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## G = C62.31D6order 432 = 24·33

### 14th 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 — C62.31D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C22×He3⋊C2 — C62.31D6
 Lower central He3 — C2×He3 — C62.31D6
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C62.31D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=ab4, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b3c5 >

Subgroups: 797 in 187 conjugacy classes, 47 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×4], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×3], C6 [×14], C2×C4, C2×C4, C23, C32 [×4], Dic3 [×4], C12 [×6], D6 [×16], C2×C6, C2×C6 [×8], C22⋊C4, C3×S3 [×8], C3×C6 [×12], C2×Dic3 [×4], C2×C12, C2×C12 [×5], C22×S3 [×4], C22×C6, He3, C3×Dic3 [×4], C3×C12 [×4], S3×C6 [×16], C62 [×4], D6⋊C4 [×4], C3×C22⋊C4, He3⋊C2 [×2], C2×He3 [×3], C6×Dic3 [×4], C6×C12 [×4], S3×C2×C6 [×4], He33C4, C4×He3, C2×He3⋊C2 [×2], C2×He3⋊C2 [×2], C22×He3, C3×D6⋊C4 [×4], C2×He33C4, C2×C4×He3, C22×He3⋊C2, C62.31D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C3⋊S3, D6⋊C4 [×4], He3⋊C2, C4×C3⋊S3, C12⋊S3, C327D4, C2×He3⋊C2, C6.11D12, C4×He3⋊C2, He35D4, He37D4, C62.31D6

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 type + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 C4×S3 D12 C3⋊D4 He3⋊C2 C2×He3⋊C2 C4×He3⋊C2 He3⋊5D4 He3⋊7D4 kernel C62.31D6 C2×He3⋊3C4 C2×C4×He3 C22×He3⋊C2 C2×He3⋊C2 C6×C12 C2×He3 C62 C3×C6 C3×C6 C3×C6 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 4 4 2 4 8 8 8 4 4 8 2 2

Matrix representation of C62.31D6 in GL7(𝔽13)

 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 3
,
 2 4 0 0 0 0 0 9 11 0 0 0 0 0 0 0 3 3 0 0 0 0 0 10 6 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0
,
 11 9 0 0 0 0 0 11 2 0 0 0 0 0 0 0 10 6 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0

`G:=sub<GL(7,GF(13))| [0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[2,9,0,0,0,0,0,4,11,0,0,0,0,0,0,0,3,10,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0],[11,11,0,0,0,0,0,9,2,0,0,0,0,0,0,0,10,3,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0] >;`

C62.31D6 in GAP, Magma, Sage, TeX

`C_6^2._{31}D_6`
`% in TeX`

`G:=Group("C6^2.31D6");`
`// GroupNames label`

`G:=SmallGroup(432,189);`
`// by ID`

`G=gap.SmallGroup(432,189);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,1124,4037,537]);`
`// 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*b^4,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

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