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## G = D4⋊8D18order 288 = 25·32

### 4th semidirect product of D4 and D18 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4⋊8D18
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — D4×D9 — D4⋊8D18
 Lower central C9 — C18 — D4⋊8D18
 Upper central C1 — C2 — C4○D4

Generators and relations for D48D18
G = < a,b,c,d | a4=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 1200 in 249 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C9, Dic3 [×2], C12, C12 [×3], D6 [×12], C2×C6 [×3], C2×D4 [×9], C4○D4, C4○D4 [×5], D9 [×6], C18, C18 [×3], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], 2+ 1+4, Dic9 [×2], C36, C36 [×3], D18 [×6], D18 [×6], C2×C18 [×3], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, Dic18, C4×D9 [×6], D36 [×9], C9⋊D4 [×6], C2×C36 [×3], D4×C9 [×3], Q8×C9, C22×D9 [×6], D4○D12, C2×D36 [×3], D365C2 [×3], D4×D9 [×6], Q83D9 [×2], C9×C4○D4, D48D18
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], D4○D12, C23×D9, D48D18

Smallest permutation representation of D48D18
On 72 points
Generators in S72
```(1 34 50 57)(2 35 51 58)(3 36 52 59)(4 19 53 60)(5 20 54 61)(6 21 37 62)(7 22 38 63)(8 23 39 64)(9 24 40 65)(10 25 41 66)(11 26 42 67)(12 27 43 68)(13 28 44 69)(14 29 45 70)(15 30 46 71)(16 31 47 72)(17 32 48 55)(18 33 49 56)
(2 51)(4 53)(6 37)(8 39)(10 41)(12 43)(14 45)(16 47)(18 49)(20 61)(22 63)(24 65)(26 67)(28 69)(30 71)(32 55)(34 57)(36 59)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 36)(17 35)(18 34)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 72)(53 71)(54 70)```

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18L 36A ··· 36F 36G ··· 36O order 1 2 2 2 2 2 ··· 2 3 4 4 4 4 4 4 6 6 6 6 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 2 2 2 18 ··· 18 2 2 2 2 2 18 18 2 4 4 4 2 2 2 2 2 4 4 4 2 2 2 4 ··· 4 2 ··· 2 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○D12 D4⋊8D18 kernel D4⋊8D18 C2×D36 D36⋊5C2 D4×D9 Q8⋊3D9 C9×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C9 C3 C1 # reps 1 3 3 6 2 1 1 3 3 1 3 9 9 3 1 2 6

Matrix representation of D48D18 in GL4(𝔽37) generated by

 36 0 5 14 0 36 23 28 6 34 1 0 3 9 0 1
,
 1 0 0 0 0 1 0 0 31 3 36 0 34 28 0 36
,
 25 33 4 21 4 29 16 20 10 0 12 4 0 10 33 8
,
 8 33 0 0 25 29 0 0 27 0 25 33 10 10 8 12
`G:=sub<GL(4,GF(37))| [36,0,6,3,0,36,34,9,5,23,1,0,14,28,0,1],[1,0,31,34,0,1,3,28,0,0,36,0,0,0,0,36],[25,4,10,0,33,29,0,10,4,16,12,33,21,20,4,8],[8,25,27,10,33,29,0,10,0,0,25,8,0,0,33,12] >;`

D48D18 in GAP, Magma, Sage, TeX

`D_4\rtimes_8D_{18}`
`% in TeX`

`G:=Group("D4:8D18");`
`// GroupNames label`

`G:=SmallGroup(288,363);`
`// by ID`

`G=gap.SmallGroup(288,363);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,6725,292,9414]);`
`// Polycyclic`

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

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