direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C5⋊D15, C10⋊D15, C5⋊2D30, C30⋊1D5, C52⋊8D6, C15⋊6D10, C6⋊(C5⋊D5), (C5×C30)⋊1C2, (C5×C10)⋊3S3, (C5×C15)⋊6C22, C3⋊2(C2×C5⋊D5), SmallGroup(300,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C2×C5⋊D15 |
Generators and relations for C2×C5⋊D15
G = < a,b,c,d | a2=b5=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 704 in 80 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, D5, C10, D6, C15, D10, C52, D15, C30, C5⋊D5, C5×C10, D30, C5×C15, C2×C5⋊D5, C5⋊D15, C5×C30, C2×C5⋊D15
Quotients: C1, C2, C22, S3, D5, D6, D10, D15, C5⋊D5, D30, C2×C5⋊D5, C5⋊D15, C2×C5⋊D15
►On 150 points
Generators in S150
(1 87)(2 88)(3 89)(4 90)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 146)(17 147)(18 148)(19 149)(20 150)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)(61 103)(62 104)(63 105)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
(1 57 26 95 109)(2 58 27 96 110)(3 59 28 97 111)(4 60 29 98 112)(5 46 30 99 113)(6 47 16 100 114)(7 48 17 101 115)(8 49 18 102 116)(9 50 19 103 117)(10 51 20 104 118)(11 52 21 105 119)(12 53 22 91 120)(13 54 23 92 106)(14 55 24 93 107)(15 56 25 94 108)(31 138 65 129 84)(32 139 66 130 85)(33 140 67 131 86)(34 141 68 132 87)(35 142 69 133 88)(36 143 70 134 89)(37 144 71 135 90)(38 145 72 121 76)(39 146 73 122 77)(40 147 74 123 78)(41 148 75 124 79)(42 149 61 125 80)(43 150 62 126 81)(44 136 63 127 82)(45 137 64 128 83)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 105)(31 134)(32 133)(33 132)(34 131)(35 130)(36 129)(37 128)(38 127)(39 126)(40 125)(41 124)(42 123)(43 122)(44 121)(45 135)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 120)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 140)(69 139)(70 138)(71 137)(72 136)(73 150)(74 149)(75 148)(76 82)(77 81)(78 80)(83 90)(84 89)(85 88)(86 87)
G:=sub<Sym(150)| (1,87)(2,88)(3,89)(4,90)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,146)(17,147)(18,148)(19,149)(20,150)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)(45,53)(61,103)(62,104)(63,105)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128), (1,57,26,95,109)(2,58,27,96,110)(3,59,28,97,111)(4,60,29,98,112)(5,46,30,99,113)(6,47,16,100,114)(7,48,17,101,115)(8,49,18,102,116)(9,50,19,103,117)(10,51,20,104,118)(11,52,21,105,119)(12,53,22,91,120)(13,54,23,92,106)(14,55,24,93,107)(15,56,25,94,108)(31,138,65,129,84)(32,139,66,130,85)(33,140,67,131,86)(34,141,68,132,87)(35,142,69,133,88)(36,143,70,134,89)(37,144,71,135,90)(38,145,72,121,76)(39,146,73,122,77)(40,147,74,123,78)(41,148,75,124,79)(42,149,61,125,80)(43,150,62,126,81)(44,136,63,127,82)(45,137,64,128,83), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,105)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,135)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,120)(61,147)(62,146)(63,145)(64,144)(65,143)(66,142)(67,141)(68,140)(69,139)(70,138)(71,137)(72,136)(73,150)(74,149)(75,148)(76,82)(77,81)(78,80)(83,90)(84,89)(85,88)(86,87)>;
G:=Group( (1,87)(2,88)(3,89)(4,90)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,146)(17,147)(18,148)(19,149)(20,150)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)(45,53)(61,103)(62,104)(63,105)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128), (1,57,26,95,109)(2,58,27,96,110)(3,59,28,97,111)(4,60,29,98,112)(5,46,30,99,113)(6,47,16,100,114)(7,48,17,101,115)(8,49,18,102,116)(9,50,19,103,117)(10,51,20,104,118)(11,52,21,105,119)(12,53,22,91,120)(13,54,23,92,106)(14,55,24,93,107)(15,56,25,94,108)(31,138,65,129,84)(32,139,66,130,85)(33,140,67,131,86)(34,141,68,132,87)(35,142,69,133,88)(36,143,70,134,89)(37,144,71,135,90)(38,145,72,121,76)(39,146,73,122,77)(40,147,74,123,78)(41,148,75,124,79)(42,149,61,125,80)(43,150,62,126,81)(44,136,63,127,82)(45,137,64,128,83), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,105)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,135)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,120)(61,147)(62,146)(63,145)(64,144)(65,143)(66,142)(67,141)(68,140)(69,139)(70,138)(71,137)(72,136)(73,150)(74,149)(75,148)(76,82)(77,81)(78,80)(83,90)(84,89)(85,88)(86,87) );
G=PermutationGroup([[(1,87),(2,88),(3,89),(4,90),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,146),(17,147),(18,148),(19,149),(20,150),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(28,143),(29,144),(30,145),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,46),(39,47),(40,48),(41,49),(42,50),(43,51),(44,52),(45,53),(61,103),(62,104),(63,105),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128)], [(1,57,26,95,109),(2,58,27,96,110),(3,59,28,97,111),(4,60,29,98,112),(5,46,30,99,113),(6,47,16,100,114),(7,48,17,101,115),(8,49,18,102,116),(9,50,19,103,117),(10,51,20,104,118),(11,52,21,105,119),(12,53,22,91,120),(13,54,23,92,106),(14,55,24,93,107),(15,56,25,94,108),(31,138,65,129,84),(32,139,66,130,85),(33,140,67,131,86),(34,141,68,132,87),(35,142,69,133,88),(36,143,70,134,89),(37,144,71,135,90),(38,145,72,121,76),(39,146,73,122,77),(40,147,74,123,78),(41,148,75,124,79),(42,149,61,125,80),(43,150,62,126,81),(44,136,63,127,82),(45,137,64,128,83)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,105),(31,134),(32,133),(33,132),(34,131),(35,130),(36,129),(37,128),(38,127),(39,126),(40,125),(41,124),(42,123),(43,122),(44,121),(45,135),(46,119),(47,118),(48,117),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,108),(58,107),(59,106),(60,120),(61,147),(62,146),(63,145),(64,144),(65,143),(66,142),(67,141),(68,140),(69,139),(70,138),(71,137),(72,136),(73,150),(74,149),(75,148),(76,82),(77,81),(78,80),(83,90),(84,89),(85,88),(86,87)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | ··· | 5L | 6 | 10A | ··· | 10L | 15A | ··· | 15X | 30A | ··· | 30X |
| order | 1 | 2 | 2 | 2 | 3 | 5 | ··· | 5 | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 75 | 75 | 2 | 2 | ··· | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D30 |
| kernel | C2×C5⋊D15 | C5⋊D15 | C5×C30 | C5×C10 | C30 | C52 | C15 | C10 | C5 |
| # reps | 1 | 2 | 1 | 1 | 12 | 1 | 12 | 24 | 24 |
Matrix representation of C2×C5⋊D15 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 30 | 1 | 0 | 0 |
| 11 | 19 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 30 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 4 |
| 0 | 0 | 27 | 22 |
| 0 | 13 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 20 | 9 |
| 0 | 0 | 28 | 11 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[30,11,0,0,1,19,0,0,0,0,0,30,0,0,1,18],[1,0,0,0,0,1,0,0,0,0,12,27,0,0,4,22],[0,12,0,0,13,0,0,0,0,0,20,28,0,0,9,11] >;
C2×C5⋊D15 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes D_{15} % in TeX
G:=Group("C2xC5:D15"); // GroupNames label
G:=SmallGroup(300,48);
// by ID
G=gap.SmallGroup(300,48);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,122,963,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^15=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