Copied to
clipboard

## G = C2×D5×D9order 360 = 23·32·5

### Direct product of C2, D5 and D9

Aliases: C2×D5×D9, C45⋊C23, C90⋊C22, D905C2, C101D18, C181D10, D45⋊C22, C30.8D6, (C5×D9)⋊C22, (C9×D5)⋊C22, C91(C22×D5), C51(C22×D9), (D5×C18)⋊3C2, (C10×D9)⋊3C2, (C6×D5).4S3, (C3×D5).6D6, C6.15(S3×D5), C15.(C22×S3), C3.(C2×S3×D5), SmallGroup(360,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — C2×D5×D9
 Chief series C1 — C3 — C15 — C45 — C9×D5 — D5×D9 — C2×D5×D9
 Lower central C45 — C2×D5×D9
 Upper central C1 — C2

Generators and relations for C2×D5×D9
G = < a,b,c,d,e | a2=b5=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 768 in 96 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, C6, C23, C9, D5, D5, C10, C10, D6, C2×C6, C15, D9, D9, C18, C18, D10, D10, C2×C10, C22×S3, C5×S3, C3×D5, D15, C30, D18, D18, C2×C18, C22×D5, C45, S3×D5, C6×D5, S3×C10, D30, C22×D9, C5×D9, C9×D5, D45, C90, C2×S3×D5, D5×D9, D5×C18, C10×D9, D90, C2×D5×D9
Quotients: C1, C2, C22, S3, C23, D5, D6, D9, D10, C22×S3, D18, C22×D5, S3×D5, C22×D9, C2×S3×D5, D5×D9, C2×D5×D9

Smallest permutation representation of C2×D5×D9
On 90 points
Generators in S90
(1 79)(2 80)(3 81)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 60)(20 61)(21 62)(22 63)(23 55)(24 56)(25 57)(26 58)(27 59)(28 71)(29 72)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)
(1 71 40 22 47)(2 72 41 23 48)(3 64 42 24 49)(4 65 43 25 50)(5 66 44 26 51)(6 67 45 27 52)(7 68 37 19 53)(8 69 38 20 54)(9 70 39 21 46)(10 76 34 82 60)(11 77 35 83 61)(12 78 36 84 62)(13 79 28 85 63)(14 80 29 86 55)(15 81 30 87 56)(16 73 31 88 57)(17 74 32 89 58)(18 75 33 90 59)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 46)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 73)(17 74)(18 75)(19 68)(20 69)(21 70)(22 71)(23 72)(24 64)(25 65)(26 66)(27 67)(28 63)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 81)(8 80)(9 79)(10 49)(11 48)(12 47)(13 46)(14 54)(15 53)(16 52)(17 51)(18 50)(19 56)(20 55)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 72)(36 71)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 90)(44 89)(45 88)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 5A 5B 6A 6B 6C 9A 9B 9C 10A 10B 10C 10D 10E 10F 15A 15B 18A 18B 18C 18D ··· 18I 30A 30B 45A ··· 45F 90A ··· 90F order 1 2 2 2 2 2 2 2 3 5 5 6 6 6 9 9 9 10 10 10 10 10 10 15 15 18 18 18 18 ··· 18 30 30 45 ··· 45 90 ··· 90 size 1 1 5 5 9 9 45 45 2 2 2 2 10 10 2 2 2 2 2 18 18 18 18 4 4 2 2 2 10 ··· 10 4 4 4 ··· 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D5 D6 D6 D9 D10 D10 D18 D18 S3×D5 C2×S3×D5 D5×D9 C2×D5×D9 kernel C2×D5×D9 D5×D9 D5×C18 C10×D9 D90 C6×D5 D18 C3×D5 C30 D10 D9 C18 D5 C10 C6 C3 C2 C1 # reps 1 4 1 1 1 1 2 2 1 3 4 2 6 3 2 2 6 6

Matrix representation of C2×D5×D9 in GL6(𝔽181)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 180 0 0 0 0 0 0 180 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 180 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 122 3 0 0 0 0 126 58 0 0 0 0 0 0 177 131 0 0 0 0 50 127
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 39 180 0 0 0 0 0 0 4 127 0 0 0 0 131 177

G:=sub<GL(6,GF(181))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,180,0,0,0,0,0,0,180,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,180,0,0,0,0,1,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,122,126,0,0,0,0,3,58,0,0,0,0,0,0,177,50,0,0,0,0,131,127],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,0,180,0,0,0,0,0,0,4,131,0,0,0,0,127,177] >;

C2×D5×D9 in GAP, Magma, Sage, TeX

C_2\times D_5\times D_9
% in TeX

G:=Group("C2xD5xD9");
// GroupNames label

G:=SmallGroup(360,45);
// by ID

G=gap.SmallGroup(360,45);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,1641,741,2884,4331]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽