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## G = C18.S4order 432 = 24·33

### 3rd non-split extension by C18 of S4 acting via S4/A4=C2

Aliases: C18.3S4, C23.D27, C22⋊Dic27, C9.A4⋊C4, C9.(A4⋊C4), (C2×C6).Dic9, (C2×C18).Dic3, C2.1(C9.S4), C3.(C6.S4), C6.3(C3.S4), (C22×C6).2D9, (C22×C18).2S3, (C2×C9.A4).C2, SmallGroup(432,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9.A4 — C18.S4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9.A4 — C2×C9.A4 — C18.S4
 Lower central C9.A4 — C18.S4
 Upper central C1 — C2

Generators and relations for C18.S4
G = < a,b,c,d,e | a18=b2=c2=1, d3=a2, e2=a9, ab=ba, ac=ca, ad=da, eae-1=a-1, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=a16d2 >

Smallest permutation representation of C18.S4
On 108 points
Generators in S108
```(1 77 4 80 7 56 10 59 13 62 16 65 19 68 22 71 25 74)(2 78 5 81 8 57 11 60 14 63 17 66 20 69 23 72 26 75)(3 79 6 55 9 58 12 61 15 64 18 67 21 70 24 73 27 76)(28 92 31 95 34 98 37 101 40 104 43 107 46 83 49 86 52 89)(29 93 32 96 35 99 38 102 41 105 44 108 47 84 50 87 53 90)(30 94 33 97 36 100 39 103 42 106 45 82 48 85 51 88 54 91)
(2 63)(3 64)(5 66)(6 67)(8 69)(9 70)(11 72)(12 73)(14 75)(15 76)(17 78)(18 79)(20 81)(21 55)(23 57)(24 58)(26 60)(27 61)(28 104)(29 105)(31 107)(32 108)(34 83)(35 84)(37 86)(38 87)(40 89)(41 90)(43 92)(44 93)(46 95)(47 96)(49 98)(50 99)(52 101)(53 102)
(1 62)(3 64)(4 65)(6 67)(7 68)(9 70)(10 71)(12 73)(13 74)(15 76)(16 77)(18 79)(19 80)(21 55)(22 56)(24 58)(25 59)(27 61)(29 105)(30 106)(32 108)(33 82)(35 84)(36 85)(38 87)(39 88)(41 90)(42 91)(44 93)(45 94)(47 96)(48 97)(50 99)(51 100)(53 102)(54 103)
(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 90 62 41)(2 89 63 40)(3 88 64 39)(4 87 65 38)(5 86 66 37)(6 85 67 36)(7 84 68 35)(8 83 69 34)(9 82 70 33)(10 108 71 32)(11 107 72 31)(12 106 73 30)(13 105 74 29)(14 104 75 28)(15 103 76 54)(16 102 77 53)(17 101 78 52)(18 100 79 51)(19 99 80 50)(20 98 81 49)(21 97 55 48)(22 96 56 47)(23 95 57 46)(24 94 58 45)(25 93 59 44)(26 92 60 43)(27 91 61 42)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 9A 9B 9C 18A 18B 18C 18D ··· 18I 27A ··· 27I 54A ··· 54I order 1 2 2 2 3 4 4 4 4 6 6 6 9 9 9 18 18 18 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 3 3 2 54 54 54 54 2 6 6 2 2 2 2 2 2 6 ··· 6 8 ··· 8 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 6 6 type + + + - + - + - + + - + - image C1 C2 C4 S3 Dic3 D9 Dic9 D27 Dic27 S4 A4⋊C4 C3.S4 C6.S4 C9.S4 C18.S4 kernel C18.S4 C2×C9.A4 C9.A4 C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 C18 C9 C6 C3 C2 C1 # reps 1 1 2 1 1 3 3 9 9 2 2 1 1 3 3

Matrix representation of C18.S4 in GL5(𝔽109)

 82 59 0 0 0 50 32 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 108 0 0 0 0 0 108 0 0 0 1 108 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 108 0 0 0 0 0 1 0 0 0 0 1 108
,
 30 16 0 0 0 93 46 0 0 0 0 0 0 108 0 0 0 1 108 2 0 0 0 0 1
,
 15 19 0 0 0 34 94 0 0 0 0 0 108 1 107 0 0 0 1 0 0 0 0 0 1

`G:=sub<GL(5,GF(109))| [82,50,0,0,0,59,32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,1,0,0,0,108,108,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,0,0,0,0,1,1,0,0,0,0,108],[30,93,0,0,0,16,46,0,0,0,0,0,0,1,0,0,0,108,108,0,0,0,0,2,1],[15,34,0,0,0,19,94,0,0,0,0,0,108,0,0,0,0,1,1,0,0,0,107,0,1] >;`

C18.S4 in GAP, Magma, Sage, TeX

`C_{18}.S_4`
`% in TeX`

`G:=Group("C18.S4");`
`// GroupNames label`

`G:=SmallGroup(432,39);`
`// by ID`

`G=gap.SmallGroup(432,39);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,422,331,1683,192,2524,9077,2287,5298,3989]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^18=b^2=c^2=1,d^3=a^2,e^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=a^16*d^2>;`
`// generators/relations`

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