direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C18×F5, C10⋊C36, D5⋊C36, C90⋊2C4, D10.C18, C30.3C12, C5⋊(C2×C36), C3.(C6×F5), C45⋊3(C2×C4), (C6×F5).C3, (C3×F5).C6, (C9×D5)⋊3C4, C15.(C2×C12), (C3×D5).C12, D5.(C2×C18), C6.3(C3×F5), (C6×D5).5C6, (D5×C18).3C2, (C9×D5).3C22, (C3×D5).3(C2×C6), SmallGroup(360,43)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C18×F5 |
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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