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## G = C62.47D6order 432 = 24·33

### 30th non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C62.47D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C4×He3⋊C2 — C62.47D6
 Lower central He3 — C2×He3 — C62.47D6
 Upper central C1 — C12 — C2×C12

Generators and relations for C62.47D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab2, dad-1=a-1b, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 797 in 220 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, He3, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, He3⋊C2, C2×He3, C2×He3, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, He33C4, C4×He3, C2×He3⋊C2, C22×He3, C3×C4○D12, He34Q8, C4×He3⋊C2, He35D4, He37D4, C2×C4×He3, C62.47D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, C4○D12, He3⋊C2, C22×C3⋊S3, C2×He3⋊C2, C12.59D6, C22×He3⋊C2, C62.47D6

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E ··· 6P 6Q 6R 6S 6T 12A 12B 12C 12D 12E 12F 12G ··· 12V 12W 12X 12Y 12Z order 1 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 2 18 18 1 1 6 6 6 6 1 1 2 18 18 1 1 2 2 6 ··· 6 18 18 18 18 1 1 1 1 2 2 6 ··· 6 18 18 18 18

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 6 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 He3⋊C2 C2×He3⋊C2 C2×He3⋊C2 C62.47D6 kernel C62.47D6 He3⋊4Q8 C4×He3⋊C2 He3⋊5D4 He3⋊7D4 C2×C4×He3 C6×C12 C3×C12 C62 He3 C32 C2×C4 C4 C22 C1 # reps 1 1 2 1 2 1 4 8 4 2 16 4 8 4 4

Matrix representation of C62.47D6 in GL5(𝔽13)

 9 2 0 0 0 11 11 0 0 0 0 0 12 12 0 0 0 1 0 0 0 0 8 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 0 5 0 0 0 8 5 0 0 0 0 0 4 0 12 0 0 0 10 10 0 0 0 0 12
,
 11 11 0 0 0 9 2 0 0 0 0 0 0 3 3 0 0 9 0 1 0 0 0 0 12

`G:=sub<GL(5,GF(13))| [9,11,0,0,0,2,11,0,0,0,0,0,12,1,8,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[0,8,0,0,0,5,5,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,12,10,12],[11,9,0,0,0,11,2,0,0,0,0,0,0,9,0,0,0,3,0,0,0,0,3,1,12] >;`

C62.47D6 in GAP, Magma, Sage, TeX

`C_6^2._{47}D_6`
`% in TeX`

`G:=Group("C6^2.47D6");`
`// GroupNames label`

`G:=SmallGroup(432,387);`
`// by ID`

`G=gap.SmallGroup(432,387);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,1124,4037,537]);`
`// Polycyclic`

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

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