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
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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