direct product, metabelian, supersoluble, monomial, A-group
Aliases: C6×C5⋊D5, C30⋊2D5, C15⋊7D10, C10⋊(C3×D5), C5⋊2(C6×D5), (C5×C10)⋊4C6, (C5×C30)⋊4C2, C52⋊5(C2×C6), (C5×C15)⋊9C22, SmallGroup(300,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C6×C5⋊D5 |
Generators and relations for C6×C5⋊D5
G = < a,b,c,d | a6=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 368 in 80 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C3, C22, C5, C6, C6, D5, C10, C2×C6, C15, D10, C52, C3×D5, C30, C5⋊D5, C5×C10, C6×D5, C5×C15, C2×C5⋊D5, C3×C5⋊D5, C5×C30, C6×C5⋊D5
Quotients: C1, C2, C3, C22, C6, D5, C2×C6, D10, C3×D5, C5⋊D5, C6×D5, C2×C5⋊D5, C3×C5⋊D5, C6×C5⋊D5
►On 150 points
Generators in S150
(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)(133 134 135 136 137 138)(139 140 141 142 143 144)(145 146 147 148 149 150)
(1 103 86 135 118)(2 104 87 136 119)(3 105 88 137 120)(4 106 89 138 115)(5 107 90 133 116)(6 108 85 134 117)(7 95 141 75 36)(8 96 142 76 31)(9 91 143 77 32)(10 92 144 78 33)(11 93 139 73 34)(12 94 140 74 35)(13 68 98 59 42)(14 69 99 60 37)(15 70 100 55 38)(16 71 101 56 39)(17 72 102 57 40)(18 67 97 58 41)(19 28 81 62 48)(20 29 82 63 43)(21 30 83 64 44)(22 25 84 65 45)(23 26 79 66 46)(24 27 80 61 47)(49 146 131 114 124)(50 147 132 109 125)(51 148 127 110 126)(52 149 128 111 121)(53 150 129 112 122)(54 145 130 113 123)
(1 72 111 9 80)(2 67 112 10 81)(3 68 113 11 82)(4 69 114 12 83)(5 70 109 7 84)(6 71 110 8 79)(13 130 34 29 120)(14 131 35 30 115)(15 132 36 25 116)(16 127 31 26 117)(17 128 32 27 118)(18 129 33 28 119)(19 136 41 150 78)(20 137 42 145 73)(21 138 37 146 74)(22 133 38 147 75)(23 134 39 148 76)(24 135 40 149 77)(43 88 59 54 139)(44 89 60 49 140)(45 90 55 50 141)(46 85 56 51 142)(47 86 57 52 143)(48 87 58 53 144)(61 103 102 121 91)(62 104 97 122 92)(63 105 98 123 93)(64 106 99 124 94)(65 107 100 125 95)(66 108 101 126 96)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 79)(7 70)(8 71)(9 72)(10 67)(11 68)(12 69)(13 93)(14 94)(15 95)(16 96)(17 91)(18 92)(19 87)(20 88)(21 89)(22 90)(23 85)(24 86)(25 107)(26 108)(27 103)(28 104)(29 105)(30 106)(31 101)(32 102)(33 97)(34 98)(35 99)(36 100)(37 140)(38 141)(39 142)(40 143)(41 144)(42 139)(43 137)(44 138)(45 133)(46 134)(47 135)(48 136)(49 146)(50 147)(51 148)(52 149)(53 150)(54 145)(55 75)(56 76)(57 77)(58 78)(59 73)(60 74)(61 118)(62 119)(63 120)(64 115)(65 116)(66 117)(121 128)(122 129)(123 130)(124 131)(125 132)(126 127)
G:=sub<Sym(150)| (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)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150), (1,103,86,135,118)(2,104,87,136,119)(3,105,88,137,120)(4,106,89,138,115)(5,107,90,133,116)(6,108,85,134,117)(7,95,141,75,36)(8,96,142,76,31)(9,91,143,77,32)(10,92,144,78,33)(11,93,139,73,34)(12,94,140,74,35)(13,68,98,59,42)(14,69,99,60,37)(15,70,100,55,38)(16,71,101,56,39)(17,72,102,57,40)(18,67,97,58,41)(19,28,81,62,48)(20,29,82,63,43)(21,30,83,64,44)(22,25,84,65,45)(23,26,79,66,46)(24,27,80,61,47)(49,146,131,114,124)(50,147,132,109,125)(51,148,127,110,126)(52,149,128,111,121)(53,150,129,112,122)(54,145,130,113,123), (1,72,111,9,80)(2,67,112,10,81)(3,68,113,11,82)(4,69,114,12,83)(5,70,109,7,84)(6,71,110,8,79)(13,130,34,29,120)(14,131,35,30,115)(15,132,36,25,116)(16,127,31,26,117)(17,128,32,27,118)(18,129,33,28,119)(19,136,41,150,78)(20,137,42,145,73)(21,138,37,146,74)(22,133,38,147,75)(23,134,39,148,76)(24,135,40,149,77)(43,88,59,54,139)(44,89,60,49,140)(45,90,55,50,141)(46,85,56,51,142)(47,86,57,52,143)(48,87,58,53,144)(61,103,102,121,91)(62,104,97,122,92)(63,105,98,123,93)(64,106,99,124,94)(65,107,100,125,95)(66,108,101,126,96), (1,80)(2,81)(3,82)(4,83)(5,84)(6,79)(7,70)(8,71)(9,72)(10,67)(11,68)(12,69)(13,93)(14,94)(15,95)(16,96)(17,91)(18,92)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,107)(26,108)(27,103)(28,104)(29,105)(30,106)(31,101)(32,102)(33,97)(34,98)(35,99)(36,100)(37,140)(38,141)(39,142)(40,143)(41,144)(42,139)(43,137)(44,138)(45,133)(46,134)(47,135)(48,136)(49,146)(50,147)(51,148)(52,149)(53,150)(54,145)(55,75)(56,76)(57,77)(58,78)(59,73)(60,74)(61,118)(62,119)(63,120)(64,115)(65,116)(66,117)(121,128)(122,129)(123,130)(124,131)(125,132)(126,127)>;
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)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150), (1,103,86,135,118)(2,104,87,136,119)(3,105,88,137,120)(4,106,89,138,115)(5,107,90,133,116)(6,108,85,134,117)(7,95,141,75,36)(8,96,142,76,31)(9,91,143,77,32)(10,92,144,78,33)(11,93,139,73,34)(12,94,140,74,35)(13,68,98,59,42)(14,69,99,60,37)(15,70,100,55,38)(16,71,101,56,39)(17,72,102,57,40)(18,67,97,58,41)(19,28,81,62,48)(20,29,82,63,43)(21,30,83,64,44)(22,25,84,65,45)(23,26,79,66,46)(24,27,80,61,47)(49,146,131,114,124)(50,147,132,109,125)(51,148,127,110,126)(52,149,128,111,121)(53,150,129,112,122)(54,145,130,113,123), (1,72,111,9,80)(2,67,112,10,81)(3,68,113,11,82)(4,69,114,12,83)(5,70,109,7,84)(6,71,110,8,79)(13,130,34,29,120)(14,131,35,30,115)(15,132,36,25,116)(16,127,31,26,117)(17,128,32,27,118)(18,129,33,28,119)(19,136,41,150,78)(20,137,42,145,73)(21,138,37,146,74)(22,133,38,147,75)(23,134,39,148,76)(24,135,40,149,77)(43,88,59,54,139)(44,89,60,49,140)(45,90,55,50,141)(46,85,56,51,142)(47,86,57,52,143)(48,87,58,53,144)(61,103,102,121,91)(62,104,97,122,92)(63,105,98,123,93)(64,106,99,124,94)(65,107,100,125,95)(66,108,101,126,96), (1,80)(2,81)(3,82)(4,83)(5,84)(6,79)(7,70)(8,71)(9,72)(10,67)(11,68)(12,69)(13,93)(14,94)(15,95)(16,96)(17,91)(18,92)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,107)(26,108)(27,103)(28,104)(29,105)(30,106)(31,101)(32,102)(33,97)(34,98)(35,99)(36,100)(37,140)(38,141)(39,142)(40,143)(41,144)(42,139)(43,137)(44,138)(45,133)(46,134)(47,135)(48,136)(49,146)(50,147)(51,148)(52,149)(53,150)(54,145)(55,75)(56,76)(57,77)(58,78)(59,73)(60,74)(61,118)(62,119)(63,120)(64,115)(65,116)(66,117)(121,128)(122,129)(123,130)(124,131)(125,132)(126,127) );
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),(133,134,135,136,137,138),(139,140,141,142,143,144),(145,146,147,148,149,150)], [(1,103,86,135,118),(2,104,87,136,119),(3,105,88,137,120),(4,106,89,138,115),(5,107,90,133,116),(6,108,85,134,117),(7,95,141,75,36),(8,96,142,76,31),(9,91,143,77,32),(10,92,144,78,33),(11,93,139,73,34),(12,94,140,74,35),(13,68,98,59,42),(14,69,99,60,37),(15,70,100,55,38),(16,71,101,56,39),(17,72,102,57,40),(18,67,97,58,41),(19,28,81,62,48),(20,29,82,63,43),(21,30,83,64,44),(22,25,84,65,45),(23,26,79,66,46),(24,27,80,61,47),(49,146,131,114,124),(50,147,132,109,125),(51,148,127,110,126),(52,149,128,111,121),(53,150,129,112,122),(54,145,130,113,123)], [(1,72,111,9,80),(2,67,112,10,81),(3,68,113,11,82),(4,69,114,12,83),(5,70,109,7,84),(6,71,110,8,79),(13,130,34,29,120),(14,131,35,30,115),(15,132,36,25,116),(16,127,31,26,117),(17,128,32,27,118),(18,129,33,28,119),(19,136,41,150,78),(20,137,42,145,73),(21,138,37,146,74),(22,133,38,147,75),(23,134,39,148,76),(24,135,40,149,77),(43,88,59,54,139),(44,89,60,49,140),(45,90,55,50,141),(46,85,56,51,142),(47,86,57,52,143),(48,87,58,53,144),(61,103,102,121,91),(62,104,97,122,92),(63,105,98,123,93),(64,106,99,124,94),(65,107,100,125,95),(66,108,101,126,96)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,79),(7,70),(8,71),(9,72),(10,67),(11,68),(12,69),(13,93),(14,94),(15,95),(16,96),(17,91),(18,92),(19,87),(20,88),(21,89),(22,90),(23,85),(24,86),(25,107),(26,108),(27,103),(28,104),(29,105),(30,106),(31,101),(32,102),(33,97),(34,98),(35,99),(36,100),(37,140),(38,141),(39,142),(40,143),(41,144),(42,139),(43,137),(44,138),(45,133),(46,134),(47,135),(48,136),(49,146),(50,147),(51,148),(52,149),(53,150),(54,145),(55,75),(56,76),(57,77),(58,78),(59,73),(60,74),(61,118),(62,119),(63,120),(64,115),(65,116),(66,117),(121,128),(122,129),(123,130),(124,131),(125,132),(126,127)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | ··· | 5L | 6A | 6B | 6C | 6D | 6E | 6F | 10A | ··· | 10L | 15A | ··· | 15X | 30A | ··· | 30X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 25 | 25 | 25 | 25 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 |
| kernel | C6×C5⋊D5 | C3×C5⋊D5 | C5×C30 | C2×C5⋊D5 | C5⋊D5 | C5×C10 | C30 | C15 | C10 | C5 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 12 | 12 | 24 | 24 |
Matrix representation of C6×C5⋊D5 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 1 |
| 0 | 0 | 17 | 13 |
| 0 | 1 | 0 | 0 |
| 30 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 |
| 0 | 0 | 18 | 13 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,30,17,0,0,1,13],[0,30,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,18,18,0,0,1,13] >;
C6×C5⋊D5 in GAP, Magma, Sage, TeX
C_6\times C_5\rtimes D_5
% in TeX
G:=Group("C6xC5:D5"); // GroupNames label
G:=SmallGroup(300,45);
// by ID
G=gap.SmallGroup(300,45);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,963,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations