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## G = C34.11S3order 486 = 2·35

### 11st non-split extension by C34 of S3 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C34.11S3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32×3- 1+2 — C34.11S3
 Lower central C32×C9 — C34.11S3
 Upper central C1

Generators and relations for C34.11S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=d, ab=ba, ac=ca, ad=da, eae-1=ad-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=d-1e2 >

Subgroups: 1736 in 294 conjugacy classes, 80 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C33, C33, C9⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32×C9, C32×C9, C3×3- 1+2, C3×3- 1+2, C34, C33.S3, C324D9, C3×C33⋊C2, C32×3- 1+2, C34.11S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C9⋊C6, C3×C3⋊S3, C33⋊C2, C33.S3, C3×C33⋊C2, C34.11S3

Smallest permutation representation of C34.11S3
On 81 points
Generators in S81
```(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 25 22)(20 23 26)(28 31 34)(30 36 33)(38 44 41)(39 42 45)(46 52 49)(47 50 53)(55 58 61)(57 63 60)(64 67 70)(66 72 69)(74 80 77)(75 78 81)
(1 43 10)(2 44 11)(3 45 12)(4 37 13)(5 38 14)(6 39 15)(7 40 16)(8 41 17)(9 42 18)(19 46 66)(20 47 67)(21 48 68)(22 49 69)(23 50 70)(24 51 71)(25 52 72)(26 53 64)(27 54 65)(28 55 81)(29 56 73)(30 57 74)(31 58 75)(32 59 76)(33 60 77)(34 61 78)(35 62 79)(36 63 80)
(1 76 51)(2 77 52)(3 78 53)(4 79 54)(5 80 46)(6 81 47)(7 73 48)(8 74 49)(9 75 50)(10 59 24)(11 60 25)(12 61 26)(13 62 27)(14 63 19)(15 55 20)(16 56 21)(17 57 22)(18 58 23)(28 67 39)(29 68 40)(30 69 41)(31 70 42)(32 71 43)(33 72 44)(34 64 45)(35 65 37)(36 66 38)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)
(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)(73 74 75 76 77 78 79 80 81)
(2 9)(3 8)(4 7)(5 6)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 28)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 66)(56 65)(57 64)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)```

`G:=sub<Sym(81)| (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,31,34)(30,36,33)(38,44,41)(39,42,45)(46,52,49)(47,50,53)(55,58,61)(57,63,60)(64,67,70)(66,72,69)(74,80,77)(75,78,81), (1,43,10)(2,44,11)(3,45,12)(4,37,13)(5,38,14)(6,39,15)(7,40,16)(8,41,17)(9,42,18)(19,46,66)(20,47,67)(21,48,68)(22,49,69)(23,50,70)(24,51,71)(25,52,72)(26,53,64)(27,54,65)(28,55,81)(29,56,73)(30,57,74)(31,58,75)(32,59,76)(33,60,77)(34,61,78)(35,62,79)(36,63,80), (1,76,51)(2,77,52)(3,78,53)(4,79,54)(5,80,46)(6,81,47)(7,73,48)(8,74,49)(9,75,50)(10,59,24)(11,60,25)(12,61,26)(13,62,27)(14,63,19)(15,55,20)(16,56,21)(17,57,22)(18,58,23)(28,67,39)(29,68,40)(30,69,41)(31,70,42)(32,71,43)(33,72,44)(34,64,45)(35,65,37)(36,66,38), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (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)(73,74,75,76,77,78,79,80,81), (2,9)(3,8)(4,7)(5,6)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,28)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,66)(56,65)(57,64)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)>;`

`G:=Group( (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,31,34)(30,36,33)(38,44,41)(39,42,45)(46,52,49)(47,50,53)(55,58,61)(57,63,60)(64,67,70)(66,72,69)(74,80,77)(75,78,81), (1,43,10)(2,44,11)(3,45,12)(4,37,13)(5,38,14)(6,39,15)(7,40,16)(8,41,17)(9,42,18)(19,46,66)(20,47,67)(21,48,68)(22,49,69)(23,50,70)(24,51,71)(25,52,72)(26,53,64)(27,54,65)(28,55,81)(29,56,73)(30,57,74)(31,58,75)(32,59,76)(33,60,77)(34,61,78)(35,62,79)(36,63,80), (1,76,51)(2,77,52)(3,78,53)(4,79,54)(5,80,46)(6,81,47)(7,73,48)(8,74,49)(9,75,50)(10,59,24)(11,60,25)(12,61,26)(13,62,27)(14,63,19)(15,55,20)(16,56,21)(17,57,22)(18,58,23)(28,67,39)(29,68,40)(30,69,41)(31,70,42)(32,71,43)(33,72,44)(34,64,45)(35,65,37)(36,66,38), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (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)(73,74,75,76,77,78,79,80,81), (2,9)(3,8)(4,7)(5,6)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,28)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,66)(56,65)(57,64)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67) );`

`G=PermutationGroup([[(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,25,22),(20,23,26),(28,31,34),(30,36,33),(38,44,41),(39,42,45),(46,52,49),(47,50,53),(55,58,61),(57,63,60),(64,67,70),(66,72,69),(74,80,77),(75,78,81)], [(1,43,10),(2,44,11),(3,45,12),(4,37,13),(5,38,14),(6,39,15),(7,40,16),(8,41,17),(9,42,18),(19,46,66),(20,47,67),(21,48,68),(22,49,69),(23,50,70),(24,51,71),(25,52,72),(26,53,64),(27,54,65),(28,55,81),(29,56,73),(30,57,74),(31,58,75),(32,59,76),(33,60,77),(34,61,78),(35,62,79),(36,63,80)], [(1,76,51),(2,77,52),(3,78,53),(4,79,54),(5,80,46),(6,81,47),(7,73,48),(8,74,49),(9,75,50),(10,59,24),(11,60,25),(12,61,26),(13,62,27),(14,63,19),(15,55,20),(16,56,21),(17,57,22),(18,58,23),(28,67,39),(29,68,40),(30,69,41),(31,70,42),(32,71,43),(33,72,44),(34,64,45),(35,65,37),(36,66,38)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81)], [(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),(73,74,75,76,77,78,79,80,81)], [(2,9),(3,8),(4,7),(5,6),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,45),(18,44),(19,28),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(46,81),(47,80),(48,79),(49,78),(50,77),(51,76),(52,75),(53,74),(54,73),(55,66),(56,65),(57,64),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67)]])`

54 conjugacy classes

 class 1 2 3A ··· 3M 3N 3O 3P ··· 3W 6A 6B 9A ··· 9AA order 1 2 3 ··· 3 3 3 3 ··· 3 6 6 9 ··· 9 size 1 81 2 ··· 2 3 3 6 ··· 6 81 81 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 6 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 C9⋊C6 kernel C34.11S3 C32×3- 1+2 C32⋊4D9 C32×C9 C3×3- 1+2 C34 C3×C9 C33 C32 # reps 1 1 2 2 12 1 24 2 9

Matrix representation of C34.11S3 in GL10(𝔽19)

 11 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 18 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 17 17 0 0 18 18
,
 0 1 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 1 0 18 18 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 17 0 0 0 18 18 0 0 0 0 0 2 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 0 0 1 18 18 0 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 17 0 0 18 18
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 17 17 0 0 18 17 0 0 0 0 0 0 0 0 1 18 0 0 0 0 18 18 0 0 0 18 0 0 0 0 0 18 0 0 0 18 0 0 0 0 1 1 1 0 0 2 0 0 0 0 1 1 0 1 0 2
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 2 1 0 0 1 1 0 0 0 0 0 18 0 0 0 18 0 0 0 0 17 0 1 1 0 0 0 0 0 0 18 1 0 18 0 0

`G:=sub<GL(10,GF(19))| [11,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,17,0,0,0,0,0,1,1,0,0,17,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18],[0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,1,1,0,17,0,0,0,0,0,18,18,0,18,0,2,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,18,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,18,18,0,2,0,0,0,0,0,1,0,0,1,0,17,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,0,18,0,1,1,0,0,0,0,17,0,18,18,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,18,1,0,0,0,0,0,0,0,0,17,18,18,18,2,2],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,1,1,2,0,17,18,0,0,0,0,0,18,1,18,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,18,0,0] >;`

C34.11S3 in GAP, Magma, Sage, TeX

`C_3^4._{11}S_3`
`% in TeX`

`G:=Group("C3^4.11S3");`
`// GroupNames label`

`G:=SmallGroup(486,244);`
`// by ID`

`G=gap.SmallGroup(486,244);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,1520,986,867,3244,11669]);`
`// Polycyclic`

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

׿
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