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## G = D18.3D6order 432 = 24·33

### 3rd non-split extension by D18 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D18.3D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — Dic3×D9 — D18.3D6
 Lower central C3×C9 — C3×C18 — D18.3D6
 Upper central C1 — C2 — C22

Generators and relations for D18.3D6
G = < a,b,c,d | a18=b2=c6=1, d2=a9, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a9b, dcd-1=c-1 >

Subgroups: 856 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×C9, Dic9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, C3×D9, C9⋊S3, C3×C18, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C9⋊D4, C2×C36, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C327D4, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×D9, C2×C9⋊S3, C6×C18, D365C2, D6.3D6, C9⋊Dic6, Dic3×D9, C18.D6, C3⋊D36, Dic3×C18, C3×C9⋊D4, C6.D18, D18.3D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, D42S3, C22×D9, C2×S32, S3×D9, D365C2, D6.3D6, C2×S3×D9, D18.3D6

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

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

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

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

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 D9 D18 D18 C4○D12 D36⋊5C2 S32 D4⋊2S3 C2×S32 S3×D9 D6.3D6 C2×S3×D9 D18.3D6 kernel D18.3D6 C9⋊Dic6 Dic3×D9 C18.D6 C3⋊D36 Dic3×C18 C3×C9⋊D4 C6.D18 C9⋊D4 C6×Dic3 Dic9 D18 C2×C18 C3×Dic3 C62 C3×C9 C2×Dic3 Dic3 C2×C6 C32 C3 C2×C6 C9 C6 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 3 6 3 4 12 1 1 1 3 2 3 6

Matrix representation of D18.3D6 in GL6(𝔽37)

 11 0 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 7
,
 0 27 0 0 0 0 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 21 0
,
 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 31 0 0 0 0 0 0 6 0 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36

`G:=sub<GL(6,GF(37))| [11,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,7],[0,11,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,21,0,0,0,0,30,0],[1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[31,0,0,0,0,0,0,6,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36] >;`

D18.3D6 in GAP, Magma, Sage, TeX

`D_{18}._3D_6`
`% in TeX`

`G:=Group("D18.3D6");`
`// GroupNames label`

`G:=SmallGroup(432,305);`
`// by ID`

`G=gap.SmallGroup(432,305);`
`# by ID`

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

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

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