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

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

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

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

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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