direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C5×He3⋊C3, He3⋊2C15, C15.4He3, (C3×C45)⋊3C3, (C3×C9)⋊3C15, (C5×He3)⋊2C3, C3.4(C5×He3), C32.3(C3×C15), (C3×C15).3C32, SmallGroup(405,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×He3⋊C3
G = < a,b,c,d,e | a5=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=bcd >
Smallest permutation representation of C5×He3⋊C3
►On 135 points
Generators in S135
(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)
(6 134 15)(7 135 11)(8 131 12)(9 132 13)(10 133 14)(56 61 68)(57 62 69)(58 63 70)(59 64 66)(60 65 67)(71 76 83)(72 77 84)(73 78 85)(74 79 81)(75 80 82)(86 91 98)(87 92 99)(88 93 100)(89 94 96)(90 95 97)(101 113 106)(102 114 107)(103 115 108)(104 111 109)(105 112 110)(116 128 121)(117 129 122)(118 130 123)(119 126 124)(120 127 125)
(1 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)(6 15 134)(7 11 135)(8 12 131)(9 13 132)(10 14 133)(26 31 38)(27 32 39)(28 33 40)(29 34 36)(30 35 37)(41 46 53)(42 47 54)(43 48 55)(44 49 51)(45 50 52)(56 61 68)(57 62 69)(58 63 70)(59 64 66)(60 65 67)(71 76 83)(72 77 84)(73 78 85)(74 79 81)(75 80 82)(86 91 98)(87 92 99)(88 93 100)(89 94 96)(90 95 97)(101 106 113)(102 107 114)(103 108 115)(104 109 111)(105 110 112)(116 121 128)(117 122 129)(118 123 130)(119 124 126)(120 125 127)
(1 8 76)(2 9 77)(3 10 78)(4 6 79)(5 7 80)(11 82 22)(12 83 23)(13 84 24)(14 85 25)(15 81 21)(16 135 75)(17 131 71)(18 132 72)(19 133 73)(20 134 74)(26 106 86)(27 107 87)(28 108 88)(29 109 89)(30 110 90)(31 113 91)(32 114 92)(33 115 93)(34 111 94)(35 112 95)(36 104 96)(37 105 97)(38 101 98)(39 102 99)(40 103 100)(41 121 61)(42 122 62)(43 123 63)(44 124 64)(45 125 65)(46 128 68)(47 129 69)(48 130 70)(49 126 66)(50 127 67)(51 119 59)(52 120 60)(53 116 56)(54 117 57)(55 118 58)
(1 128 91)(2 129 92)(3 130 93)(4 126 94)(5 127 95)(6 64 29)(7 65 30)(8 61 26)(9 62 27)(10 63 28)(11 67 35)(12 68 31)(13 69 32)(14 70 33)(15 66 34)(16 125 90)(17 121 86)(18 122 87)(19 123 88)(20 124 89)(21 119 96)(22 120 97)(23 116 98)(24 117 99)(25 118 100)(36 134 59)(37 135 60)(38 131 56)(39 132 57)(40 133 58)(41 113 76)(42 114 77)(43 115 78)(44 111 79)(45 112 80)(46 101 83)(47 102 84)(48 103 85)(49 104 81)(50 105 82)(51 109 74)(52 110 75)(53 106 71)(54 107 72)(55 108 73)
G:=sub<Sym(135)| (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), (6,134,15)(7,135,11)(8,131,12)(9,132,13)(10,133,14)(56,61,68)(57,62,69)(58,63,70)(59,64,66)(60,65,67)(71,76,83)(72,77,84)(73,78,85)(74,79,81)(75,80,82)(86,91,98)(87,92,99)(88,93,100)(89,94,96)(90,95,97)(101,113,106)(102,114,107)(103,115,108)(104,111,109)(105,112,110)(116,128,121)(117,129,122)(118,130,123)(119,126,124)(120,127,125), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,134)(7,11,135)(8,12,131)(9,13,132)(10,14,133)(26,31,38)(27,32,39)(28,33,40)(29,34,36)(30,35,37)(41,46,53)(42,47,54)(43,48,55)(44,49,51)(45,50,52)(56,61,68)(57,62,69)(58,63,70)(59,64,66)(60,65,67)(71,76,83)(72,77,84)(73,78,85)(74,79,81)(75,80,82)(86,91,98)(87,92,99)(88,93,100)(89,94,96)(90,95,97)(101,106,113)(102,107,114)(103,108,115)(104,109,111)(105,110,112)(116,121,128)(117,122,129)(118,123,130)(119,124,126)(120,125,127), (1,8,76)(2,9,77)(3,10,78)(4,6,79)(5,7,80)(11,82,22)(12,83,23)(13,84,24)(14,85,25)(15,81,21)(16,135,75)(17,131,71)(18,132,72)(19,133,73)(20,134,74)(26,106,86)(27,107,87)(28,108,88)(29,109,89)(30,110,90)(31,113,91)(32,114,92)(33,115,93)(34,111,94)(35,112,95)(36,104,96)(37,105,97)(38,101,98)(39,102,99)(40,103,100)(41,121,61)(42,122,62)(43,123,63)(44,124,64)(45,125,65)(46,128,68)(47,129,69)(48,130,70)(49,126,66)(50,127,67)(51,119,59)(52,120,60)(53,116,56)(54,117,57)(55,118,58), (1,128,91)(2,129,92)(3,130,93)(4,126,94)(5,127,95)(6,64,29)(7,65,30)(8,61,26)(9,62,27)(10,63,28)(11,67,35)(12,68,31)(13,69,32)(14,70,33)(15,66,34)(16,125,90)(17,121,86)(18,122,87)(19,123,88)(20,124,89)(21,119,96)(22,120,97)(23,116,98)(24,117,99)(25,118,100)(36,134,59)(37,135,60)(38,131,56)(39,132,57)(40,133,58)(41,113,76)(42,114,77)(43,115,78)(44,111,79)(45,112,80)(46,101,83)(47,102,84)(48,103,85)(49,104,81)(50,105,82)(51,109,74)(52,110,75)(53,106,71)(54,107,72)(55,108,73)>;
G:=Group( (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), (6,134,15)(7,135,11)(8,131,12)(9,132,13)(10,133,14)(56,61,68)(57,62,69)(58,63,70)(59,64,66)(60,65,67)(71,76,83)(72,77,84)(73,78,85)(74,79,81)(75,80,82)(86,91,98)(87,92,99)(88,93,100)(89,94,96)(90,95,97)(101,113,106)(102,114,107)(103,115,108)(104,111,109)(105,112,110)(116,128,121)(117,129,122)(118,130,123)(119,126,124)(120,127,125), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,134)(7,11,135)(8,12,131)(9,13,132)(10,14,133)(26,31,38)(27,32,39)(28,33,40)(29,34,36)(30,35,37)(41,46,53)(42,47,54)(43,48,55)(44,49,51)(45,50,52)(56,61,68)(57,62,69)(58,63,70)(59,64,66)(60,65,67)(71,76,83)(72,77,84)(73,78,85)(74,79,81)(75,80,82)(86,91,98)(87,92,99)(88,93,100)(89,94,96)(90,95,97)(101,106,113)(102,107,114)(103,108,115)(104,109,111)(105,110,112)(116,121,128)(117,122,129)(118,123,130)(119,124,126)(120,125,127), (1,8,76)(2,9,77)(3,10,78)(4,6,79)(5,7,80)(11,82,22)(12,83,23)(13,84,24)(14,85,25)(15,81,21)(16,135,75)(17,131,71)(18,132,72)(19,133,73)(20,134,74)(26,106,86)(27,107,87)(28,108,88)(29,109,89)(30,110,90)(31,113,91)(32,114,92)(33,115,93)(34,111,94)(35,112,95)(36,104,96)(37,105,97)(38,101,98)(39,102,99)(40,103,100)(41,121,61)(42,122,62)(43,123,63)(44,124,64)(45,125,65)(46,128,68)(47,129,69)(48,130,70)(49,126,66)(50,127,67)(51,119,59)(52,120,60)(53,116,56)(54,117,57)(55,118,58), (1,128,91)(2,129,92)(3,130,93)(4,126,94)(5,127,95)(6,64,29)(7,65,30)(8,61,26)(9,62,27)(10,63,28)(11,67,35)(12,68,31)(13,69,32)(14,70,33)(15,66,34)(16,125,90)(17,121,86)(18,122,87)(19,123,88)(20,124,89)(21,119,96)(22,120,97)(23,116,98)(24,117,99)(25,118,100)(36,134,59)(37,135,60)(38,131,56)(39,132,57)(40,133,58)(41,113,76)(42,114,77)(43,115,78)(44,111,79)(45,112,80)(46,101,83)(47,102,84)(48,103,85)(49,104,81)(50,105,82)(51,109,74)(52,110,75)(53,106,71)(54,107,72)(55,108,73) );
G=PermutationGroup([[(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)], [(6,134,15),(7,135,11),(8,131,12),(9,132,13),(10,133,14),(56,61,68),(57,62,69),(58,63,70),(59,64,66),(60,65,67),(71,76,83),(72,77,84),(73,78,85),(74,79,81),(75,80,82),(86,91,98),(87,92,99),(88,93,100),(89,94,96),(90,95,97),(101,113,106),(102,114,107),(103,115,108),(104,111,109),(105,112,110),(116,128,121),(117,129,122),(118,130,123),(119,126,124),(120,127,125)], [(1,23,17),(2,24,18),(3,25,19),(4,21,20),(5,22,16),(6,15,134),(7,11,135),(8,12,131),(9,13,132),(10,14,133),(26,31,38),(27,32,39),(28,33,40),(29,34,36),(30,35,37),(41,46,53),(42,47,54),(43,48,55),(44,49,51),(45,50,52),(56,61,68),(57,62,69),(58,63,70),(59,64,66),(60,65,67),(71,76,83),(72,77,84),(73,78,85),(74,79,81),(75,80,82),(86,91,98),(87,92,99),(88,93,100),(89,94,96),(90,95,97),(101,106,113),(102,107,114),(103,108,115),(104,109,111),(105,110,112),(116,121,128),(117,122,129),(118,123,130),(119,124,126),(120,125,127)], [(1,8,76),(2,9,77),(3,10,78),(4,6,79),(5,7,80),(11,82,22),(12,83,23),(13,84,24),(14,85,25),(15,81,21),(16,135,75),(17,131,71),(18,132,72),(19,133,73),(20,134,74),(26,106,86),(27,107,87),(28,108,88),(29,109,89),(30,110,90),(31,113,91),(32,114,92),(33,115,93),(34,111,94),(35,112,95),(36,104,96),(37,105,97),(38,101,98),(39,102,99),(40,103,100),(41,121,61),(42,122,62),(43,123,63),(44,124,64),(45,125,65),(46,128,68),(47,129,69),(48,130,70),(49,126,66),(50,127,67),(51,119,59),(52,120,60),(53,116,56),(54,117,57),(55,118,58)], [(1,128,91),(2,129,92),(3,130,93),(4,126,94),(5,127,95),(6,64,29),(7,65,30),(8,61,26),(9,62,27),(10,63,28),(11,67,35),(12,68,31),(13,69,32),(14,70,33),(15,66,34),(16,125,90),(17,121,86),(18,122,87),(19,123,88),(20,124,89),(21,119,96),(22,120,97),(23,116,98),(24,117,99),(25,118,100),(36,134,59),(37,135,60),(38,131,56),(39,132,57),(40,133,58),(41,113,76),(42,114,77),(43,115,78),(44,111,79),(45,112,80),(46,101,83),(47,102,84),(48,103,85),(49,104,81),(50,105,82),(51,109,74),(52,110,75),(53,106,71),(54,107,72),(55,108,73)]])
85 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 15A | ··· | 15H | 15I | ··· | 15P | 15Q | ··· | 15AN | 45A | ··· | 45X |
| order | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 15 | ··· | 15 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 3 | 3 | 9 | ··· | 9 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C3 | C5 | C15 | C15 | He3 | He3⋊C3 | C5×He3 | C5×He3⋊C3 |
| kernel | C5×He3⋊C3 | C3×C45 | C5×He3 | He3⋊C3 | C3×C9 | He3 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 6 | 4 | 8 | 24 | 2 | 6 | 8 | 24 |
Matrix representation of C5×He3⋊C3 ►in GL3(𝔽181) generated by
| 135 | 0 | 0 |
| 0 | 135 | 0 |
| 0 | 0 | 135 |
| 1 | 0 | 0 |
| 48 | 48 | 0 |
| 133 | 0 | 132 |
| 48 | 0 | 0 |
| 0 | 48 | 0 |
| 0 | 0 | 48 |
| 39 | 140 | 0 |
| 0 | 142 | 80 |
| 80 | 119 | 0 |
| 43 | 159 | 0 |
| 0 | 138 | 65 |
| 108 | 108 | 0 |
G:=sub<GL(3,GF(181))| [135,0,0,0,135,0,0,0,135],[1,48,133,0,48,0,0,0,132],[48,0,0,0,48,0,0,0,48],[39,0,80,140,142,119,0,80,0],[43,0,108,159,138,108,0,65,0] >;
C5×He3⋊C3 in GAP, Magma, Sage, TeX
C_5\times {\rm He}_3\rtimes C_3 % in TeX
G:=Group("C5xHe3:C3"); // GroupNames label
G:=SmallGroup(405,9);
// by ID
G=gap.SmallGroup(405,9);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,481,906,3603]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations
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