direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C5×C9○He3, C45.C32, He3.2C15, C15.3C33, 3- 1+2⋊3C15, C9.(C3×C15), (C3×C45)⋊5C3, (C3×C9)⋊5C15, (C5×He3).2C3, C32.6(C3×C15), (C3×C15).5C32, C3.3(C32×C15), (C5×3- 1+2)⋊3C3, SmallGroup(405,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C9○He3
G = < a,b,c,d,e | a5=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >
Subgroups: 82 in 66 conjugacy classes, 58 normal (10 characteristic)
C1, C3, C3, C5, C9, C9, C32, C15, C15, C3×C9, He3, 3- 1+2, C45, C45, C3×C15, C9○He3, C3×C45, C5×He3, C5×3- 1+2, C5×C9○He3
Quotients: C1, C3, C5, C32, C15, C33, C3×C15, C9○He3, C32×C15, C5×C9○He3
►On 135 points
Generators in S135
(1 65 43 26 48)(2 66 44 27 49)(3 67 45 19 50)(4 68 37 20 51)(5 69 38 21 52)(6 70 39 22 53)(7 71 40 23 54)(8 72 41 24 46)(9 64 42 25 47)(10 122 36 128 104)(11 123 28 129 105)(12 124 29 130 106)(13 125 30 131 107)(14 126 31 132 108)(15 118 32 133 100)(16 119 33 134 101)(17 120 34 135 102)(18 121 35 127 103)(55 93 73 110 88)(56 94 74 111 89)(57 95 75 112 90)(58 96 76 113 82)(59 97 77 114 83)(60 98 78 115 84)(61 99 79 116 85)(62 91 80 117 86)(63 92 81 109 87)
(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)
(1 73 124)(2 74 125)(3 75 126)(4 76 118)(5 77 119)(6 78 120)(7 79 121)(8 80 122)(9 81 123)(10 46 91)(11 47 92)(12 48 93)(13 49 94)(14 50 95)(15 51 96)(16 52 97)(17 53 98)(18 54 99)(19 57 108)(20 58 100)(21 59 101)(22 60 102)(23 61 103)(24 62 104)(25 63 105)(26 55 106)(27 56 107)(28 64 109)(29 65 110)(30 66 111)(31 67 112)(32 68 113)(33 69 114)(34 70 115)(35 71 116)(36 72 117)(37 82 133)(38 83 134)(39 84 135)(40 85 127)(41 86 128)(42 87 129)(43 88 130)(44 89 131)(45 90 132)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)
(10 13 16)(11 14 17)(12 15 18)(28 31 34)(29 32 35)(30 33 36)(55 61 58)(56 62 59)(57 63 60)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 103 106)(101 104 107)(102 105 108)(109 115 112)(110 116 113)(111 117 114)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)
G:=sub<Sym(135)| (1,65,43,26,48)(2,66,44,27,49)(3,67,45,19,50)(4,68,37,20,51)(5,69,38,21,52)(6,70,39,22,53)(7,71,40,23,54)(8,72,41,24,46)(9,64,42,25,47)(10,122,36,128,104)(11,123,28,129,105)(12,124,29,130,106)(13,125,30,131,107)(14,126,31,132,108)(15,118,32,133,100)(16,119,33,134,101)(17,120,34,135,102)(18,121,35,127,103)(55,93,73,110,88)(56,94,74,111,89)(57,95,75,112,90)(58,96,76,113,82)(59,97,77,114,83)(60,98,78,115,84)(61,99,79,116,85)(62,91,80,117,86)(63,92,81,109,87), (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), (1,73,124)(2,74,125)(3,75,126)(4,76,118)(5,77,119)(6,78,120)(7,79,121)(8,80,122)(9,81,123)(10,46,91)(11,47,92)(12,48,93)(13,49,94)(14,50,95)(15,51,96)(16,52,97)(17,53,98)(18,54,99)(19,57,108)(20,58,100)(21,59,101)(22,60,102)(23,61,103)(24,62,104)(25,63,105)(26,55,106)(27,56,107)(28,64,109)(29,65,110)(30,66,111)(31,67,112)(32,68,113)(33,69,114)(34,70,115)(35,71,116)(36,72,117)(37,82,133)(38,83,134)(39,84,135)(40,85,127)(41,86,128)(42,87,129)(43,88,130)(44,89,131)(45,90,132), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132), (10,13,16)(11,14,17)(12,15,18)(28,31,34)(29,32,35)(30,33,36)(55,61,58)(56,62,59)(57,63,60)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,103,106)(101,104,107)(102,105,108)(109,115,112)(110,116,113)(111,117,114)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)>;
G:=Group( (1,65,43,26,48)(2,66,44,27,49)(3,67,45,19,50)(4,68,37,20,51)(5,69,38,21,52)(6,70,39,22,53)(7,71,40,23,54)(8,72,41,24,46)(9,64,42,25,47)(10,122,36,128,104)(11,123,28,129,105)(12,124,29,130,106)(13,125,30,131,107)(14,126,31,132,108)(15,118,32,133,100)(16,119,33,134,101)(17,120,34,135,102)(18,121,35,127,103)(55,93,73,110,88)(56,94,74,111,89)(57,95,75,112,90)(58,96,76,113,82)(59,97,77,114,83)(60,98,78,115,84)(61,99,79,116,85)(62,91,80,117,86)(63,92,81,109,87), (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), (1,73,124)(2,74,125)(3,75,126)(4,76,118)(5,77,119)(6,78,120)(7,79,121)(8,80,122)(9,81,123)(10,46,91)(11,47,92)(12,48,93)(13,49,94)(14,50,95)(15,51,96)(16,52,97)(17,53,98)(18,54,99)(19,57,108)(20,58,100)(21,59,101)(22,60,102)(23,61,103)(24,62,104)(25,63,105)(26,55,106)(27,56,107)(28,64,109)(29,65,110)(30,66,111)(31,67,112)(32,68,113)(33,69,114)(34,70,115)(35,71,116)(36,72,117)(37,82,133)(38,83,134)(39,84,135)(40,85,127)(41,86,128)(42,87,129)(43,88,130)(44,89,131)(45,90,132), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132), (10,13,16)(11,14,17)(12,15,18)(28,31,34)(29,32,35)(30,33,36)(55,61,58)(56,62,59)(57,63,60)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,103,106)(101,104,107)(102,105,108)(109,115,112)(110,116,113)(111,117,114)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135) );
G=PermutationGroup([[(1,65,43,26,48),(2,66,44,27,49),(3,67,45,19,50),(4,68,37,20,51),(5,69,38,21,52),(6,70,39,22,53),(7,71,40,23,54),(8,72,41,24,46),(9,64,42,25,47),(10,122,36,128,104),(11,123,28,129,105),(12,124,29,130,106),(13,125,30,131,107),(14,126,31,132,108),(15,118,32,133,100),(16,119,33,134,101),(17,120,34,135,102),(18,121,35,127,103),(55,93,73,110,88),(56,94,74,111,89),(57,95,75,112,90),(58,96,76,113,82),(59,97,77,114,83),(60,98,78,115,84),(61,99,79,116,85),(62,91,80,117,86),(63,92,81,109,87)], [(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)], [(1,73,124),(2,74,125),(3,75,126),(4,76,118),(5,77,119),(6,78,120),(7,79,121),(8,80,122),(9,81,123),(10,46,91),(11,47,92),(12,48,93),(13,49,94),(14,50,95),(15,51,96),(16,52,97),(17,53,98),(18,54,99),(19,57,108),(20,58,100),(21,59,101),(22,60,102),(23,61,103),(24,62,104),(25,63,105),(26,55,106),(27,56,107),(28,64,109),(29,65,110),(30,66,111),(31,67,112),(32,68,113),(33,69,114),(34,70,115),(35,71,116),(36,72,117),(37,82,133),(38,83,134),(39,84,135),(40,85,127),(41,86,128),(42,87,129),(43,88,130),(44,89,131),(45,90,132)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105),(109,115,112),(110,116,113),(111,117,114),(118,124,121),(119,125,122),(120,126,123),(127,133,130),(128,134,131),(129,135,132)], [(10,13,16),(11,14,17),(12,15,18),(28,31,34),(29,32,35),(30,33,36),(55,61,58),(56,62,59),(57,63,60),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,103,106),(101,104,107),(102,105,108),(109,115,112),(110,116,113),(111,117,114),(118,121,124),(119,122,125),(120,123,126),(127,130,133),(128,131,134),(129,132,135)]])
165 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 9G | ··· | 9V | 15A | ··· | 15H | 15I | ··· | 15AN | 45A | ··· | 45X | 45Y | ··· | 45CJ |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 9 | ··· | 9 | 15 | ··· | 15 | 15 | ··· | 15 | 45 | ··· | 45 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C3 | C3 | C5 | C15 | C15 | C15 | C9○He3 | C5×C9○He3 |
| kernel | C5×C9○He3 | C3×C45 | C5×He3 | C5×3- 1+2 | C9○He3 | C3×C9 | He3 | 3- 1+2 | C5 | C1 |
| # reps | 1 | 8 | 2 | 16 | 4 | 32 | 8 | 64 | 6 | 24 |
Matrix representation of C5×C9○He3 ►in GL3(𝔽181) generated by
| 125 | 0 | 0 |
| 0 | 125 | 0 |
| 0 | 0 | 125 |
| 65 | 0 | 0 |
| 0 | 65 | 0 |
| 0 | 0 | 65 |
| 55 | 106 | 92 |
| 131 | 126 | 57 |
| 0 | 132 | 0 |
| 132 | 0 | 0 |
| 0 | 132 | 0 |
| 0 | 0 | 132 |
| 1 | 55 | 21 |
| 0 | 132 | 0 |
| 0 | 0 | 48 |
G:=sub<GL(3,GF(181))| [125,0,0,0,125,0,0,0,125],[65,0,0,0,65,0,0,0,65],[55,131,0,106,126,132,92,57,0],[132,0,0,0,132,0,0,0,132],[1,0,0,55,132,0,21,0,48] >;
C5×C9○He3 in GAP, Magma, Sage, TeX
C_5\times C_9\circ {\rm He}_3 % in TeX
G:=Group("C5xC9oHe3"); // GroupNames label
G:=SmallGroup(405,14);
// by ID
G=gap.SmallGroup(405,14);
# by ID
G:=PCGroup([5,-3,-3,-3,-5,-3,1381,237]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^9=c^3=e^3=1,d^1=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations