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

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