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## G = C62⋊24D6order 432 = 24·33

### 5th semidirect product of C62 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62⋊24D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C2×C32⋊4D6 — C62⋊24D6
 Lower central C33 — C32×C6 — C62⋊24D6
 Upper central C1 — C2 — C22

Generators and relations for C6224D6
G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1640 in 290 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×16], C6, C6 [×2], C6 [×18], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×6], C12 [×2], D6 [×26], C2×C6, C2×C6 [×2], C2×C6 [×10], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×3], C3×C6, C3×C6 [×2], C3×C6 [×13], C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×10], C3×D4 [×2], C22×S3 [×6], C22×C6, C33, C3×Dic3 [×6], C3⋊Dic3 [×2], S32 [×6], S3×C6 [×20], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×4], S3×D4 [×2], C2×C3⋊D4, C3×C3⋊S3 [×2], C3×C3⋊S3 [×3], C32×C6, C32×C6, S3×Dic3 [×2], C6.D6, D6⋊S3 [×2], C3⋊D12 [×4], C3×C3⋊D4 [×6], C327D4 [×2], C2×S32 [×3], S3×C2×C6 [×3], C22×C3⋊S3, C3×C3⋊Dic3 [×2], C324D6 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C3×C62, S3×C3⋊D4 [×2], Dic3⋊D6, C339(C2×C4), C339D4 [×2], C3×C327D4 [×2], C2×C324D6, C2×C6×C3⋊S3, C6224D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, C3⋊D4 [×2], C22×S3 [×3], S32 [×3], S3×D4 [×2], C2×C3⋊D4, C2×S32 [×3], C324D6, S3×C3⋊D4 [×2], Dic3⋊D6, C2×C324D6, C6224D6

Permutation representations of C6224D6
On 24 points - transitive group 24T1284
Generators in S24
```(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 3 5 2 4)(7 12 8 10 9 11)(13 18 17 16 15 14)(19 20 21 22 23 24)
(1 20 2 22 3 24)(4 21 5 23 6 19)(7 18 8 14 9 16)(10 15 11 17 12 13)
(1 14)(2 18)(3 16)(4 13)(5 17)(6 15)(7 22)(8 20)(9 24)(10 19)(11 23)(12 21)```

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

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

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

`G:=TransitiveGroup(24,1284);`

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D ··· 3H 4A 4B 6A ··· 6E 6F ··· 6V 6W 6X 6Y 6Z 6AA 6AB 12A 12B order 1 2 2 2 2 2 2 2 3 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 6 6 12 12 size 1 1 2 9 9 18 18 18 2 2 2 4 ··· 4 18 18 2 ··· 2 4 ··· 4 18 18 18 18 36 36 36 36

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C3⋊D4 S32 S3×D4 C2×S32 C32⋊4D6 S3×C3⋊D4 Dic3⋊D6 C2×C32⋊4D6 C62⋊24D6 kernel C62⋊24D6 C33⋊9(C2×C4) C33⋊9D4 C3×C32⋊7D4 C2×C32⋊4D6 C2×C6×C3⋊S3 C32⋊7D4 C22×C3⋊S3 C3×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C62 C3⋊S3 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 2 2 1 1 2 1 2 2 4 3 4 3 2 3 2 4 2 2 4

Matrix representation of C6224D6 in GL4(𝔽7) generated by

 2 0 4 2 5 0 1 6 4 4 1 6 0 0 0 4
,
 4 6 3 2 6 4 4 2 0 0 3 0 0 0 0 5
,
 2 4 2 4 1 1 0 6 5 2 1 4 2 2 6 3
,
 1 4 5 4 3 4 4 0 5 2 1 4 3 3 4 1
`G:=sub<GL(4,GF(7))| [2,5,4,0,0,0,4,0,4,1,1,0,2,6,6,4],[4,6,0,0,6,4,0,0,3,4,3,0,2,2,0,5],[2,1,5,2,4,1,2,2,2,0,1,6,4,6,4,3],[1,3,5,3,4,4,2,3,5,4,1,4,4,0,4,1] >;`

C6224D6 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_{24}D_6`
`% in TeX`

`G:=Group("C6^2:24D6");`
`// GroupNames label`

`G:=SmallGroup(432,696);`
`// by ID`

`G=gap.SmallGroup(432,696);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,1124,571,2028,14118]);`
`// Polycyclic`

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

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