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G = C18×F5order 360 = 23·32·5

Direct product of C18 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C18×F5
 Chief series C1 — C5 — C15 — C3×D5 — C9×D5 — C9×F5 — C18×F5
 Lower central C5 — C18×F5
 Upper central C1 — C18

Generators and relations for C18×F5
G = < a,b,c | a18=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C18×F5
On 90 points
Generators in S90
(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)
(1 65 20 40 83)(2 66 21 41 84)(3 67 22 42 85)(4 68 23 43 86)(5 69 24 44 87)(6 70 25 45 88)(7 71 26 46 89)(8 72 27 47 90)(9 55 28 48 73)(10 56 29 49 74)(11 57 30 50 75)(12 58 31 51 76)(13 59 32 52 77)(14 60 33 53 78)(15 61 34 54 79)(16 62 35 37 80)(17 63 36 38 81)(18 64 19 39 82)
(19 82 39 64)(20 83 40 65)(21 84 41 66)(22 85 42 67)(23 86 43 68)(24 87 44 69)(25 88 45 70)(26 89 46 71)(27 90 47 72)(28 73 48 55)(29 74 49 56)(30 75 50 57)(31 76 51 58)(32 77 52 59)(33 78 53 60)(34 79 54 61)(35 80 37 62)(36 81 38 63)

G:=sub<Sym(90)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90), (1,65,20,40,83)(2,66,21,41,84)(3,67,22,42,85)(4,68,23,43,86)(5,69,24,44,87)(6,70,25,45,88)(7,71,26,46,89)(8,72,27,47,90)(9,55,28,48,73)(10,56,29,49,74)(11,57,30,50,75)(12,58,31,51,76)(13,59,32,52,77)(14,60,33,53,78)(15,61,34,54,79)(16,62,35,37,80)(17,63,36,38,81)(18,64,19,39,82), (19,82,39,64)(20,83,40,65)(21,84,41,66)(22,85,42,67)(23,86,43,68)(24,87,44,69)(25,88,45,70)(26,89,46,71)(27,90,47,72)(28,73,48,55)(29,74,49,56)(30,75,50,57)(31,76,51,58)(32,77,52,59)(33,78,53,60)(34,79,54,61)(35,80,37,62)(36,81,38,63)>;

G:=Group( (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90), (1,65,20,40,83)(2,66,21,41,84)(3,67,22,42,85)(4,68,23,43,86)(5,69,24,44,87)(6,70,25,45,88)(7,71,26,46,89)(8,72,27,47,90)(9,55,28,48,73)(10,56,29,49,74)(11,57,30,50,75)(12,58,31,51,76)(13,59,32,52,77)(14,60,33,53,78)(15,61,34,54,79)(16,62,35,37,80)(17,63,36,38,81)(18,64,19,39,82), (19,82,39,64)(20,83,40,65)(21,84,41,66)(22,85,42,67)(23,86,43,68)(24,87,44,69)(25,88,45,70)(26,89,46,71)(27,90,47,72)(28,73,48,55)(29,74,49,56)(30,75,50,57)(31,76,51,58)(32,77,52,59)(33,78,53,60)(34,79,54,61)(35,80,37,62)(36,81,38,63) );

G=PermutationGroup([(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)], [(1,65,20,40,83),(2,66,21,41,84),(3,67,22,42,85),(4,68,23,43,86),(5,69,24,44,87),(6,70,25,45,88),(7,71,26,46,89),(8,72,27,47,90),(9,55,28,48,73),(10,56,29,49,74),(11,57,30,50,75),(12,58,31,51,76),(13,59,32,52,77),(14,60,33,53,78),(15,61,34,54,79),(16,62,35,37,80),(17,63,36,38,81),(18,64,19,39,82)], [(19,82,39,64),(20,83,40,65),(21,84,41,66),(22,85,42,67),(23,86,43,68),(24,87,44,69),(25,88,45,70),(26,89,46,71),(27,90,47,72),(28,73,48,55),(29,74,49,56),(30,75,50,57),(31,76,51,58),(32,77,52,59),(33,78,53,60),(34,79,54,61),(35,80,37,62),(36,81,38,63)])

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5 6A 6B 6C 6D 6E 6F 9A ··· 9F 10 12A ··· 12H 15A 15B 18A ··· 18F 18G ··· 18R 30A 30B 36A ··· 36X 45A ··· 45F 90A ··· 90F order 1 2 2 2 3 3 4 4 4 4 5 6 6 6 6 6 6 9 ··· 9 10 12 ··· 12 15 15 18 ··· 18 18 ··· 18 30 30 36 ··· 36 45 ··· 45 90 ··· 90 size 1 1 5 5 1 1 5 5 5 5 4 1 1 5 5 5 5 1 ··· 1 4 5 ··· 5 4 4 1 ··· 1 5 ··· 5 4 4 5 ··· 5 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C9 C12 C12 C18 C18 C36 C36 F5 C2×F5 C3×F5 C6×F5 C9×F5 C18×F5 kernel C18×F5 C9×F5 D5×C18 C6×F5 C9×D5 C90 C3×F5 C6×D5 C2×F5 C3×D5 C30 F5 D10 D5 C10 C18 C9 C6 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 6 4 4 12 6 12 12 1 1 2 2 6 6

Matrix representation of C18×F5 in GL5(𝔽181)

 180 0 0 0 0 0 73 0 0 0 0 0 73 0 0 0 0 0 73 0 0 0 0 0 73
,
 1 0 0 0 0 0 0 0 0 180 0 1 0 0 180 0 0 1 0 180 0 0 0 1 180
,
 162 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(181))| [180,0,0,0,0,0,73,0,0,0,0,0,73,0,0,0,0,0,73,0,0,0,0,0,73],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,180,180,180,180],[162,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

C18×F5 in GAP, Magma, Sage, TeX

C_{18}\times F_5
% in TeX

G:=Group("C18xF5");
// GroupNames label

G:=SmallGroup(360,43);
// by ID

G=gap.SmallGroup(360,43);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,122,5189,887]);
// Polycyclic

G:=Group<a,b,c|a^18=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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