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## G = C62.79D6order 432 = 24·33

### 27th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.79D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — Dic3×C3×C6 — C62.79D6
 Lower central C33 — C32×C6 — C62.79D6
 Upper central C1 — C22

Generators and relations for C62.79D6
G = < a,b,c,d | a6=b6=d2=1, c6=a3, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=b-1, dcd=b3c5 >

Subgroups: 2456 in 332 conjugacy classes, 72 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×26], C6 [×3], C6 [×12], C6 [×12], C2×C4 [×2], C23, C32, C32 [×4], C32 [×4], Dic3 [×5], C12 [×5], D6 [×44], C2×C6, C2×C6 [×4], C2×C6 [×4], C22⋊C4, C3⋊S3 [×26], C3×C6 [×3], C3×C6 [×12], C3×C6 [×12], C2×Dic3, C2×Dic3 [×4], C2×C12 [×5], C22×S3 [×9], C33, C3×Dic3 [×8], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×44], C62, C62 [×4], C62 [×4], D6⋊C4 [×5], C33⋊C2 [×2], C32×C6 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3 [×9], C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2 [×2], C2×C33⋊C2 [×2], C3×C62, C6.D12 [×4], C6.11D12, Dic3×C3×C6, C6×C3⋊Dic3, C22×C33⋊C2, C62.79D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D4 [×2], D6 [×5], C22⋊C4, C3⋊S3, C4×S3 [×5], D12 [×5], C3⋊D4 [×5], S32 [×4], C2×C3⋊S3, D6⋊C4 [×5], C6.D6 [×4], C3⋊D12 [×8], C4×C3⋊S3, C12⋊S3, C327D4, S3×C3⋊S3, C6.D12 [×4], C6.11D12, C338(C2×C4), C337D4, C338D4, C62.79D6

Smallest permutation representation of C62.79D6
On 72 points
Generators in S72
```(1 17 50 7 23 56)(2 18 51 8 24 57)(3 19 52 9 13 58)(4 20 53 10 14 59)(5 21 54 11 15 60)(6 22 55 12 16 49)(25 40 71 31 46 65)(26 41 72 32 47 66)(27 42 61 33 48 67)(28 43 62 34 37 68)(29 44 63 35 38 69)(30 45 64 36 39 70)
(1 30 9 26 5 34)(2 35 6 27 10 31)(3 32 11 28 7 36)(4 25 8 29 12 33)(13 41 21 37 17 45)(14 46 18 38 22 42)(15 43 23 39 19 47)(16 48 20 40 24 44)(49 67 53 71 57 63)(50 64 58 72 54 68)(51 69 55 61 59 65)(52 66 60 62 56 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 28)(2 8)(3 26)(4 6)(5 36)(7 34)(9 32)(10 12)(11 30)(13 72)(14 55)(15 70)(16 53)(17 68)(18 51)(19 66)(20 49)(21 64)(22 59)(23 62)(24 57)(25 35)(27 33)(29 31)(37 50)(38 65)(39 60)(40 63)(41 58)(42 61)(43 56)(44 71)(45 54)(46 69)(47 52)(48 67)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6O 6P ··· 6AA 12A ··· 12P 12Q 12R 12S 12T order 1 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 54 54 2 ··· 2 4 4 4 4 6 6 18 18 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 S3 D4 D6 C4×S3 D12 C3⋊D4 S32 C6.D6 C3⋊D12 kernel C62.79D6 Dic3×C3×C6 C6×C3⋊Dic3 C22×C33⋊C2 C2×C33⋊C2 C6×Dic3 C2×C3⋊Dic3 C32×C6 C62 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 1 1 1 4 4 1 2 5 10 10 10 4 4 8

Matrix representation of C62.79D6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 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 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 1 0 0 0 0 0 0 1 0 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 0 8 0 0 0 0 0 0 5 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 12 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 12 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C62.79D6 in GAP, Magma, Sage, TeX

`C_6^2._{79}D_6`
`% in TeX`

`G:=Group("C6^2.79D6");`
`// GroupNames label`

`G:=SmallGroup(432,451);`
`// by ID`

`G=gap.SmallGroup(432,451);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,92,571,2028,14118]);`
`// Polycyclic`

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

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