direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×C33, C5⋊(C32×C6), (C3×C15)⋊7C6, C15⋊2(C3×C6), (C32×C15)⋊4C2, SmallGroup(270,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C33 |
Generators and relations for D5×C33
G = < a,b,c,d,e | a3=b3=c3=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 224 in 112 conjugacy classes, 84 normal (6 characteristic)
C1, C2, C3, C5, C6, C32, D5, C15, C3×C6, C33, C3×D5, C3×C15, C32×C6, C32×D5, C32×C15, D5×C33
Quotients: C1, C2, C3, C6, C32, D5, C3×C6, C33, C3×D5, C32×C6, C32×D5, D5×C33
►On 135 points
Generators in S135
(1 134 69)(2 135 70)(3 131 66)(4 132 67)(5 133 68)(6 121 71)(7 122 72)(8 123 73)(9 124 74)(10 125 75)(11 126 61)(12 127 62)(13 128 63)(14 129 64)(15 130 65)(16 101 81)(17 102 82)(18 103 83)(19 104 84)(20 105 85)(21 91 86)(22 92 87)(23 93 88)(24 94 89)(25 95 90)(26 96 76)(27 97 77)(28 98 78)(29 99 79)(30 100 80)(31 116 51)(32 117 52)(33 118 53)(34 119 54)(35 120 55)(36 106 56)(37 107 57)(38 108 58)(39 109 59)(40 110 60)(41 111 46)(42 112 47)(43 113 48)(44 114 49)(45 115 50)
(1 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)(46 76 61)(47 77 62)(48 78 63)(49 79 64)(50 80 65)(51 81 66)(52 82 67)(53 83 68)(54 84 69)(55 85 70)(56 86 71)(57 87 72)(58 88 73)(59 89 74)(60 90 75)(91 121 106)(92 122 107)(93 123 108)(94 124 109)(95 125 110)(96 126 111)(97 127 112)(98 128 113)(99 129 114)(100 130 115)(101 131 116)(102 132 117)(103 133 118)(104 134 119)(105 135 120)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(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)(121 131 126)(122 132 127)(123 133 128)(124 134 129)(125 135 130)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)(122 125)(123 124)(127 130)(128 129)(132 135)(133 134)
G:=sub<Sym(135)| (1,134,69)(2,135,70)(3,131,66)(4,132,67)(5,133,68)(6,121,71)(7,122,72)(8,123,73)(9,124,74)(10,125,75)(11,126,61)(12,127,62)(13,128,63)(14,129,64)(15,130,65)(16,101,81)(17,102,82)(18,103,83)(19,104,84)(20,105,85)(21,91,86)(22,92,87)(23,93,88)(24,94,89)(25,95,90)(26,96,76)(27,97,77)(28,98,78)(29,99,79)(30,100,80)(31,116,51)(32,117,52)(33,118,53)(34,119,54)(35,120,55)(36,106,56)(37,107,57)(38,108,58)(39,109,59)(40,110,60)(41,111,46)(42,112,47)(43,113,48)(44,114,49)(45,115,50), (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30)(46,76,61)(47,77,62)(48,78,63)(49,79,64)(50,80,65)(51,81,66)(52,82,67)(53,83,68)(54,84,69)(55,85,70)(56,86,71)(57,87,72)(58,88,73)(59,89,74)(60,90,75)(91,121,106)(92,122,107)(93,123,108)(94,124,109)(95,125,110)(96,126,111)(97,127,112)(98,128,113)(99,129,114)(100,130,115)(101,131,116)(102,132,117)(103,133,118)(104,134,119)(105,135,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(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)(121,131,126)(122,132,127)(123,133,128)(124,134,129)(125,135,130), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(122,125)(123,124)(127,130)(128,129)(132,135)(133,134)>;
G:=Group( (1,134,69)(2,135,70)(3,131,66)(4,132,67)(5,133,68)(6,121,71)(7,122,72)(8,123,73)(9,124,74)(10,125,75)(11,126,61)(12,127,62)(13,128,63)(14,129,64)(15,130,65)(16,101,81)(17,102,82)(18,103,83)(19,104,84)(20,105,85)(21,91,86)(22,92,87)(23,93,88)(24,94,89)(25,95,90)(26,96,76)(27,97,77)(28,98,78)(29,99,79)(30,100,80)(31,116,51)(32,117,52)(33,118,53)(34,119,54)(35,120,55)(36,106,56)(37,107,57)(38,108,58)(39,109,59)(40,110,60)(41,111,46)(42,112,47)(43,113,48)(44,114,49)(45,115,50), (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30)(46,76,61)(47,77,62)(48,78,63)(49,79,64)(50,80,65)(51,81,66)(52,82,67)(53,83,68)(54,84,69)(55,85,70)(56,86,71)(57,87,72)(58,88,73)(59,89,74)(60,90,75)(91,121,106)(92,122,107)(93,123,108)(94,124,109)(95,125,110)(96,126,111)(97,127,112)(98,128,113)(99,129,114)(100,130,115)(101,131,116)(102,132,117)(103,133,118)(104,134,119)(105,135,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(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)(121,131,126)(122,132,127)(123,133,128)(124,134,129)(125,135,130), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(122,125)(123,124)(127,130)(128,129)(132,135)(133,134) );
G=PermutationGroup([[(1,134,69),(2,135,70),(3,131,66),(4,132,67),(5,133,68),(6,121,71),(7,122,72),(8,123,73),(9,124,74),(10,125,75),(11,126,61),(12,127,62),(13,128,63),(14,129,64),(15,130,65),(16,101,81),(17,102,82),(18,103,83),(19,104,84),(20,105,85),(21,91,86),(22,92,87),(23,93,88),(24,94,89),(25,95,90),(26,96,76),(27,97,77),(28,98,78),(29,99,79),(30,100,80),(31,116,51),(32,117,52),(33,118,53),(34,119,54),(35,120,55),(36,106,56),(37,107,57),(38,108,58),(39,109,59),(40,110,60),(41,111,46),(42,112,47),(43,113,48),(44,114,49),(45,115,50)], [(1,34,19),(2,35,20),(3,31,16),(4,32,17),(5,33,18),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30),(46,76,61),(47,77,62),(48,78,63),(49,79,64),(50,80,65),(51,81,66),(52,82,67),(53,83,68),(54,84,69),(55,85,70),(56,86,71),(57,87,72),(58,88,73),(59,89,74),(60,90,75),(91,121,106),(92,122,107),(93,123,108),(94,124,109),(95,125,110),(96,126,111),(97,127,112),(98,128,113),(99,129,114),(100,130,115),(101,131,116),(102,132,117),(103,133,118),(104,134,119),(105,135,120)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(61,71,66),(62,72,67),(63,73,68),(64,74,69),(65,75,70),(76,86,81),(77,87,82),(78,88,83),(79,89,84),(80,90,85),(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),(121,131,126),(122,132,127),(123,133,128),(124,134,129),(125,135,130)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(62,65),(63,64),(67,70),(68,69),(72,75),(73,74),(77,80),(78,79),(82,85),(83,84),(87,90),(88,89),(92,95),(93,94),(97,100),(98,99),(102,105),(103,104),(107,110),(108,109),(112,115),(113,114),(117,120),(118,119),(122,125),(123,124),(127,130),(128,129),(132,135),(133,134)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 5A | 5B | 6A | ··· | 6Z | 15A | ··· | 15AZ |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 6 | ··· | 6 | 15 | ··· | 15 |
| size | 1 | 5 | 1 | ··· | 1 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D5 | C3×D5 |
| kernel | D5×C33 | C32×C15 | C32×D5 | C3×C15 | C33 | C32 |
| # reps | 1 | 1 | 26 | 26 | 2 | 52 |
Matrix representation of D5×C33 ►in GL4(𝔽31) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 25 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 25 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 30 | 18 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(31))| [25,0,0,0,0,25,0,0,0,0,5,0,0,0,0,5],[25,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[25,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,0,30,0,0,1,18],[30,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0] >;
D5×C33 in GAP, Magma, Sage, TeX
D_5\times C_3^3
% in TeX
G:=Group("D5xC3^3"); // GroupNames label
G:=SmallGroup(270,23);
// by ID
G=gap.SmallGroup(270,23);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-5,5404]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^5=e^2=1,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,c*e=e*c,e*d*e=d^-1>;
// generators/relations