Copied to
clipboard

## G = 2+ 1+4⋊C9order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — 2+ 1+4⋊C9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — Q8.C18 — 2+ 1+4⋊C9
 Lower central Q8 — 2+ 1+4⋊C9
 Upper central C1 — C6 — C3×Q8

Generators and relations for 2+ 1+4⋊C9
G = < a,b,c,d,e | a4=b2=d2=e9=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a-1bcd, dcd=a2c, ece-1=a-1d, ede-1=cd >

Subgroups: 249 in 81 conjugacy classes, 27 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C23, C9, C12, C12, C2×C6, C2×D4, C4○D4, C4○D4, C18, C2×C12, C3×D4, C3×Q8, C22×C6, 2+ 1+4, C36, C6×D4, C3×C4○D4, C3×C4○D4, Q8⋊C9, Q8×C9, C3×2+ 1+4, Q8.C18, 2+ 1+4⋊C9
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C22×A4, C2×C3.A4, Q8.A4, C22×C3.A4, 2+ 1+4⋊C9

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 6A 6B 6C ··· 6H 9A ··· 9F 12A ··· 12F 12G 12H 18A ··· 18F 36A ··· 36R order 1 2 2 2 2 3 3 4 4 4 4 6 6 6 ··· 6 9 ··· 9 12 ··· 12 12 12 18 ··· 18 36 ··· 36 size 1 1 6 6 6 1 1 2 2 2 6 1 1 6 ··· 6 4 ··· 4 2 ··· 2 6 6 4 ··· 4 8 ··· 8

57 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 4 4 4 type + + + + + image C1 C2 C3 C6 C9 C18 A4 C2×A4 C3.A4 C2×C3.A4 Q8.A4 Q8.A4 2+ 1+4⋊C9 kernel 2+ 1+4⋊C9 Q8.C18 C3×2+ 1+4 C3×C4○D4 2+ 1+4 C4○D4 C3×Q8 C12 Q8 C4 C3 C3 C1 # reps 1 3 2 6 6 18 1 3 2 6 1 2 6

Matrix representation of 2+ 1+4⋊C9 in GL7(𝔽37)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 2 0 0 0 0 1 0 0 35 0 0 0 0 0 0 36 0 0 0 0 0 1 0
,
 1 0 3 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 3 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 2 0 0 0 0 36 0 0 2 0 0 0 36 0 0 1 0 0 0 0 36 1 0
,
 36 0 34 0 0 0 0 0 36 21 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 36 0 0 0 0 1 36 0
,
 0 16 32 0 0 0 0 30 36 0 0 0 0 0 0 14 1 0 0 0 0 0 0 0 0 10 27 27 0 0 0 0 27 10 27 0 0 0 5 32 0 27 0 0 0 5 5 0 27

`G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,2,0,0,1,0,0,0,0,35,36,0],[1,0,0,0,0,0,0,0,36,0,0,0,0,0,3,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,36,0,0,0,0,0,3,0,36,0,0,0,0,0,0,0,0,36,36,0,0,0,0,36,0,0,36,0,0,0,2,0,0,1,0,0,0,0,2,1,0],[36,0,0,0,0,0,0,0,36,0,0,0,0,0,34,21,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,36,0],[0,30,0,0,0,0,0,16,36,14,0,0,0,0,32,0,1,0,0,0,0,0,0,0,0,0,5,5,0,0,0,10,27,32,5,0,0,0,27,10,0,0,0,0,0,27,27,27,27] >;`

2+ 1+4⋊C9 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes C_9`
`% in TeX`

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

`G:=SmallGroup(288,348);`
`// by ID`

`G=gap.SmallGroup(288,348);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,79,648,172,1153,285,124]);`
`// 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,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b*c*d,d*c*d=a^2*c,e*c*e^-1=a^-1*d,e*d*e^-1=c*d>;`
`// generators/relations`

׿
×
𝔽