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G = C18xF5order 360 = 23·32·5

Direct product of C18 and F5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C18xF5, C10:C36, D5:C36, C90:2C4, D10.C18, C30.3C12, C5:(C2xC36), C3.(C6xF5), C45:3(C2xC4), (C6xF5).C3, (C3xF5).C6, (C9xD5):3C4, C15.(C2xC12), (C3xD5).C12, D5.(C2xC18), C6.3(C3xF5), (C6xD5).5C6, (D5xC18).3C2, (C9xD5).3C22, (C3xD5).3(C2xC6), SmallGroup(360,43)

Series: Derived Chief Lower central Upper central

C1C5 — C18xF5
C1C5C15C3xD5C9xD5C9xF5 — C18xF5
C5 — C18xF5
C1C18

Generators and relations for C18xF5
 G = < a,b,c | a18=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 120 in 48 conjugacy classes, 30 normal (24 characteristic)
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C9, C12, C2xC6, C18, F5, C2xC12, C36, C2xC18, C2xF5, C3xF5, C2xC36, C6xF5, C9xF5, C18xF5
5C2
5C2
5C4
5C22
5C4
5C6
5C6
5C2xC4
5C12
5C12
5C2xC6
5C18
5C18
5C2xC12
5C2xC18
5C36
5C36
5C2xC36

Smallest permutation representation of C18xF5
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 2A2B2C3A3B4A4B4C4D 5 6A6B6C6D6E6F9A···9F 10 12A···12H15A15B18A···18F18G···18R30A30B36A···36X45A···45F90A···90F
order122233444456666669···91012···12151518···1818···18303036···3645···4590···90
size115511555541155551···145···5441···15···5445···54···44···4

90 irreducible representations

dim111111111111111444444
type+++++
imageC1C2C2C3C4C4C6C6C9C12C12C18C18C36C36F5C2xF5C3xF5C6xF5C9xF5C18xF5
kernelC18xF5C9xF5D5xC18C6xF5C9xD5C90C3xF5C6xD5C2xF5C3xD5C30F5D10D5C10C18C9C6C3C2C1
# reps121222426441261212112266

Matrix representation of C18xF5 in GL5(F181)

1800000
073000
007300
000730
000073
,
10000
0000180
0100180
0010180
0001180
,
1620000
00010
01000
00001
00100

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] >;

C18xF5 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

Export

Subgroup lattice of C18xF5 in TeX

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