direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C5×He3.C3, He3.C15, C15.3He3, 3- 1+2⋊2C15, (C3×C45)⋊2C3, (C3×C9)⋊2C15, (C5×He3).C3, C3.3(C5×He3), C32.2(C3×C15), (C3×C15).2C32, (C5×3- 1+2)⋊2C3, SmallGroup(405,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×He3.C3
G = < a,b,c,d,e | a5=b3=c3=d3=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >
Smallest permutation representation of C5×He3.C3
►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)(10 124 36 131 102)(11 125 28 132 103)(12 126 29 133 104)(13 118 30 134 105)(14 119 31 135 106)(15 120 32 127 107)(16 121 33 128 108)(17 122 34 129 100)(18 123 35 130 101)(55 94 73 109 88)(56 95 74 110 89)(57 96 75 111 90)(58 97 76 112 82)(59 98 77 113 83)(60 99 78 114 84)(61 91 79 115 85)(62 92 80 116 86)(63 93 81 117 87)
(10 16 13)(11 17 14)(12 18 15)(28 34 31)(29 35 32)(30 36 33)(55 58 61)(56 59 62)(57 60 63)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 106 103)(101 107 104)(102 108 105)(109 112 115)(110 113 116)(111 114 117)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)
(1 123 73)(2 121 74)(3 119 75)(4 126 76)(5 124 77)(6 122 78)(7 120 79)(8 118 80)(9 125 81)(10 98 52)(11 93 47)(12 97 51)(13 92 46)(14 96 50)(15 91 54)(16 95 49)(17 99 53)(18 94 48)(19 106 57)(20 104 58)(21 102 59)(22 100 60)(23 107 61)(24 105 62)(25 103 63)(26 101 55)(27 108 56)(28 117 64)(29 112 68)(30 116 72)(31 111 67)(32 115 71)(33 110 66)(34 114 70)(35 109 65)(36 113 69)(37 133 82)(38 131 83)(39 129 84)(40 127 85)(41 134 86)(42 132 87)(43 130 88)(44 128 89)(45 135 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)(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)| (1,65,43,26,48)(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)(10,124,36,131,102)(11,125,28,132,103)(12,126,29,133,104)(13,118,30,134,105)(14,119,31,135,106)(15,120,32,127,107)(16,121,33,128,108)(17,122,34,129,100)(18,123,35,130,101)(55,94,73,109,88)(56,95,74,110,89)(57,96,75,111,90)(58,97,76,112,82)(59,98,77,113,83)(60,99,78,114,84)(61,91,79,115,85)(62,92,80,116,86)(63,93,81,117,87), (10,16,13)(11,17,14)(12,18,15)(28,34,31)(29,35,32)(30,36,33)(55,58,61)(56,59,62)(57,60,63)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,106,103)(101,107,104)(102,108,105)(109,112,115)(110,113,116)(111,114,117)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135), (1,123,73)(2,121,74)(3,119,75)(4,126,76)(5,124,77)(6,122,78)(7,120,79)(8,118,80)(9,125,81)(10,98,52)(11,93,47)(12,97,51)(13,92,46)(14,96,50)(15,91,54)(16,95,49)(17,99,53)(18,94,48)(19,106,57)(20,104,58)(21,102,59)(22,100,60)(23,107,61)(24,105,62)(25,103,63)(26,101,55)(27,108,56)(28,117,64)(29,112,68)(30,116,72)(31,111,67)(32,115,71)(33,110,66)(34,114,70)(35,109,65)(36,113,69)(37,133,82)(38,131,83)(39,129,84)(40,127,85)(41,134,86)(42,132,87)(43,130,88)(44,128,89)(45,135,90), (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)(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( (1,65,43,26,48)(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)(10,124,36,131,102)(11,125,28,132,103)(12,126,29,133,104)(13,118,30,134,105)(14,119,31,135,106)(15,120,32,127,107)(16,121,33,128,108)(17,122,34,129,100)(18,123,35,130,101)(55,94,73,109,88)(56,95,74,110,89)(57,96,75,111,90)(58,97,76,112,82)(59,98,77,113,83)(60,99,78,114,84)(61,91,79,115,85)(62,92,80,116,86)(63,93,81,117,87), (10,16,13)(11,17,14)(12,18,15)(28,34,31)(29,35,32)(30,36,33)(55,58,61)(56,59,62)(57,60,63)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,106,103)(101,107,104)(102,108,105)(109,112,115)(110,113,116)(111,114,117)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135), (1,123,73)(2,121,74)(3,119,75)(4,126,76)(5,124,77)(6,122,78)(7,120,79)(8,118,80)(9,125,81)(10,98,52)(11,93,47)(12,97,51)(13,92,46)(14,96,50)(15,91,54)(16,95,49)(17,99,53)(18,94,48)(19,106,57)(20,104,58)(21,102,59)(22,100,60)(23,107,61)(24,105,62)(25,103,63)(26,101,55)(27,108,56)(28,117,64)(29,112,68)(30,116,72)(31,111,67)(32,115,71)(33,110,66)(34,114,70)(35,109,65)(36,113,69)(37,133,82)(38,131,83)(39,129,84)(40,127,85)(41,134,86)(42,132,87)(43,130,88)(44,128,89)(45,135,90), (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)(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([[(1,65,43,26,48),(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),(10,124,36,131,102),(11,125,28,132,103),(12,126,29,133,104),(13,118,30,134,105),(14,119,31,135,106),(15,120,32,127,107),(16,121,33,128,108),(17,122,34,129,100),(18,123,35,130,101),(55,94,73,109,88),(56,95,74,110,89),(57,96,75,111,90),(58,97,76,112,82),(59,98,77,113,83),(60,99,78,114,84),(61,91,79,115,85),(62,92,80,116,86),(63,93,81,117,87)], [(10,16,13),(11,17,14),(12,18,15),(28,34,31),(29,35,32),(30,36,33),(55,58,61),(56,59,62),(57,60,63),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,106,103),(101,107,104),(102,108,105),(109,112,115),(110,113,116),(111,114,117),(118,124,121),(119,125,122),(120,126,123),(127,133,130),(128,134,131),(129,135,132)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108),(109,112,115),(110,113,116),(111,114,117),(118,121,124),(119,122,125),(120,123,126),(127,130,133),(128,131,134),(129,132,135)], [(1,123,73),(2,121,74),(3,119,75),(4,126,76),(5,124,77),(6,122,78),(7,120,79),(8,118,80),(9,125,81),(10,98,52),(11,93,47),(12,97,51),(13,92,46),(14,96,50),(15,91,54),(16,95,49),(17,99,53),(18,94,48),(19,106,57),(20,104,58),(21,102,59),(22,100,60),(23,107,61),(24,105,62),(25,103,63),(26,101,55),(27,108,56),(28,117,64),(29,112,68),(30,116,72),(31,111,67),(32,115,71),(33,110,66),(34,114,70),(35,109,65),(36,113,69),(37,133,82),(38,131,83),(39,129,84),(40,127,85),(41,134,86),(42,132,87),(43,130,88),(44,128,89),(45,135,90)], [(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),(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)]])
85 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 15A | ··· | 15H | 15I | ··· | 15P | 15Q | ··· | 15X | 45A | ··· | 45X | 45Y | ··· | 45AN |
| order | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 15 | ··· | 15 | 15 | ··· | 15 | 15 | ··· | 15 | 45 | ··· | 45 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 3 | 3 | 9 | 9 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||||
| image | C1 | C3 | C3 | C3 | C5 | C15 | C15 | C15 | He3 | He3.C3 | C5×He3 | C5×He3.C3 |
| kernel | C5×He3.C3 | C3×C45 | C5×He3 | C5×3- 1+2 | He3.C3 | C3×C9 | He3 | 3- 1+2 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 2 | 6 | 8 | 24 |
Matrix representation of C5×He3.C3 ►in GL3(𝔽181) generated by
| 59 | 0 | 0 |
| 0 | 59 | 0 |
| 0 | 0 | 59 |
| 1 | 0 | 0 |
| 0 | 132 | 0 |
| 0 | 0 | 48 |
| 132 | 0 | 0 |
| 0 | 132 | 0 |
| 0 | 0 | 132 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 62 | 0 | 0 |
| 0 | 62 | 0 |
| 0 | 0 | 39 |
G:=sub<GL(3,GF(181))| [59,0,0,0,59,0,0,0,59],[1,0,0,0,132,0,0,0,48],[132,0,0,0,132,0,0,0,132],[0,0,1,1,0,0,0,1,0],[62,0,0,0,62,0,0,0,39] >;
C5×He3.C3 in GAP, Magma, Sage, TeX
C_5\times {\rm He}_3.C_3 % in TeX
G:=Group("C5xHe3.C3"); // GroupNames label
G:=SmallGroup(405,8);
// by ID
G=gap.SmallGroup(405,8);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,481,456,3603]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^3=1,e^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c^-1*d>;
// generators/relations
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