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G = D6.D18order 432 = 24·33

1st non-split extension by D6 of D18 acting via D18/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D6.D18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — D6⋊D9 — D6.D18
 Lower central C3×C9 — C3×C18 — D6.D18
 Upper central C1 — C4

Generators and relations for D6.D18
G = < a,b,c,d | a18=b2=1, c6=d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a9b, dcd-1=c5 >

Subgroups: 860 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×C9, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C4○D12, C3×D9, S3×C9, C9⋊S3, C3×C18, Dic18, C4×D9, C4×D9, D36, C9⋊D4, C2×C36, D6⋊S3, C3⋊D12, C322Q8, S3×C12, S3×C12, C4×C3⋊S3, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, C6×D9, S3×C18, C2×C9⋊S3, D365C2, D6.D6, C9⋊Dic6, C3⋊D36, D6⋊D9, C9⋊D12, C12×D9, S3×C36, C4×C9⋊S3, D6.D18
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, C22×D9, C2×S32, S3×D9, D365C2, D6.D6, C2×S3×D9, D6.D18

Smallest permutation representation of D6.D18
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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)
(1 32 16 29 13 26 10 23 7 20 4 35)(2 33 17 30 14 27 11 24 8 21 5 36)(3 34 18 31 15 28 12 25 9 22 6 19)(37 61 40 64 43 67 46 70 49 55 52 58)(38 62 41 65 44 68 47 71 50 56 53 59)(39 63 42 66 45 69 48 72 51 57 54 60)
(1 39 10 48)(2 40 11 49)(3 41 12 50)(4 42 13 51)(5 43 14 52)(6 44 15 53)(7 45 16 54)(8 46 17 37)(9 47 18 38)(19 56 28 65)(20 57 29 66)(21 58 30 67)(22 59 31 68)(23 60 32 69)(24 61 33 70)(25 62 34 71)(26 63 35 72)(27 64 36 55)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36F 36G ··· 36L 36M ··· 36R order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 6 18 54 2 2 4 1 1 6 18 54 2 2 4 6 6 18 18 2 2 2 4 4 4 2 2 2 2 4 4 6 6 18 18 2 2 2 4 4 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 D6 C4○D4 D9 D18 D18 D18 C4○D12 C4○D12 D36⋊5C2 S32 C2×S32 S3×D9 D6.D6 C2×S3×D9 D6.D18 kernel D6.D18 C9⋊Dic6 C3⋊D36 D6⋊D9 C9⋊D12 C12×D9 S3×C36 C4×C9⋊S3 C4×D9 S3×C12 Dic9 C36 D18 C3×Dic3 C3×C12 S3×C6 C3×C9 C4×S3 Dic3 C12 D6 C9 C32 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 3 3 4 4 12 1 1 3 2 3 6

Matrix representation of D6.D18 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 11 0 0 0 0 26 6
,
 20 2 0 0 0 0 4 17 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 20 26 0 0 0 0 6 17
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 0 36 0 0 0 0 1 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 20 2 0 0 0 0 3 17 0 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,26,0,0,0,0,11,6],[20,4,0,0,0,0,2,17,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,20,6,0,0,0,0,26,17],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[20,3,0,0,0,0,2,17,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

D6.D18 in GAP, Magma, Sage, TeX

`D_6.D_{18}`
`% in TeX`

`G:=Group("D6.D18");`
`// GroupNames label`

`G:=SmallGroup(432,287);`
`// by ID`

`G=gap.SmallGroup(432,287);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^18=b^2=1,c^6=d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^5>;`
`// generators/relations`

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