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## G = C62.84D6order 432 = 24·33

### 32nd non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.84D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — C2×C6×C3⋊S3 — C62.84D6
 Lower central C33 — C32×C6 — C62.84D6
 Upper central C1 — C22

Generators and relations for C62.84D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 952 in 210 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C3×C3⋊S3, C32×C6, C32×C6, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C6×C3⋊S3, C6×C3⋊S3, C3×C62, D6⋊Dic3, C6.D12, C6×C3⋊Dic3, C2×C6×C3⋊S3, C62.84D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, S32, D6⋊C4, C6.D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C324D6, D6⋊Dic3, C6.D12, C339(C2×C4), C339D4, C62.84D6

Smallest permutation representation of C62.84D6
On 48 points
Generators in S48
```(1 3 5 7 9 11)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 47 45 43 41 39)(38 40 42 44 46 48)
(1 15 5 19 9 23)(2 24 10 20 6 16)(3 17 7 21 11 13)(4 14 12 22 8 18)(25 42 33 38 29 46)(26 47 30 39 34 43)(27 44 35 40 31 48)(28 37 32 41 36 45)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 39 13 32)(2 31 14 38)(3 37 15 30)(4 29 16 48)(5 47 17 28)(6 27 18 46)(7 45 19 26)(8 25 20 44)(9 43 21 36)(10 35 22 42)(11 41 23 34)(12 33 24 40)```

`G:=sub<Sym(48)| (1,3,5,7,9,11)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,47,45,43,41,39)(38,40,42,44,46,48), (1,15,5,19,9,23)(2,24,10,20,6,16)(3,17,7,21,11,13)(4,14,12,22,8,18)(25,42,33,38,29,46)(26,47,30,39,34,43)(27,44,35,40,31,48)(28,37,32,41,36,45), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,39,13,32)(2,31,14,38)(3,37,15,30)(4,29,16,48)(5,47,17,28)(6,27,18,46)(7,45,19,26)(8,25,20,44)(9,43,21,36)(10,35,22,42)(11,41,23,34)(12,33,24,40)>;`

`G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,47,45,43,41,39)(38,40,42,44,46,48), (1,15,5,19,9,23)(2,24,10,20,6,16)(3,17,7,21,11,13)(4,14,12,22,8,18)(25,42,33,38,29,46)(26,47,30,39,34,43)(27,44,35,40,31,48)(28,37,32,41,36,45), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,39,13,32)(2,31,14,38)(3,37,15,30)(4,29,16,48)(5,47,17,28)(6,27,18,46)(7,45,19,26)(8,25,20,44)(9,43,21,36)(10,35,22,42)(11,41,23,34)(12,33,24,40) );`

`G=PermutationGroup([[(1,3,5,7,9,11),(2,12,10,8,6,4),(13,15,17,19,21,23),(14,24,22,20,18,16),(25,27,29,31,33,35),(26,36,34,32,30,28),(37,47,45,43,41,39),(38,40,42,44,46,48)], [(1,15,5,19,9,23),(2,24,10,20,6,16),(3,17,7,21,11,13),(4,14,12,22,8,18),(25,42,33,38,29,46),(26,47,30,39,34,43),(27,44,35,40,31,48),(28,37,32,41,36,45)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,39,13,32),(2,31,14,38),(3,37,15,30),(4,29,16,48),(5,47,17,28),(6,27,18,46),(7,45,19,26),(8,25,20,44),(9,43,21,36),(10,35,22,42),(11,41,23,34),(12,33,24,40)]])`

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D ··· 3H 4A 4B 4C 4D 6A ··· 6I 6J ··· 6X 6Y 6Z 6AA 6AB 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 1 1 18 18 2 2 2 4 ··· 4 18 18 18 18 2 ··· 2 4 ··· 4 18 18 18 18 18 ··· 18

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + + + - + - + image C1 C2 C2 C4 S3 S3 D4 Dic3 D6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 C6.D6 D6⋊S3 C3⋊D12 C32⋊4D6 C33⋊9(C2×C4) C33⋊9D4 kernel C62.84D6 C6×C3⋊Dic3 C2×C6×C3⋊S3 C6×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C32×C6 C2×C3⋊S3 C62 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C6 C22 C2 C2 # reps 1 2 1 4 2 1 2 2 3 4 4 8 3 2 1 2 4 2 2 4

Matrix representation of C62.84D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 11 11 0 0 0 0 2 9
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 2 9 0 0 0 0 11 11

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,11,2,0,0,0,0,11,9],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,11,0,0,0,0,9,11] >;`

C62.84D6 in GAP, Magma, Sage, TeX

`C_6^2._{84}D_6`
`% in TeX`

`G:=Group("C6^2.84D6");`
`// GroupNames label`

`G:=SmallGroup(432,461);`
`// by ID`

`G=gap.SmallGroup(432,461);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,1124,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

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