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## G = C62.27D6order 432 = 24·33

### 10th non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C62.27D6
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×3- 1+2 — C2×C9⋊C12 — C62.27D6
 Lower central C9 — C18 — C62.27D6
 Upper central C1 — C22 — C23

Generators and relations for C62.27D6
G = < a,b,c,d | a6=b6=1, c6=b4, d2=a3b3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >

Subgroups: 342 in 118 conjugacy classes, 46 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, 3- 1+2, Dic9, C2×C18, C2×C18, C2×C18, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C9⋊C12, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, C18.D4, C3×C6.D4, C2×C9⋊C12, C23×3- 1+2, C62.27D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C9⋊C6, C6×Dic3, C3×C3⋊D4, C9⋊C12, C2×C9⋊C6, C3×C6.D4, C2×C9⋊C12, Dic9⋊C6, C62.27D6

Smallest permutation representation of C62.27D6
On 72 points
Generators in S72
```(1 10)(2 5 8 11 14 17)(3 18 15 12 9 6)(4 13)(7 16)(19 70 31 64 25 58)(20 65)(21 60 27 66 33 72)(22 55 34 67 28 61)(23 68)(24 63 30 69 36 57)(26 71)(29 56)(32 59)(35 62)(37 52 49 46 43 40)(38 47)(39 42 45 48 51 54)(41 50)(44 53)
(1 47 13 41 7 53)(2 48 14 42 8 54)(3 49 15 43 9 37)(4 50 16 44 10 38)(5 51 17 45 11 39)(6 52 18 46 12 40)(19 61 31 55 25 67)(20 62 32 56 26 68)(21 63 33 57 27 69)(22 64 34 58 28 70)(23 65 35 59 29 71)(24 66 36 60 30 72)
(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 25 50 34)(2 60 51 69)(3 23 52 32)(4 58 53 67)(5 21 54 30)(6 56 37 65)(7 19 38 28)(8 72 39 63)(9 35 40 26)(10 70 41 61)(11 33 42 24)(12 68 43 59)(13 31 44 22)(14 66 45 57)(15 29 46 20)(16 64 47 55)(17 27 48 36)(18 62 49 71)```

`G:=sub<Sym(72)| (1,10)(2,5,8,11,14,17)(3,18,15,12,9,6)(4,13)(7,16)(19,70,31,64,25,58)(20,65)(21,60,27,66,33,72)(22,55,34,67,28,61)(23,68)(24,63,30,69,36,57)(26,71)(29,56)(32,59)(35,62)(37,52,49,46,43,40)(38,47)(39,42,45,48,51,54)(41,50)(44,53), (1,47,13,41,7,53)(2,48,14,42,8,54)(3,49,15,43,9,37)(4,50,16,44,10,38)(5,51,17,45,11,39)(6,52,18,46,12,40)(19,61,31,55,25,67)(20,62,32,56,26,68)(21,63,33,57,27,69)(22,64,34,58,28,70)(23,65,35,59,29,71)(24,66,36,60,30,72), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,25,50,34)(2,60,51,69)(3,23,52,32)(4,58,53,67)(5,21,54,30)(6,56,37,65)(7,19,38,28)(8,72,39,63)(9,35,40,26)(10,70,41,61)(11,33,42,24)(12,68,43,59)(13,31,44,22)(14,66,45,57)(15,29,46,20)(16,64,47,55)(17,27,48,36)(18,62,49,71)>;`

`G:=Group( (1,10)(2,5,8,11,14,17)(3,18,15,12,9,6)(4,13)(7,16)(19,70,31,64,25,58)(20,65)(21,60,27,66,33,72)(22,55,34,67,28,61)(23,68)(24,63,30,69,36,57)(26,71)(29,56)(32,59)(35,62)(37,52,49,46,43,40)(38,47)(39,42,45,48,51,54)(41,50)(44,53), (1,47,13,41,7,53)(2,48,14,42,8,54)(3,49,15,43,9,37)(4,50,16,44,10,38)(5,51,17,45,11,39)(6,52,18,46,12,40)(19,61,31,55,25,67)(20,62,32,56,26,68)(21,63,33,57,27,69)(22,64,34,58,28,70)(23,65,35,59,29,71)(24,66,36,60,30,72), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,25,50,34)(2,60,51,69)(3,23,52,32)(4,58,53,67)(5,21,54,30)(6,56,37,65)(7,19,38,28)(8,72,39,63)(9,35,40,26)(10,70,41,61)(11,33,42,24)(12,68,43,59)(13,31,44,22)(14,66,45,57)(15,29,46,20)(16,64,47,55)(17,27,48,36)(18,62,49,71) );`

`G=PermutationGroup([[(1,10),(2,5,8,11,14,17),(3,18,15,12,9,6),(4,13),(7,16),(19,70,31,64,25,58),(20,65),(21,60,27,66,33,72),(22,55,34,67,28,61),(23,68),(24,63,30,69,36,57),(26,71),(29,56),(32,59),(35,62),(37,52,49,46,43,40),(38,47),(39,42,45,48,51,54),(41,50),(44,53)], [(1,47,13,41,7,53),(2,48,14,42,8,54),(3,49,15,43,9,37),(4,50,16,44,10,38),(5,51,17,45,11,39),(6,52,18,46,12,40),(19,61,31,55,25,67),(20,62,32,56,26,68),(21,63,33,57,27,69),(22,64,34,58,28,70),(23,65,35,59,29,71),(24,66,36,60,30,72)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,25,50,34),(2,60,51,69),(3,23,52,32),(4,58,53,67),(5,21,54,30),(6,56,37,65),(7,19,38,28),(8,72,39,63),(9,35,40,26),(10,70,41,61),(11,33,42,24),(12,68,43,59),(13,31,44,22),(14,66,45,57),(15,29,46,20),(16,64,47,55),(17,27,48,36),(18,62,49,71)]])`

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A ··· 6G 6H ··· 6M 6N 6O 6P 6Q 9A 9B 9C 12A ··· 12H 18A ··· 18U order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 9 9 9 12 ··· 12 18 ··· 18 size 1 1 1 1 2 2 2 3 3 18 18 18 18 2 ··· 2 3 ··· 3 6 6 6 6 6 6 6 18 ··· 18 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 C3×S3 C3×D4 C3⋊D4 C3×Dic3 S3×C6 C3×C3⋊D4 C9⋊C6 C9⋊C12 C2×C9⋊C6 Dic9⋊C6 kernel C62.27D6 C2×C9⋊C12 C23×3- 1+2 C18.D4 C22×3- 1+2 C2×Dic9 C22×C18 C2×C18 C2×C62 C2×3- 1+2 C62 C62 C22×C6 C18 C3×C6 C2×C6 C2×C6 C6 C23 C22 C22 C2 # reps 1 2 1 2 4 4 2 8 1 2 2 1 2 4 4 4 2 8 1 2 1 4

Matrix representation of C62.27D6 in GL8(𝔽37)

 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 36 26 36 0 8 31 0 0 0 11 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 27
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 27 0 0 14 14 14 0 0 0 27 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11
,
 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 36 0 26 32 26 26 0 0 28 1 1 6 6 6 0 0 0 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 10 0 0
,
 6 0 0 0 0 0 0 0 31 31 0 0 0 0 0 0 0 0 14 0 6 7 33 33 0 0 0 0 0 0 0 11 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 21 10 10 23 23 23 0 0 0 27 0 0 0 0

`G:=sub<GL(8,GF(37))| [11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,26,11,0,0,0,0,0,0,36,0,27,0,0,0,0,0,0,0,0,36,0,0,0,0,8,0,0,0,11,0,0,0,31,0,0,0,0,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,14,0,0,11,0,0,0,0,14,0,0,0,11,0,0,0,14,0,0,0,0,11],[36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,28,0,0,0,0,0,0,0,1,36,0,0,0,0,0,26,1,0,0,0,0,0,0,32,6,0,0,0,10,0,0,26,6,0,1,0,0,0,0,26,6,0,0,1,0],[6,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,14,0,0,0,21,0,0,0,0,0,0,0,10,27,0,0,6,0,0,36,10,0,0,0,7,0,36,0,23,0,0,0,33,0,0,0,23,0,0,0,33,11,0,0,23,0] >;`

C62.27D6 in GAP, Magma, Sage, TeX

`C_6^2._{27}D_6`
`% in TeX`

`G:=Group("C6^2.27D6");`
`// GroupNames label`

`G:=SmallGroup(432,167);`
`// by ID`

`G=gap.SmallGroup(432,167);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,10085,2035,292,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^4,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^5>;`
`// generators/relations`

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