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## G = C5×C32⋊C9order 405 = 34·5

### Direct product of C5 and C32⋊C9

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C5×C32⋊C9, C32⋊C45, C15.1He3, C33.1C15, C15.13- 1+2, (C3×C15)⋊C9, (C3×C9)⋊1C15, (C3×C45)⋊1C3, C3.1(C3×C45), C15.1(C3×C9), C3.1(C5×He3), (C32×C15).1C3, C32.7(C3×C15), (C3×C15).6C32, C3.1(C5×3- 1+2), SmallGroup(405,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C5×C32⋊C9
 Chief series C1 — C3 — C32 — C3×C15 — C3×C45 — C5×C32⋊C9
 Lower central C1 — C3 — C5×C32⋊C9
 Upper central C1 — C3×C15 — C5×C32⋊C9

Generators and relations for C5×C32⋊C9
G = < a,b,c,d | a5=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C5×C32⋊C9
On 135 points
Generators in S135
(1 36 65 122 77)(2 28 66 123 78)(3 29 67 124 79)(4 30 68 125 80)(5 31 69 126 81)(6 32 70 118 73)(7 33 71 119 74)(8 34 72 120 75)(9 35 64 121 76)(10 23 106 117 60)(11 24 107 109 61)(12 25 108 110 62)(13 26 100 111 63)(14 27 101 112 55)(15 19 102 113 56)(16 20 103 114 57)(17 21 104 115 58)(18 22 105 116 59)(37 96 49 129 84)(38 97 50 130 85)(39 98 51 131 86)(40 99 52 132 87)(41 91 53 133 88)(42 92 54 134 89)(43 93 46 135 90)(44 94 47 127 82)(45 95 48 128 83)
(1 7 4)(2 56 37)(3 41 63)(5 59 40)(6 44 57)(8 62 43)(9 38 60)(10 35 97)(11 17 14)(12 93 34)(13 29 91)(15 96 28)(16 32 94)(18 99 31)(19 49 66)(20 70 47)(21 27 24)(22 52 69)(23 64 50)(25 46 72)(26 67 53)(30 36 33)(39 45 42)(48 54 51)(55 61 58)(65 71 68)(73 82 114)(74 80 77)(75 110 90)(76 85 117)(78 113 84)(79 88 111)(81 116 87)(83 89 86)(92 98 95)(100 124 133)(101 107 104)(102 129 123)(103 118 127)(105 132 126)(106 121 130)(108 135 120)(109 115 112)(119 125 122)(128 134 131)
(1 42 58)(2 43 59)(3 44 60)(4 45 61)(5 37 62)(6 38 63)(7 39 55)(8 40 56)(9 41 57)(10 29 94)(11 30 95)(12 31 96)(13 32 97)(14 33 98)(15 34 99)(16 35 91)(17 36 92)(18 28 93)(19 72 52)(20 64 53)(21 65 54)(22 66 46)(23 67 47)(24 68 48)(25 69 49)(26 70 50)(27 71 51)(73 85 111)(74 86 112)(75 87 113)(76 88 114)(77 89 115)(78 90 116)(79 82 117)(80 83 109)(81 84 110)(100 118 130)(101 119 131)(102 120 132)(103 121 133)(104 122 134)(105 123 135)(106 124 127)(107 125 128)(108 126 129)
(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)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)

G:=sub<Sym(135)| (1,36,65,122,77)(2,28,66,123,78)(3,29,67,124,79)(4,30,68,125,80)(5,31,69,126,81)(6,32,70,118,73)(7,33,71,119,74)(8,34,72,120,75)(9,35,64,121,76)(10,23,106,117,60)(11,24,107,109,61)(12,25,108,110,62)(13,26,100,111,63)(14,27,101,112,55)(15,19,102,113,56)(16,20,103,114,57)(17,21,104,115,58)(18,22,105,116,59)(37,96,49,129,84)(38,97,50,130,85)(39,98,51,131,86)(40,99,52,132,87)(41,91,53,133,88)(42,92,54,134,89)(43,93,46,135,90)(44,94,47,127,82)(45,95,48,128,83), (1,7,4)(2,56,37)(3,41,63)(5,59,40)(6,44,57)(8,62,43)(9,38,60)(10,35,97)(11,17,14)(12,93,34)(13,29,91)(15,96,28)(16,32,94)(18,99,31)(19,49,66)(20,70,47)(21,27,24)(22,52,69)(23,64,50)(25,46,72)(26,67,53)(30,36,33)(39,45,42)(48,54,51)(55,61,58)(65,71,68)(73,82,114)(74,80,77)(75,110,90)(76,85,117)(78,113,84)(79,88,111)(81,116,87)(83,89,86)(92,98,95)(100,124,133)(101,107,104)(102,129,123)(103,118,127)(105,132,126)(106,121,130)(108,135,120)(109,115,112)(119,125,122)(128,134,131), (1,42,58)(2,43,59)(3,44,60)(4,45,61)(5,37,62)(6,38,63)(7,39,55)(8,40,56)(9,41,57)(10,29,94)(11,30,95)(12,31,96)(13,32,97)(14,33,98)(15,34,99)(16,35,91)(17,36,92)(18,28,93)(19,72,52)(20,64,53)(21,65,54)(22,66,46)(23,67,47)(24,68,48)(25,69,49)(26,70,50)(27,71,51)(73,85,111)(74,86,112)(75,87,113)(76,88,114)(77,89,115)(78,90,116)(79,82,117)(80,83,109)(81,84,110)(100,118,130)(101,119,131)(102,120,132)(103,121,133)(104,122,134)(105,123,135)(106,124,127)(107,125,128)(108,126,129), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)>;

G:=Group( (1,36,65,122,77)(2,28,66,123,78)(3,29,67,124,79)(4,30,68,125,80)(5,31,69,126,81)(6,32,70,118,73)(7,33,71,119,74)(8,34,72,120,75)(9,35,64,121,76)(10,23,106,117,60)(11,24,107,109,61)(12,25,108,110,62)(13,26,100,111,63)(14,27,101,112,55)(15,19,102,113,56)(16,20,103,114,57)(17,21,104,115,58)(18,22,105,116,59)(37,96,49,129,84)(38,97,50,130,85)(39,98,51,131,86)(40,99,52,132,87)(41,91,53,133,88)(42,92,54,134,89)(43,93,46,135,90)(44,94,47,127,82)(45,95,48,128,83), (1,7,4)(2,56,37)(3,41,63)(5,59,40)(6,44,57)(8,62,43)(9,38,60)(10,35,97)(11,17,14)(12,93,34)(13,29,91)(15,96,28)(16,32,94)(18,99,31)(19,49,66)(20,70,47)(21,27,24)(22,52,69)(23,64,50)(25,46,72)(26,67,53)(30,36,33)(39,45,42)(48,54,51)(55,61,58)(65,71,68)(73,82,114)(74,80,77)(75,110,90)(76,85,117)(78,113,84)(79,88,111)(81,116,87)(83,89,86)(92,98,95)(100,124,133)(101,107,104)(102,129,123)(103,118,127)(105,132,126)(106,121,130)(108,135,120)(109,115,112)(119,125,122)(128,134,131), (1,42,58)(2,43,59)(3,44,60)(4,45,61)(5,37,62)(6,38,63)(7,39,55)(8,40,56)(9,41,57)(10,29,94)(11,30,95)(12,31,96)(13,32,97)(14,33,98)(15,34,99)(16,35,91)(17,36,92)(18,28,93)(19,72,52)(20,64,53)(21,65,54)(22,66,46)(23,67,47)(24,68,48)(25,69,49)(26,70,50)(27,71,51)(73,85,111)(74,86,112)(75,87,113)(76,88,114)(77,89,115)(78,90,116)(79,82,117)(80,83,109)(81,84,110)(100,118,130)(101,119,131)(102,120,132)(103,121,133)(104,122,134)(105,123,135)(106,124,127)(107,125,128)(108,126,129), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135) );

G=PermutationGroup([[(1,36,65,122,77),(2,28,66,123,78),(3,29,67,124,79),(4,30,68,125,80),(5,31,69,126,81),(6,32,70,118,73),(7,33,71,119,74),(8,34,72,120,75),(9,35,64,121,76),(10,23,106,117,60),(11,24,107,109,61),(12,25,108,110,62),(13,26,100,111,63),(14,27,101,112,55),(15,19,102,113,56),(16,20,103,114,57),(17,21,104,115,58),(18,22,105,116,59),(37,96,49,129,84),(38,97,50,130,85),(39,98,51,131,86),(40,99,52,132,87),(41,91,53,133,88),(42,92,54,134,89),(43,93,46,135,90),(44,94,47,127,82),(45,95,48,128,83)], [(1,7,4),(2,56,37),(3,41,63),(5,59,40),(6,44,57),(8,62,43),(9,38,60),(10,35,97),(11,17,14),(12,93,34),(13,29,91),(15,96,28),(16,32,94),(18,99,31),(19,49,66),(20,70,47),(21,27,24),(22,52,69),(23,64,50),(25,46,72),(26,67,53),(30,36,33),(39,45,42),(48,54,51),(55,61,58),(65,71,68),(73,82,114),(74,80,77),(75,110,90),(76,85,117),(78,113,84),(79,88,111),(81,116,87),(83,89,86),(92,98,95),(100,124,133),(101,107,104),(102,129,123),(103,118,127),(105,132,126),(106,121,130),(108,135,120),(109,115,112),(119,125,122),(128,134,131)], [(1,42,58),(2,43,59),(3,44,60),(4,45,61),(5,37,62),(6,38,63),(7,39,55),(8,40,56),(9,41,57),(10,29,94),(11,30,95),(12,31,96),(13,32,97),(14,33,98),(15,34,99),(16,35,91),(17,36,92),(18,28,93),(19,72,52),(20,64,53),(21,65,54),(22,66,46),(23,67,47),(24,68,48),(25,69,49),(26,70,50),(27,71,51),(73,85,111),(74,86,112),(75,87,113),(76,88,114),(77,89,115),(78,90,116),(79,82,117),(80,83,109),(81,84,110),(100,118,130),(101,119,131),(102,120,132),(103,121,133),(104,122,134),(105,123,135),(106,124,127),(107,125,128),(108,126,129)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135)]])

165 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 5A 5B 5C 5D 9A ··· 9R 15A ··· 15AF 15AG ··· 15BD 45A ··· 45BT order 1 3 ··· 3 3 ··· 3 5 5 5 5 9 ··· 9 15 ··· 15 15 ··· 15 45 ··· 45 size 1 1 ··· 1 3 ··· 3 1 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3

165 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + image C1 C3 C3 C5 C9 C15 C15 C45 He3 3- 1+2 C5×He3 C5×3- 1+2 kernel C5×C32⋊C9 C3×C45 C32×C15 C32⋊C9 C3×C15 C3×C9 C33 C32 C15 C15 C3 C3 # reps 1 6 2 4 18 24 8 72 2 4 8 16

Matrix representation of C5×C32⋊C9 in GL4(𝔽181) generated by

 1 0 0 0 0 135 0 0 0 0 135 0 0 0 0 135
,
 1 0 0 0 0 132 0 0 0 0 48 0 0 0 0 1
,
 1 0 0 0 0 132 0 0 0 0 132 0 0 0 0 132
,
 39 0 0 0 0 0 1 0 0 0 0 1 0 48 0 0
G:=sub<GL(4,GF(181))| [1,0,0,0,0,135,0,0,0,0,135,0,0,0,0,135],[1,0,0,0,0,132,0,0,0,0,48,0,0,0,0,1],[1,0,0,0,0,132,0,0,0,0,132,0,0,0,0,132],[39,0,0,0,0,0,0,48,0,1,0,0,0,0,1,0] >;

C5×C32⋊C9 in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes C_9
% in TeX

G:=Group("C5xC3^2:C9");
// GroupNames label

G:=SmallGroup(405,3);
// by ID

G=gap.SmallGroup(405,3);
# by ID

G:=PCGroup([5,-3,-3,-5,-3,-3,675,481]);
// Polycyclic

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

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