direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C18, C10⋊C18, C90⋊3C2, C30.3C6, C45⋊4C22, 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
| C5 — D5×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 48 68 34 86)(2 49 69 35 87)(3 50 70 36 88)(4 51 71 19 89)(5 52 72 20 90)(6 53 55 21 73)(7 54 56 22 74)(8 37 57 23 75)(9 38 58 24 76)(10 39 59 25 77)(11 40 60 26 78)(12 41 61 27 79)(13 42 62 28 80)(14 43 63 29 81)(15 44 64 30 82)(16 45 65 31 83)(17 46 66 32 84)(18 47 67 33 85)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 73)(16 74)(17 75)(18 76)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 37)(33 38)(34 39)(35 40)(36 41)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)
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), (1,48,68,34,86)(2,49,69,35,87)(3,50,70,36,88)(4,51,71,19,89)(5,52,72,20,90)(6,53,55,21,73)(7,54,56,22,74)(8,37,57,23,75)(9,38,58,24,76)(10,39,59,25,77)(11,40,60,26,78)(12,41,61,27,79)(13,42,62,28,80)(14,43,63,29,81)(15,44,64,30,82)(16,45,65,31,83)(17,46,66,32,84)(18,47,67,33,85), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,73)(16,74)(17,75)(18,76)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,37)(33,38)(34,39)(35,40)(36,41)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)>;
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), (1,48,68,34,86)(2,49,69,35,87)(3,50,70,36,88)(4,51,71,19,89)(5,52,72,20,90)(6,53,55,21,73)(7,54,56,22,74)(8,37,57,23,75)(9,38,58,24,76)(10,39,59,25,77)(11,40,60,26,78)(12,41,61,27,79)(13,42,62,28,80)(14,43,63,29,81)(15,44,64,30,82)(16,45,65,31,83)(17,46,66,32,84)(18,47,67,33,85), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,73)(16,74)(17,75)(18,76)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,37)(33,38)(34,39)(35,40)(36,41)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72) );
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)], [(1,48,68,34,86),(2,49,69,35,87),(3,50,70,36,88),(4,51,71,19,89),(5,52,72,20,90),(6,53,55,21,73),(7,54,56,22,74),(8,37,57,23,75),(9,38,58,24,76),(10,39,59,25,77),(11,40,60,26,78),(12,41,61,27,79),(13,42,62,28,80),(14,43,63,29,81),(15,44,64,30,82),(16,45,65,31,83),(17,46,66,32,84),(18,47,67,33,85)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,73),(16,74),(17,75),(18,76),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,37),(33,38),(34,39),(35,40),(36,41),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)]])
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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