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## G = D18⋊C4order 144 = 24·32

### The semidirect product of D18 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D18⋊C4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — D18⋊C4
 Lower central C9 — C18 — D18⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for D18⋊C4
G = < a,b,c | a18=b2=c4=1, bab=a-1, ac=ca, cbc-1=a9b >

Smallest permutation representation of D18⋊C4
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 36)(34 35)(37 41)(38 40)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(55 59)(56 58)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
(1 62 26 44)(2 63 27 45)(3 64 28 46)(4 65 29 47)(5 66 30 48)(6 67 31 49)(7 68 32 50)(8 69 33 51)(9 70 34 52)(10 71 35 53)(11 72 36 54)(12 55 19 37)(13 56 20 38)(14 57 21 39)(15 58 22 40)(16 59 23 41)(17 60 24 42)(18 61 25 43)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 D9 C4×S3 D12 C3⋊D4 D18 C4×D9 D36 C9⋊D4 kernel D18⋊C4 C2×Dic9 C2×C36 C22×D9 D18 C2×C12 C18 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 1 3 2 2 2 3 6 6 6

Matrix representation of D18⋊C4 in GL3(𝔽37) generated by

 1 0 0 0 6 17 0 20 26
,
 36 0 0 0 6 17 0 11 31
,
 6 0 0 0 5 10 0 27 32
`G:=sub<GL(3,GF(37))| [1,0,0,0,6,20,0,17,26],[36,0,0,0,6,11,0,17,31],[6,0,0,0,5,27,0,10,32] >;`

D18⋊C4 in GAP, Magma, Sage, TeX

`D_{18}\rtimes C_4`
`% in TeX`

`G:=Group("D18:C4");`
`// GroupNames label`

`G:=SmallGroup(144,14);`
`// by ID`

`G=gap.SmallGroup(144,14);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,2404,208,3461]);`
`// Polycyclic`

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

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