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## G = C62.85D6order 432 = 24·33

### 33rd non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.85D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — C6×C3⋊Dic3 — C62.85D6
 Lower central C33 — C32×C6 — C62.85D6
 Upper central C1 — C22

Generators and relations for C62.85D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 600 in 162 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C32×C6, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, Dic3⋊Dic3, C62.C22, C6×C3⋊Dic3, C6×C3⋊Dic3, C62.85D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C324D6, Dic3⋊Dic3, C62.C22, C339(C2×C4), C339D4, C335Q8, C62.85D6

Smallest permutation representation of C62.85D6
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 39 41 43 45 47)(38 48 46 44 42 40)
(1 13 5 17 9 21)(2 22 10 18 6 14)(3 15 7 19 11 23)(4 24 12 20 8 16)(25 43 33 39 29 47)(26 48 30 40 34 44)(27 45 35 41 31 37)(28 38 32 42 36 46)
(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 44 7 38)(2 35 8 29)(3 42 9 48)(4 33 10 27)(5 40 11 46)(6 31 12 25)(13 34 19 28)(14 41 20 47)(15 32 21 26)(16 39 22 45)(17 30 23 36)(18 37 24 43)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D ··· 3H 4A ··· 4F 6A ··· 6I 6J ··· 6X 12A ··· 12L order 1 2 2 2 3 3 3 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 4 ··· 4 18 ··· 18 2 ··· 2 4 ··· 4 18 ··· 18

54 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + - - + - + + - + - + - image C1 C2 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 S32 S3×Dic3 C6.D6 D6⋊S3 C3⋊D12 C32⋊2Q8 C32⋊4D6 C33⋊9(C2×C4) C33⋊9D4 C33⋊5Q8 kernel C62.85D6 C6×C3⋊Dic3 C3×C3⋊Dic3 C2×C3⋊Dic3 C32×C6 C32×C6 C3⋊Dic3 C62 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 3 4 3 1 1 2 3 6 4 2 4 3 2 1 1 2 3 2 2 2 2

Matrix representation of C62.85D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 5 0 0 0 0 5 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 0 8 0 0 0 0 8 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))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,11,0,0,0,0,9,11] >;`

C62.85D6 in GAP, Magma, Sage, TeX

`C_6^2._{85}D_6`
`% in TeX`

`G:=Group("C6^2.85D6");`
`// GroupNames label`

`G:=SmallGroup(432,462);`
`// by ID`

`G=gap.SmallGroup(432,462);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,1124,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=a^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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