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## G = D18.D6order 432 = 24·33

### 1st non-split extension by D18 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D18.D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — Dic3×D9 — D18.D6
 Lower central C3×C9 — C3×C18 — D18.D6
 Upper central C1 — C2 — C4

Generators and relations for D18.D6
G = < a,b,c,d | a36=b2=c3=d2=1, bab=a-1, ac=ca, dad=a17, bc=cb, dbd=a34b, dcd=c-1 >

Subgroups: 888 in 136 conjugacy classes, 41 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, D18, D18, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, D42S3, Q83S3, C3×D9, C9⋊S3, C3×C18, C4×D9, D36, D36, Q8×C9, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C9×Dic3, C9⋊Dic3, C3×C36, C6×D9, C2×C9⋊S3, Q83D9, D12⋊S3, Dic3×D9, C3⋊D36, C9×Dic6, C3×D36, C4×C9⋊S3, D18.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, D42S3, Q83S3, C22×D9, C2×S32, S3×D9, Q83D9, D12⋊S3, C2×S3×D9, D18.D6

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

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 18A 18B 18C 18D 18E 18F 36A ··· 36I 36J ··· 36O order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 18 18 18 18 18 18 36 ··· 36 36 ··· 36 size 1 1 18 18 54 2 2 4 2 6 6 27 27 2 2 4 36 36 2 2 2 4 4 4 4 4 4 4 12 12 2 2 2 4 4 4 4 ··· 4 12 ··· 12

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 C4○D4 D9 D18 D18 S32 D4⋊2S3 Q8⋊3S3 C2×S32 S3×D9 Q8⋊3D9 D12⋊S3 C2×S3×D9 D18.D6 kernel D18.D6 Dic3×D9 C3⋊D36 C9×Dic6 C3×D36 C4×C9⋊S3 D36 C3×Dic6 C36 D18 C3×Dic3 C3×C12 C3×C9 Dic6 Dic3 C12 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 1 2 2 1 2 3 6 3 1 1 1 1 3 3 2 3 6

Matrix representation of D18.D6 in GL6(𝔽37)

 31 0 0 0 0 0 18 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 26 6 0 0 0 0 31 20
,
 31 33 0 0 0 0 18 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 26 6 0 0 0 0 17 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 34 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 1 1

`G:=sub<GL(6,GF(37))| [31,18,0,0,0,0,0,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,26,31,0,0,0,0,6,20],[31,18,0,0,0,0,33,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,26,17,0,0,0,0,6,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,34,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,1,0,0,0,0,0,1] >;`

D18.D6 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(432,281);`
`// by ID`

`G=gap.SmallGroup(432,281);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,135,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^36=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^17,b*c=c*b,d*b*d=a^34*b,d*c*d=c^-1>;`
`// generators/relations`

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