Copied to
clipboard

## G = C23⋊2D18order 288 = 25·32

### 1st semidirect product of C23 and D18 acting via D18/C9=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C23⋊2D18
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C23×D9 — C23⋊2D18
 Lower central C9 — C2×C18 — C23⋊2D18
 Upper central C1 — C22 — C2×D4

Generators and relations for C232D18
G = < a,b,c,d,e | a2=b2=c2=d18=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1028 in 195 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×4], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], C9, Dic3 [×2], C12, D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, D9 [×4], C18, C18 [×2], C18 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×8], C22×C6 [×2], C22≀C2, Dic9 [×2], C36, D18 [×4], D18 [×12], C2×C18, C2×C18 [×2], C2×C18 [×5], D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C2×Dic9 [×2], C9⋊D4 [×4], C2×C36, D4×C9 [×2], C22×D9 [×2], C22×D9 [×6], C22×C18 [×2], C232D6, D18⋊C4 [×2], C18.D4, C2×C9⋊D4 [×2], D4×C18, C23×D9, C232D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D9, C3⋊D4 [×2], C22×S3, C22≀C2, D18 [×3], S3×D4 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C232D6, D4×D9 [×2], C2×C9⋊D4, C232D18

Smallest permutation representation of C232D18
On 72 points
Generators in S72
```(1 39)(2 49)(3 41)(4 51)(5 43)(6 53)(7 45)(8 37)(9 47)(10 48)(11 40)(12 50)(13 42)(14 52)(15 44)(16 54)(17 46)(18 38)(19 66)(20 58)(21 68)(22 60)(23 70)(24 62)(25 72)(26 64)(27 56)(28 57)(29 67)(30 59)(31 69)(32 61)(33 71)(34 63)(35 55)(36 65)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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 5)(2 4)(6 9)(7 8)(10 14)(11 13)(15 18)(16 17)(19 23)(20 22)(24 27)(25 26)(28 32)(29 31)(33 36)(34 35)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

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

Matrix representation of C232D18 in GL4(𝔽37) generated by

 0 1 0 0 1 0 0 0 0 0 30 14 0 0 23 7
,
 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 36 0 0 0 0 6 11 0 0 26 17
,
 1 0 0 0 0 1 0 0 0 0 26 17 0 0 6 11
`G:=sub<GL(4,GF(37))| [0,1,0,0,1,0,0,0,0,0,30,23,0,0,14,7],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,6,26,0,0,11,17],[1,0,0,0,0,1,0,0,0,0,26,6,0,0,17,11] >;`

C232D18 in GAP, Magma, Sage, TeX

`C_2^3\rtimes_2D_{18}`
`% in TeX`

`G:=Group("C2^3:2D18");`
`// GroupNames label`

`G:=SmallGroup(288,147);`
`// by ID`

`G=gap.SmallGroup(288,147);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,6725,292,9414]);`
`// Polycyclic`

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

׿
×
𝔽