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G = C18.D4order 144 = 24·32

7th non-split extension by C18 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C18.D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C18.D4
 Lower central C9 — C18 — C18.D4
 Upper central C1 — C22 — C23

Generators and relations for C18.D4
G = < a,b,c | a18=b4=1, c2=a9, bab-1=cac-1=a-1, cbc-1=a9b-1 >

Smallest permutation representation of C18.D4
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 23 42 57)(2 22 43 56)(3 21 44 55)(4 20 45 72)(5 19 46 71)(6 36 47 70)(7 35 48 69)(8 34 49 68)(9 33 50 67)(10 32 51 66)(11 31 52 65)(12 30 53 64)(13 29 54 63)(14 28 37 62)(15 27 38 61)(16 26 39 60)(17 25 40 59)(18 24 41 58)
(1 66 10 57)(2 65 11 56)(3 64 12 55)(4 63 13 72)(5 62 14 71)(6 61 15 70)(7 60 16 69)(8 59 17 68)(9 58 18 67)(19 46 28 37)(20 45 29 54)(21 44 30 53)(22 43 31 52)(23 42 32 51)(24 41 33 50)(25 40 34 49)(26 39 35 48)(27 38 36 47)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A ··· 6G 9A 9B 9C 18A ··· 18U order 1 2 2 2 2 2 3 4 4 4 4 6 ··· 6 9 9 9 18 ··· 18 size 1 1 1 1 2 2 2 18 18 18 18 2 ··· 2 2 2 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + image C1 C2 C2 C4 S3 D4 Dic3 D6 D9 C3⋊D4 Dic9 D18 C9⋊D4 kernel C18.D4 C2×Dic9 C22×C18 C2×C18 C22×C6 C18 C2×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 4 1 2 2 1 3 4 6 3 12

Matrix representation of C18.D4 in GL3(𝔽37) generated by

 36 0 0 0 3 0 0 0 25
,
 31 0 0 0 0 1 0 1 0
,
 6 0 0 0 0 1 0 36 0
`G:=sub<GL(3,GF(37))| [36,0,0,0,3,0,0,0,25],[31,0,0,0,0,1,0,1,0],[6,0,0,0,0,36,0,1,0] >;`

C18.D4 in GAP, Magma, Sage, TeX

`C_{18}.D_4`
`% in TeX`

`G:=Group("C18.D4");`
`// GroupNames label`

`G:=SmallGroup(144,19);`
`// by ID`

`G=gap.SmallGroup(144,19);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,2404,208,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^18=b^4=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

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