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## G = D5×C18order 180 = 22·32·5

### Direct product of C18 and D5

Aliases: D5×C18, C10⋊C18, C903C2, C30.3C6, C454C22, C5⋊(C2×C18), C3.(C6×D5), C15.(C2×C6), (C6×D5).C3, C6.3(C3×D5), (C3×D5).4C6, SmallGroup(180,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C18
 Chief series C1 — C5 — C15 — C45 — C9×D5 — D5×C18
 Lower central C5 — D5×C18
 Upper central C1 — C18

Generators and relations for D5×C18
G = < a,b,c | a18=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C18
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 47 26 69 75)(2 48 27 70 76)(3 49 28 71 77)(4 50 29 72 78)(5 51 30 55 79)(6 52 31 56 80)(7 53 32 57 81)(8 54 33 58 82)(9 37 34 59 83)(10 38 35 60 84)(11 39 36 61 85)(12 40 19 62 86)(13 41 20 63 87)(14 42 21 64 88)(15 43 22 65 89)(16 44 23 66 90)(17 45 24 67 73)(18 46 25 68 74)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 68)(38 69)(39 70)(40 71)(41 72)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)

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

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

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

D5×C18 is a maximal subgroup of   C45⋊D4  C9⋊D20

72 conjugacy classes

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

72 irreducible representations

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

Matrix representation of D5×C18 in GL2(𝔽19) generated by

 13 0 0 13
,
 18 8 17 15
,
 4 17 17 15
G:=sub<GL(2,GF(19))| [13,0,0,13],[18,17,8,15],[4,17,17,15] >;

D5×C18 in GAP, Magma, Sage, TeX

D_5\times C_{18}
% in TeX

G:=Group("D5xC18");
// GroupNames label

G:=SmallGroup(180,9);
// by ID

G=gap.SmallGroup(180,9);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,57,3604]);
// Polycyclic

G:=Group<a,b,c|a^18=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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