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## G = C62.21D6order 432 = 24·33

### 4th non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.21D6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×He3 — C22×C32⋊C6 — C62.21D6
 Lower central C32 — C3×C6 — C62.21D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.21D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 677 in 135 conjugacy classes, 40 normal (36 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×3], C4 [×2], C22, C22 [×4], S3 [×4], C6 [×3], C6 [×11], C2×C4, C2×C4, C23, C32 [×2], C32, Dic3 [×2], C12 [×6], D6 [×8], C2×C6, C2×C6 [×7], C22⋊C4, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×6], C3×C6 [×3], C2×Dic3 [×2], C2×C12, C2×C12 [×4], C22×S3 [×2], C22×C6, He3, C3×Dic3, C3⋊Dic3, C3×C12 [×4], S3×C6 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62 [×2], C62, D6⋊C4 [×2], C3×C22⋊C4, C32⋊C6 [×2], C2×He3 [×3], C6×Dic3, C2×C3⋊Dic3, C6×C12 [×2], C6×C12, S3×C2×C6, C22×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6 [×2], C2×C32⋊C6 [×2], C22×He3, C3×D6⋊C4, C6.11D12, C2×C32⋊C12, C2×C4×He3, C22×C32⋊C6, C62.21D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], S3×C6, D6⋊C4, C3×C22⋊C4, C32⋊C6, S3×C12, C3×D12, C3×C3⋊D4, C2×C32⋊C6, C3×D6⋊C4, C4×C32⋊C6, He34D4, He36D4, C62.21D6

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B 6C 6D ··· 6I 6J ··· 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 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 1 1 18 18 2 3 3 6 6 6 2 2 18 18 2 2 2 3 ··· 3 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 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 C32⋊C6 C2×C32⋊C6 C4×C32⋊C6 He3⋊4D4 He3⋊6D4 kernel C62.21D6 C2×C32⋊C12 C2×C4×He3 C22×C32⋊C6 C6.11D12 C2×C32⋊C6 C2×C3⋊Dic3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C6×C12 C2×He3 C62 C2×C12 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 2 2 2 2 4 2 4 4 4 1 1 2 2 2

Matrix representation of C62.21D6 in GL8(𝔽13)

 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1
,
 1 11 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 9 2 0 0 11 4 0 0 0 0 0 0 9 2 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 9 2 0 0
,
 12 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 11 0 0 0 0 0 0 2 9 0 0 0 0 4 11 0 0 0 0 0 0 2 9 0 0 0 0 4 11 0 0 0 0 0 0 2 9 0 0 0 0

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

C62.21D6 in GAP, Magma, Sage, TeX

`C_6^2._{21}D_6`
`% in TeX`

`G:=Group("C6^2.21D6");`
`// GroupNames label`

`G:=SmallGroup(432,141);`
`// by ID`

`G=gap.SmallGroup(432,141);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,4037,2035,14118]);`
`// 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=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

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