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## G = C36⋊S3order 216 = 23·33

### 1st semidirect product of C36 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C36⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×C9⋊S3 — C36⋊S3
 Lower central C3×C9 — C3×C18 — C36⋊S3
 Upper central C1 — C2 — C4

Generators and relations for C36⋊S3
G = < a,b,c | a36=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 562 in 80 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C9, C32, C12, C12, D6, D9, C18, C3⋊S3, C3×C6, D12, C3×C9, C36, D18, C3×C12, C2×C3⋊S3, C9⋊S3, C3×C18, D36, C12⋊S3, C3×C36, C2×C9⋊S3, C36⋊S3
Quotients: C1, C2, C22, S3, D4, D6, D9, C3⋊S3, D12, D18, C2×C3⋊S3, C9⋊S3, D36, C12⋊S3, C2×C9⋊S3, C36⋊S3

Smallest permutation representation of C36⋊S3
On 108 points
Generators in S108
```(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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)
(1 60 106)(2 61 107)(3 62 108)(4 63 73)(5 64 74)(6 65 75)(7 66 76)(8 67 77)(9 68 78)(10 69 79)(11 70 80)(12 71 81)(13 72 82)(14 37 83)(15 38 84)(16 39 85)(17 40 86)(18 41 87)(19 42 88)(20 43 89)(21 44 90)(22 45 91)(23 46 92)(24 47 93)(25 48 94)(26 49 95)(27 50 96)(28 51 97)(29 52 98)(30 53 99)(31 54 100)(32 55 101)(33 56 102)(34 57 103)(35 58 104)(36 59 105)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 96)(71 95)(72 94)```

`G:=sub<Sym(108)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,60,106)(2,61,107)(3,62,108)(4,63,73)(5,64,74)(6,65,75)(7,66,76)(8,67,77)(9,68,78)(10,69,79)(11,70,80)(12,71,81)(13,72,82)(14,37,83)(15,38,84)(16,39,85)(17,40,86)(18,41,87)(19,42,88)(20,43,89)(21,44,90)(22,45,91)(23,46,92)(24,47,93)(25,48,94)(26,49,95)(27,50,96)(28,51,97)(29,52,98)(30,53,99)(31,54,100)(32,55,101)(33,56,102)(34,57,103)(35,58,104)(36,59,105), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94)>;`

`G:=Group( (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,60,106)(2,61,107)(3,62,108)(4,63,73)(5,64,74)(6,65,75)(7,66,76)(8,67,77)(9,68,78)(10,69,79)(11,70,80)(12,71,81)(13,72,82)(14,37,83)(15,38,84)(16,39,85)(17,40,86)(18,41,87)(19,42,88)(20,43,89)(21,44,90)(22,45,91)(23,46,92)(24,47,93)(25,48,94)(26,49,95)(27,50,96)(28,51,97)(29,52,98)(30,53,99)(31,54,100)(32,55,101)(33,56,102)(34,57,103)(35,58,104)(36,59,105), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94) );`

`G=PermutationGroup([[(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)], [(1,60,106),(2,61,107),(3,62,108),(4,63,73),(5,64,74),(6,65,75),(7,66,76),(8,67,77),(9,68,78),(10,69,79),(11,70,80),(12,71,81),(13,72,82),(14,37,83),(15,38,84),(16,39,85),(17,40,86),(18,41,87),(19,42,88),(20,43,89),(21,44,90),(22,45,91),(23,46,92),(24,47,93),(25,48,94),(26,49,95),(27,50,96),(28,51,97),(29,52,98),(30,53,99),(31,54,100),(32,55,101),(33,56,102),(34,57,103),(35,58,104),(36,59,105)], [(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,108),(59,107),(60,106),(61,105),(62,104),(63,103),(64,102),(65,101),(66,100),(67,99),(68,98),(69,97),(70,96),(71,95),(72,94)]])`

C36⋊S3 is a maximal subgroup of
C6.D36  C3⋊D72  C9⋊D24  C18.D12  C24⋊D9  C721S3  C36.18D6  C36.20D6  Dic65D9  Dic9.D6  S3×D36  D9×D12  C36.70D6  D4×C9⋊S3  C36.29D6
C36⋊S3 is a maximal quotient of
C24.D9  C24⋊D9  C721S3  C36⋊Dic3  C6.11D36

57 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4 6A 6B 6C 6D 9A ··· 9I 12A ··· 12H 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 3 3 3 4 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 54 54 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

57 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 S3 S3 D4 D6 D6 D9 D12 D12 D18 D36 kernel C36⋊S3 C3×C36 C2×C9⋊S3 C36 C3×C12 C3×C9 C18 C3×C6 C12 C9 C32 C6 C3 # reps 1 1 2 3 1 1 3 1 9 6 2 9 18

Matrix representation of C36⋊S3 in GL4(𝔽37) generated by

 6 17 0 0 20 26 0 0 0 0 4 25 0 0 12 29
,
 1 0 0 0 0 1 0 0 0 0 0 36 0 0 1 36
,
 11 17 0 0 6 26 0 0 0 0 36 1 0 0 0 1
`G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,4,12,0,0,25,29],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,36,36],[11,6,0,0,17,26,0,0,0,0,36,0,0,0,1,1] >;`

C36⋊S3 in GAP, Magma, Sage, TeX

`C_{36}\rtimes S_3`
`% in TeX`

`G:=Group("C36:S3");`
`// GroupNames label`

`G:=SmallGroup(216,65);`
`// by ID`

`G=gap.SmallGroup(216,65);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,2115,453,1444,5189]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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