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## G = 2+ 1+4⋊2C9order 288 = 25·32

### 2nd semidirect product of 2+ 1+4 and C9 acting via C9/C3=C3

Aliases: 2+ 1+42C9, (C3×Q8).2A4, C3.(C23⋊A4), Q82(C3.A4), (C22×C6).4A4, C232(C3.A4), C6.2(C22⋊A4), C2.2(C24⋊C9), (C3×2+ 1+4).2C3, SmallGroup(288,351)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — 2+ 1+4⋊2C9
 Chief series C1 — C2 — C23 — 2+ 1+4 — C3×2+ 1+4 — 2+ 1+4⋊2C9
 Lower central 2+ 1+4 — 2+ 1+4⋊2C9
 Upper central C1 — C6

Generators and relations for 2+ 1+42C9
G = < a,b,c,d,e | a4=b2=d2=e9=1, c2=a2, bab=a-1, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, ebe-1=a2cd, dcd=a2c, ece-1=acd, ede-1=a-1b >

Subgroups: 285 in 81 conjugacy classes, 17 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, D4, Q8, C23, C23, C9, C12, C2×C6, C2×D4, C4○D4, C18, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, 2+ 1+4, C3.A4, C6×D4, C3×C4○D4, Q8⋊C9, C2×C3.A4, C3×2+ 1+4, 2+ 1+42C9
Quotients: C1, C3, C9, A4, C3.A4, C22⋊A4, C23⋊A4, C24⋊C9, 2+ 1+42C9

Smallest permutation representation of 2+ 1+42C9
On 72 points
Generators in S72
```(1 27 63 39)(2 10 55 29)(3 47 56 66)(4 21 57 42)(5 13 58 32)(6 50 59 69)(7 24 60 45)(8 16 61 35)(9 53 62 72)(11 20 30 41)(12 67 31 48)(14 23 33 44)(15 70 34 51)(17 26 36 38)(18 64 28 54)(19 46 40 65)(22 49 43 68)(25 52 37 71)
(1 39)(3 41)(4 42)(6 44)(7 45)(9 38)(10 29)(11 66)(12 48)(13 32)(14 69)(15 51)(16 35)(17 72)(18 54)(20 56)(21 57)(23 59)(24 60)(26 62)(27 63)(28 64)(30 47)(31 67)(33 50)(34 70)(36 53)(46 65)(49 68)(52 71)
(1 28 63 18)(2 40 55 19)(3 47 56 66)(4 31 57 12)(5 43 58 22)(6 50 59 69)(7 34 60 15)(8 37 61 25)(9 53 62 72)(10 65 29 46)(11 41 30 20)(13 68 32 49)(14 44 33 23)(16 71 35 52)(17 38 36 26)(21 48 42 67)(24 51 45 70)(27 54 39 64)
(1 18)(3 11)(4 12)(6 14)(7 15)(9 17)(19 40)(20 47)(21 67)(22 43)(23 50)(24 70)(25 37)(26 53)(27 64)(28 63)(30 56)(31 57)(33 59)(34 60)(36 62)(38 72)(39 54)(41 66)(42 48)(44 69)(45 51)(46 65)(49 68)(52 71)
(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)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 6A 6B 6C ··· 6H 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F order 1 2 2 2 2 3 3 4 4 6 6 6 ··· 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 6 6 6 1 1 6 6 1 1 6 ··· 6 16 ··· 16 6 6 6 6 16 ··· 16

33 irreducible representations

 dim 1 1 1 3 3 3 3 4 4 4 type + + + + image C1 C3 C9 A4 A4 C3.A4 C3.A4 C23⋊A4 C23⋊A4 2+ 1+4⋊2C9 kernel 2+ 1+4⋊2C9 C3×2+ 1+4 2+ 1+4 C3×Q8 C22×C6 Q8 C23 C3 C3 C1 # reps 1 2 6 2 3 4 6 1 2 6

Matrix representation of 2+ 1+42C9 in GL4(𝔽37) generated by

 0 0 1 0 36 1 1 35 36 0 0 0 36 1 0 36
,
 0 0 1 0 36 1 1 35 1 0 0 0 0 0 0 36
,
 0 1 0 0 36 0 0 0 36 1 1 35 36 0 1 36
,
 0 1 0 0 1 0 0 0 36 1 1 35 0 0 0 36
,
 16 0 0 16 0 16 0 21 0 0 16 21 21 16 16 5
`G:=sub<GL(4,GF(37))| [0,36,36,36,0,1,0,1,1,1,0,0,0,35,0,36],[0,36,1,0,0,1,0,0,1,1,0,0,0,35,0,36],[0,36,36,36,1,0,1,0,0,0,1,1,0,0,35,36],[0,1,36,0,1,0,1,0,0,0,1,0,0,0,35,36],[16,0,0,21,0,16,0,16,0,0,16,16,16,21,21,5] >;`

2+ 1+42C9 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes_2C_9`
`% in TeX`

`G:=Group("ES+(2,2):2C9");`
`// GroupNames label`

`G:=SmallGroup(288,351);`
`// by ID`

`G=gap.SmallGroup(288,351);`
`# by ID`

`G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,21,380,759,2524,375,4541,1027]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^2=d^2=e^9=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*c*d,d*c*d=a^2*c,e*c*e^-1=a*c*d,e*d*e^-1=a^-1*b>;`
`// generators/relations`

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