direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C15×He3, C33⋊2C15, C15.1C33, C32⋊(C3×C15), (C3×C15)⋊C32, (C32×C15)⋊2C3, C3.1(C32×C15), SmallGroup(405,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×He3
G = < a,b,c,d | a15=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Subgroups: 208 in 112 conjugacy classes, 64 normal (8 characteristic)
C1, C3, C3, C3, C5, C32, C32, C32, C15, C15, C15, He3, C33, C3×C15, C3×C15, C3×C15, C3×He3, C5×He3, C32×C15, C15×He3
Quotients: C1, C3, C5, C32, C15, He3, C33, C3×C15, C3×He3, C5×He3, C32×C15, C15×He3
►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)
(1 86 105)(2 87 91)(3 88 92)(4 89 93)(5 90 94)(6 76 95)(7 77 96)(8 78 97)(9 79 98)(10 80 99)(11 81 100)(12 82 101)(13 83 102)(14 84 103)(15 85 104)(16 33 126)(17 34 127)(18 35 128)(19 36 129)(20 37 130)(21 38 131)(22 39 132)(23 40 133)(24 41 134)(25 42 135)(26 43 121)(27 44 122)(28 45 123)(29 31 124)(30 32 125)(46 116 66)(47 117 67)(48 118 68)(49 119 69)(50 120 70)(51 106 71)(52 107 72)(53 108 73)(54 109 74)(55 110 75)(56 111 61)(57 112 62)(58 113 63)(59 114 64)(60 115 65)
(1 42 62)(2 43 63)(3 44 64)(4 45 65)(5 31 66)(6 32 67)(7 33 68)(8 34 69)(9 35 70)(10 36 71)(11 37 72)(12 38 73)(13 39 74)(14 40 75)(15 41 61)(16 118 96)(17 119 97)(18 120 98)(19 106 99)(20 107 100)(21 108 101)(22 109 102)(23 110 103)(24 111 104)(25 112 105)(26 113 91)(27 114 92)(28 115 93)(29 116 94)(30 117 95)(46 90 124)(47 76 125)(48 77 126)(49 78 127)(50 79 128)(51 80 129)(52 81 130)(53 82 131)(54 83 132)(55 84 133)(56 85 134)(57 86 135)(58 87 121)(59 88 122)(60 89 123)
(1 37 67)(2 38 68)(3 39 69)(4 40 70)(5 41 71)(6 42 72)(7 43 73)(8 44 74)(9 45 75)(10 31 61)(11 32 62)(12 33 63)(13 34 64)(14 35 65)(15 36 66)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(46 134 80)(47 135 81)(48 121 82)(49 122 83)(50 123 84)(51 124 85)(52 125 86)(53 126 87)(54 127 88)(55 128 89)(56 129 90)(57 130 76)(58 131 77)(59 132 78)(60 133 79)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
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), (1,86,105)(2,87,91)(3,88,92)(4,89,93)(5,90,94)(6,76,95)(7,77,96)(8,78,97)(9,79,98)(10,80,99)(11,81,100)(12,82,101)(13,83,102)(14,84,103)(15,85,104)(16,33,126)(17,34,127)(18,35,128)(19,36,129)(20,37,130)(21,38,131)(22,39,132)(23,40,133)(24,41,134)(25,42,135)(26,43,121)(27,44,122)(28,45,123)(29,31,124)(30,32,125)(46,116,66)(47,117,67)(48,118,68)(49,119,69)(50,120,70)(51,106,71)(52,107,72)(53,108,73)(54,109,74)(55,110,75)(56,111,61)(57,112,62)(58,113,63)(59,114,64)(60,115,65), (1,42,62)(2,43,63)(3,44,64)(4,45,65)(5,31,66)(6,32,67)(7,33,68)(8,34,69)(9,35,70)(10,36,71)(11,37,72)(12,38,73)(13,39,74)(14,40,75)(15,41,61)(16,118,96)(17,119,97)(18,120,98)(19,106,99)(20,107,100)(21,108,101)(22,109,102)(23,110,103)(24,111,104)(25,112,105)(26,113,91)(27,114,92)(28,115,93)(29,116,94)(30,117,95)(46,90,124)(47,76,125)(48,77,126)(49,78,127)(50,79,128)(51,80,129)(52,81,130)(53,82,131)(54,83,132)(55,84,133)(56,85,134)(57,86,135)(58,87,121)(59,88,122)(60,89,123), (1,37,67)(2,38,68)(3,39,69)(4,40,70)(5,41,71)(6,42,72)(7,43,73)(8,44,74)(9,45,75)(10,31,61)(11,32,62)(12,33,63)(13,34,64)(14,35,65)(15,36,66)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(46,134,80)(47,135,81)(48,121,82)(49,122,83)(50,123,84)(51,124,85)(52,125,86)(53,126,87)(54,127,88)(55,128,89)(56,129,90)(57,130,76)(58,131,77)(59,132,78)(60,133,79)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115)>;
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), (1,86,105)(2,87,91)(3,88,92)(4,89,93)(5,90,94)(6,76,95)(7,77,96)(8,78,97)(9,79,98)(10,80,99)(11,81,100)(12,82,101)(13,83,102)(14,84,103)(15,85,104)(16,33,126)(17,34,127)(18,35,128)(19,36,129)(20,37,130)(21,38,131)(22,39,132)(23,40,133)(24,41,134)(25,42,135)(26,43,121)(27,44,122)(28,45,123)(29,31,124)(30,32,125)(46,116,66)(47,117,67)(48,118,68)(49,119,69)(50,120,70)(51,106,71)(52,107,72)(53,108,73)(54,109,74)(55,110,75)(56,111,61)(57,112,62)(58,113,63)(59,114,64)(60,115,65), (1,42,62)(2,43,63)(3,44,64)(4,45,65)(5,31,66)(6,32,67)(7,33,68)(8,34,69)(9,35,70)(10,36,71)(11,37,72)(12,38,73)(13,39,74)(14,40,75)(15,41,61)(16,118,96)(17,119,97)(18,120,98)(19,106,99)(20,107,100)(21,108,101)(22,109,102)(23,110,103)(24,111,104)(25,112,105)(26,113,91)(27,114,92)(28,115,93)(29,116,94)(30,117,95)(46,90,124)(47,76,125)(48,77,126)(49,78,127)(50,79,128)(51,80,129)(52,81,130)(53,82,131)(54,83,132)(55,84,133)(56,85,134)(57,86,135)(58,87,121)(59,88,122)(60,89,123), (1,37,67)(2,38,68)(3,39,69)(4,40,70)(5,41,71)(6,42,72)(7,43,73)(8,44,74)(9,45,75)(10,31,61)(11,32,62)(12,33,63)(13,34,64)(14,35,65)(15,36,66)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(46,134,80)(47,135,81)(48,121,82)(49,122,83)(50,123,84)(51,124,85)(52,125,86)(53,126,87)(54,127,88)(55,128,89)(56,129,90)(57,130,76)(58,131,77)(59,132,78)(60,133,79)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115) );
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)], [(1,86,105),(2,87,91),(3,88,92),(4,89,93),(5,90,94),(6,76,95),(7,77,96),(8,78,97),(9,79,98),(10,80,99),(11,81,100),(12,82,101),(13,83,102),(14,84,103),(15,85,104),(16,33,126),(17,34,127),(18,35,128),(19,36,129),(20,37,130),(21,38,131),(22,39,132),(23,40,133),(24,41,134),(25,42,135),(26,43,121),(27,44,122),(28,45,123),(29,31,124),(30,32,125),(46,116,66),(47,117,67),(48,118,68),(49,119,69),(50,120,70),(51,106,71),(52,107,72),(53,108,73),(54,109,74),(55,110,75),(56,111,61),(57,112,62),(58,113,63),(59,114,64),(60,115,65)], [(1,42,62),(2,43,63),(3,44,64),(4,45,65),(5,31,66),(6,32,67),(7,33,68),(8,34,69),(9,35,70),(10,36,71),(11,37,72),(12,38,73),(13,39,74),(14,40,75),(15,41,61),(16,118,96),(17,119,97),(18,120,98),(19,106,99),(20,107,100),(21,108,101),(22,109,102),(23,110,103),(24,111,104),(25,112,105),(26,113,91),(27,114,92),(28,115,93),(29,116,94),(30,117,95),(46,90,124),(47,76,125),(48,77,126),(49,78,127),(50,79,128),(51,80,129),(52,81,130),(53,82,131),(54,83,132),(55,84,133),(56,85,134),(57,86,135),(58,87,121),(59,88,122),(60,89,123)], [(1,37,67),(2,38,68),(3,39,69),(4,40,70),(5,41,71),(6,42,72),(7,43,73),(8,44,74),(9,45,75),(10,31,61),(11,32,62),(12,33,63),(13,34,64),(14,35,65),(15,36,66),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(46,134,80),(47,135,81),(48,121,82),(49,122,83),(50,123,84),(51,124,85),(52,125,86),(53,126,87),(54,127,88),(55,128,89),(56,129,90),(57,130,76),(58,131,77),(59,132,78),(60,133,79),(91,101,96),(92,102,97),(93,103,98),(94,104,99),(95,105,100),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115)]])
165 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3AF | 5A | 5B | 5C | 5D | 15A | ··· | 15AF | 15AG | ··· | 15DX |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 15 | ··· | 15 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C5 | C15 | C15 | He3 | C5×He3 |
| kernel | C15×He3 | C5×He3 | C32×C15 | C3×He3 | He3 | C33 | C15 | C3 |
| # reps | 1 | 18 | 8 | 4 | 72 | 32 | 6 | 24 |
Matrix representation of C15×He3 ►in GL4(𝔽31) generated by
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 25 |
| 0 | 11 | 6 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 5 | 6 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 25 |
G:=sub<GL(4,GF(31))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[25,0,0,0,0,25,0,11,0,0,0,6,0,0,25,6],[1,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,5,0,0,0,5,1,0,0,6,0,25] >;
C15×He3 in GAP, Magma, Sage, TeX
C_{15}\times {\rm He}_3 % in TeX
G:=Group("C15xHe3"); // GroupNames label
G:=SmallGroup(405,12);
// by ID
G=gap.SmallGroup(405,12);
# by ID
G:=PCGroup([5,-3,-3,-3,-5,-3,1381]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^3=c^3=d^3=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