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## G = C18.A4order 216 = 23·33

### The non-split extension by C18 of A4 acting via A4/C22=C3

Aliases: C18.A4, C9⋊SL2(𝔽3), Q813- 1+2, C2.(C9⋊A4), Q8⋊C91C3, C6.9(C3×A4), (Q8×C9)⋊2C3, (C3×Q8).2C32, (C3×SL2(𝔽3)).C3, C3.3(C3×SL2(𝔽3)), SmallGroup(216,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C18.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C18.A4
 Lower central Q8 — C3×Q8 — C18.A4
 Upper central C1 — C6 — C18

Generators and relations for C18.A4
G = < a,b,c,d | a18=d3=1, b2=c2=a9, ab=ba, ac=ca, dad-1=a7, cbc-1=a9b, dbd-1=a9bc, dcd-1=b >

Smallest permutation representation of C18.A4
On 72 points
Generators in S72
```(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 29 10 20)(2 30 11 21)(3 31 12 22)(4 32 13 23)(5 33 14 24)(6 34 15 25)(7 35 16 26)(8 36 17 27)(9 19 18 28)(37 65 46 56)(38 66 47 57)(39 67 48 58)(40 68 49 59)(41 69 50 60)(42 70 51 61)(43 71 52 62)(44 72 53 63)(45 55 54 64)
(1 56 10 65)(2 57 11 66)(3 58 12 67)(4 59 13 68)(5 60 14 69)(6 61 15 70)(7 62 16 71)(8 63 17 72)(9 64 18 55)(19 54 28 45)(20 37 29 46)(21 38 30 47)(22 39 31 48)(23 40 32 49)(24 41 33 50)(25 42 34 51)(26 43 35 52)(27 44 36 53)
(2 14 8)(3 9 15)(5 17 11)(6 12 18)(19 70 39)(20 65 46)(21 60 53)(22 55 42)(23 68 49)(24 63 38)(25 58 45)(26 71 52)(27 66 41)(28 61 48)(29 56 37)(30 69 44)(31 64 51)(32 59 40)(33 72 47)(34 67 54)(35 62 43)(36 57 50)```

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

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

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

C18.A4 is a maximal subgroup of   Dic9.A4  D18.A4  C36.A4

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 4 6A 6B 6C 6D 9A 9B 9C 9D 9E 9F 12A 12B 18A 18B 18C 18D 18E 18F 36A ··· 36F order 1 2 3 3 3 3 4 6 6 6 6 9 9 9 9 9 9 12 12 18 18 18 18 18 18 36 ··· 36 size 1 1 1 1 12 12 6 1 1 12 12 3 3 12 12 12 12 6 6 3 3 12 12 12 12 6 ··· 6

31 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 3 3 6 type + - + image C1 C3 C3 C3 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 3- 1+2 C3×A4 C9⋊A4 C18.A4 kernel C18.A4 Q8⋊C9 Q8×C9 C3×SL2(𝔽3) C9 C9 C3 C18 Q8 C6 C2 C1 # reps 1 4 2 2 1 2 6 1 2 2 6 2

Matrix representation of C18.A4 in GL5(𝔽37)

 36 0 0 0 0 0 36 0 0 0 0 0 10 0 4 0 0 3 27 27 0 0 0 10 0
,
 27 26 0 0 0 26 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 1 0 0 0 27 0 0 0 0 0 0 1 30 34 0 0 0 10 0 0 0 0 0 26

`G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,10,3,0,0,0,0,27,10,0,0,4,27,0],[27,26,0,0,0,26,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,27,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,30,10,0,0,0,34,0,26] >;`

C18.A4 in GAP, Magma, Sage, TeX

`C_{18}.A_4`
`% in TeX`

`G:=Group("C18.A4");`
`// GroupNames label`

`G:=SmallGroup(216,39);`
`// by ID`

`G=gap.SmallGroup(216,39);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-2,2,-2,145,43,1299,117,2434,202,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^18=d^3=1,b^2=c^2=a^9,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,c*b*c^-1=a^9*b,d*b*d^-1=a^9*b*c,d*c*d^-1=b>;`
`// generators/relations`

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