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## G = D4⋊6D18order 288 = 25·32

### 2nd semidirect product of D4 and D18 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4⋊6D18
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — D4×D9 — D4⋊6D18
 Lower central C9 — C18 — D4⋊6D18
 Upper central C1 — C2 — C2×D4

Generators and relations for D46D18
G = < a,b,c,d | a4=b2=c18=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1080 in 249 conjugacy classes, 102 normal (16 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C9, Dic3 [×4], C12 [×2], D6 [×8], C2×C6, C2×C6 [×4], C2×C6 [×2], C2×D4, C2×D4 [×8], C4○D4 [×6], D9 [×4], C18, C18 [×5], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], 2+ 1+4, Dic9 [×4], C36 [×2], D18 [×4], D18 [×4], C2×C18, C2×C18 [×4], C2×C18 [×2], C4○D12 [×2], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4, Dic18 [×2], C4×D9 [×4], D36 [×2], C2×Dic9 [×4], C9⋊D4 [×12], C2×C36, D4×C9 [×4], C22×D9 [×4], C22×C18 [×2], D46D6, D365C2 [×2], D4×D9 [×4], D42D9 [×4], C2×C9⋊D4 [×4], D4×C18, D46D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2+ 1+4, D18 [×7], S3×C23, C22×D9 [×7], D46D6, C23×D9, D46D18

Smallest permutation representation of D46D18
On 72 points
Generators in S72
```(1 72 10 63)(2 64 11 55)(3 56 12 65)(4 66 13 57)(5 58 14 67)(6 68 15 59)(7 60 16 69)(8 70 17 61)(9 62 18 71)(19 54 28 45)(20 46 29 37)(21 38 30 47)(22 48 31 39)(23 40 32 49)(24 50 33 41)(25 42 34 51)(26 52 35 43)(27 44 36 53)
(1 45)(2 37)(3 47)(4 39)(5 49)(6 41)(7 51)(8 43)(9 53)(10 54)(11 46)(12 38)(13 48)(14 40)(15 50)(16 42)(17 52)(18 44)(19 63)(20 55)(21 65)(22 57)(23 67)(24 59)(25 69)(26 61)(27 71)(28 72)(29 64)(30 56)(31 66)(32 58)(33 68)(34 60)(35 70)(36 62)
(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 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(53 54)(55 61)(56 60)(57 59)(62 72)(63 71)(64 70)(65 69)(66 68)```

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 D9 D18 D18 D18 2+ 1+4 D4⋊6D6 D4⋊6D18 kernel D4⋊6D18 D36⋊5C2 D4×D9 D4⋊2D9 C2×C9⋊D4 D4×C18 C6×D4 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C9 C3 C1 # reps 1 2 4 4 4 1 1 1 4 2 3 3 12 6 1 2 6

Matrix representation of D46D18 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 30 23 0 0 0 0 14 7 0 0 7 14 0 0 0 0 23 30 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 6 17 0 0 0 0 20 26 0 0 0 0 0 0 0 1 0 0 0 0 36 36 0 0 0 0 0 0 0 36 0 0 0 0 1 1
,
 6 17 0 0 0 0 11 31 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 36 0

`G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,23,0,0,0,0,14,30,0,0,30,14,0,0,0,0,23,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,20,0,0,0,0,17,26,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1],[6,11,0,0,0,0,17,31,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0] >;`

D46D18 in GAP, Magma, Sage, TeX

`D_4\rtimes_6D_{18}`
`% in TeX`

`G:=Group("D4:6D18");`
`// GroupNames label`

`G:=SmallGroup(288,358);`
`// by ID`

`G=gap.SmallGroup(288,358);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,6725,292,9414]);`
`// Polycyclic`

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

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