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## G = C2×C9⋊F5order 360 = 23·32·5

### Direct product of C2 and C9⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — C2×C9⋊F5
 Chief series C1 — C3 — C15 — C45 — C9×D5 — C9⋊F5 — C2×C9⋊F5
 Lower central C45 — C2×C9⋊F5
 Upper central C1 — C2

Generators and relations for C2×C9⋊F5
G = < a,b,c,d | a2=b9=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C2×C9⋊F5
On 90 points
Generators in S90
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 78)(20 79)(21 80)(22 81)(23 73)(24 74)(25 75)(26 76)(27 77)(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 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 54 41 22 70)(2 46 42 23 71)(3 47 43 24 72)(4 48 44 25 64)(5 49 45 26 65)(6 50 37 27 66)(7 51 38 19 67)(8 52 39 20 68)(9 53 40 21 69)(10 85 80 35 63)(11 86 81 36 55)(12 87 73 28 56)(13 88 74 29 57)(14 89 75 30 58)(15 90 76 31 59)(16 82 77 32 60)(17 83 78 33 61)(18 84 79 34 62)
(2 9)(3 8)(4 7)(5 6)(10 87 35 73)(11 86 36 81)(12 85 28 80)(13 84 29 79)(14 83 30 78)(15 82 31 77)(16 90 32 76)(17 89 33 75)(18 88 34 74)(19 48 38 64)(20 47 39 72)(21 46 40 71)(22 54 41 70)(23 53 42 69)(24 52 43 68)(25 51 44 67)(26 50 45 66)(27 49 37 65)(56 63)(57 62)(58 61)(59 60)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + - + - + - + + image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 D9 Dic9 D18 Dic9 F5 C2×F5 C3⋊F5 C2×C3⋊F5 C9⋊F5 C2×C9⋊F5 kernel C2×C9⋊F5 C9⋊F5 D5×C18 C9×D5 C90 C6×D5 C3×D5 C3×D5 C30 D10 D5 D5 C10 C18 C9 C6 C3 C2 C1 # reps 1 2 1 2 2 1 1 1 1 3 3 3 3 1 1 2 2 6 6

Matrix representation of C2×C9⋊F5 in GL6(𝔽181)

 180 0 0 0 0 0 0 180 0 0 0 0 0 0 180 0 0 0 0 0 0 180 0 0 0 0 0 0 180 0 0 0 0 0 0 180
,
 127 50 0 0 0 0 131 177 0 0 0 0 0 0 8 164 0 17 0 0 0 172 164 17 0 0 17 164 172 0 0 0 17 0 164 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 180 1 0 0 0 0 180 0 1 0 0 0 180 0 0 1 0 0 180 0 0 0
,
 32 76 0 0 0 0 108 149 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(181))| [180,0,0,0,0,0,0,180,0,0,0,0,0,0,180,0,0,0,0,0,0,180,0,0,0,0,0,0,180,0,0,0,0,0,0,180],[127,131,0,0,0,0,50,177,0,0,0,0,0,0,8,0,17,17,0,0,164,172,164,0,0,0,0,164,172,164,0,0,17,17,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,180,180,180,180,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[32,108,0,0,0,0,76,149,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C2×C9⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes F_5
% in TeX

G:=Group("C2xC9:F5");
// GroupNames label

G:=SmallGroup(360,44);
// by ID

G=gap.SmallGroup(360,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,3267,741,2164,736,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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