metabelian, supersoluble, monomial
Aliases: C62⋊3D5, C6.18D30, C30.50D6, (C6×C30)⋊2C2, (C2×C30)⋊6S3, (C2×C6)⋊4D15, (C3×C15)⋊22D4, C3⋊3(C15⋊7D4), C3⋊Dic15⋊1C2, (C3×C6).36D10, C5⋊3(C32⋊7D4), C15⋊12(C3⋊D4), C22⋊2(C3⋊D15), C32⋊10(C5⋊D4), (C3×C30).36C22, (C2×C3⋊D15)⋊2C2, C2.5(C2×C3⋊D15), (C2×C10)⋊4(C3⋊S3), C10.12(C2×C3⋊S3), SmallGroup(360,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊D5
G = < a,b,c,d | a6=b6=c5=d2=1, ab=ba, ac=ca, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 720 in 96 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, D6, C2×C6, C15, C3⋊S3, C3×C6, C3×C6, Dic5, D10, C2×C10, C3⋊D4, D15, C30, C30, C3⋊Dic3, C2×C3⋊S3, C62, C5⋊D4, C3×C15, Dic15, D30, C2×C30, C32⋊7D4, C3⋊D15, C3×C30, C3×C30, C15⋊7D4, C3⋊Dic15, C2×C3⋊D15, C6×C30, C62⋊D5
Quotients: C1, C2, C22, S3, D4, D5, D6, C3⋊S3, D10, C3⋊D4, D15, C2×C3⋊S3, C5⋊D4, D30, C32⋊7D4, C3⋊D15, C15⋊7D4, C2×C3⋊D15, C62⋊D5
►On 180 points
Generators in S180
(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)(151 152 153 154 155 156)(157 158 159 160 161 162)(163 164 165 166 167 168)(169 170 171 172 173 174)(175 176 177 178 179 180)
(1 126 83 43 145 141)(2 121 84 44 146 142)(3 122 79 45 147 143)(4 123 80 46 148 144)(5 124 81 47 149 139)(6 125 82 48 150 140)(7 51 168 99 96 62)(8 52 163 100 91 63)(9 53 164 101 92 64)(10 54 165 102 93 65)(11 49 166 97 94 66)(12 50 167 98 95 61)(13 158 68 89 131 30)(14 159 69 90 132 25)(15 160 70 85 127 26)(16 161 71 86 128 27)(17 162 72 87 129 28)(18 157 67 88 130 29)(19 155 77 178 115 104)(20 156 78 179 116 105)(21 151 73 180 117 106)(22 152 74 175 118 107)(23 153 75 176 119 108)(24 154 76 177 120 103)(31 111 136 172 58 42)(32 112 137 173 59 37)(33 113 138 174 60 38)(34 114 133 169 55 39)(35 109 134 170 56 40)(36 110 135 171 57 41)
(1 22 37 53 160)(2 23 38 54 161)(3 24 39 49 162)(4 19 40 50 157)(5 20 41 51 158)(6 21 42 52 159)(7 13 139 105 57)(8 14 140 106 58)(9 15 141 107 59)(10 16 142 108 60)(11 17 143 103 55)(12 18 144 104 56)(25 150 117 172 63)(26 145 118 173 64)(27 146 119 174 65)(28 147 120 169 66)(29 148 115 170 61)(30 149 116 171 62)(31 163 69 125 151)(32 164 70 126 152)(33 165 71 121 153)(34 166 72 122 154)(35 167 67 123 155)(36 168 68 124 156)(43 175 137 92 127)(44 176 138 93 128)(45 177 133 94 129)(46 178 134 95 130)(47 179 135 96 131)(48 180 136 91 132)(73 111 100 90 82)(74 112 101 85 83)(75 113 102 86 84)(76 114 97 87 79)(77 109 98 88 80)(78 110 99 89 81)
(1 160)(2 132)(3 158)(4 130)(5 162)(6 128)(7 154)(8 119)(9 152)(10 117)(11 156)(12 115)(13 122)(14 146)(15 126)(16 150)(17 124)(18 148)(19 95)(20 49)(21 93)(22 53)(23 91)(24 51)(25 142)(26 83)(27 140)(28 81)(29 144)(30 79)(31 113)(32 59)(33 111)(34 57)(35 109)(36 55)(38 136)(39 41)(40 134)(42 138)(43 127)(44 159)(45 131)(46 157)(47 129)(48 161)(50 178)(52 176)(54 180)(56 170)(58 174)(60 172)(61 104)(62 76)(63 108)(64 74)(65 106)(66 78)(67 80)(68 143)(69 84)(70 141)(71 82)(72 139)(73 165)(75 163)(77 167)(85 145)(86 125)(87 149)(88 123)(89 147)(90 121)(92 175)(94 179)(96 177)(97 116)(98 155)(99 120)(100 153)(101 118)(102 151)(103 168)(105 166)(107 164)(110 169)(112 173)(114 171)(133 135)
G:=sub<Sym(180)| (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)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180), (1,126,83,43,145,141)(2,121,84,44,146,142)(3,122,79,45,147,143)(4,123,80,46,148,144)(5,124,81,47,149,139)(6,125,82,48,150,140)(7,51,168,99,96,62)(8,52,163,100,91,63)(9,53,164,101,92,64)(10,54,165,102,93,65)(11,49,166,97,94,66)(12,50,167,98,95,61)(13,158,68,89,131,30)(14,159,69,90,132,25)(15,160,70,85,127,26)(16,161,71,86,128,27)(17,162,72,87,129,28)(18,157,67,88,130,29)(19,155,77,178,115,104)(20,156,78,179,116,105)(21,151,73,180,117,106)(22,152,74,175,118,107)(23,153,75,176,119,108)(24,154,76,177,120,103)(31,111,136,172,58,42)(32,112,137,173,59,37)(33,113,138,174,60,38)(34,114,133,169,55,39)(35,109,134,170,56,40)(36,110,135,171,57,41), (1,22,37,53,160)(2,23,38,54,161)(3,24,39,49,162)(4,19,40,50,157)(5,20,41,51,158)(6,21,42,52,159)(7,13,139,105,57)(8,14,140,106,58)(9,15,141,107,59)(10,16,142,108,60)(11,17,143,103,55)(12,18,144,104,56)(25,150,117,172,63)(26,145,118,173,64)(27,146,119,174,65)(28,147,120,169,66)(29,148,115,170,61)(30,149,116,171,62)(31,163,69,125,151)(32,164,70,126,152)(33,165,71,121,153)(34,166,72,122,154)(35,167,67,123,155)(36,168,68,124,156)(43,175,137,92,127)(44,176,138,93,128)(45,177,133,94,129)(46,178,134,95,130)(47,179,135,96,131)(48,180,136,91,132)(73,111,100,90,82)(74,112,101,85,83)(75,113,102,86,84)(76,114,97,87,79)(77,109,98,88,80)(78,110,99,89,81), (1,160)(2,132)(3,158)(4,130)(5,162)(6,128)(7,154)(8,119)(9,152)(10,117)(11,156)(12,115)(13,122)(14,146)(15,126)(16,150)(17,124)(18,148)(19,95)(20,49)(21,93)(22,53)(23,91)(24,51)(25,142)(26,83)(27,140)(28,81)(29,144)(30,79)(31,113)(32,59)(33,111)(34,57)(35,109)(36,55)(38,136)(39,41)(40,134)(42,138)(43,127)(44,159)(45,131)(46,157)(47,129)(48,161)(50,178)(52,176)(54,180)(56,170)(58,174)(60,172)(61,104)(62,76)(63,108)(64,74)(65,106)(66,78)(67,80)(68,143)(69,84)(70,141)(71,82)(72,139)(73,165)(75,163)(77,167)(85,145)(86,125)(87,149)(88,123)(89,147)(90,121)(92,175)(94,179)(96,177)(97,116)(98,155)(99,120)(100,153)(101,118)(102,151)(103,168)(105,166)(107,164)(110,169)(112,173)(114,171)(133,135)>;
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)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180), (1,126,83,43,145,141)(2,121,84,44,146,142)(3,122,79,45,147,143)(4,123,80,46,148,144)(5,124,81,47,149,139)(6,125,82,48,150,140)(7,51,168,99,96,62)(8,52,163,100,91,63)(9,53,164,101,92,64)(10,54,165,102,93,65)(11,49,166,97,94,66)(12,50,167,98,95,61)(13,158,68,89,131,30)(14,159,69,90,132,25)(15,160,70,85,127,26)(16,161,71,86,128,27)(17,162,72,87,129,28)(18,157,67,88,130,29)(19,155,77,178,115,104)(20,156,78,179,116,105)(21,151,73,180,117,106)(22,152,74,175,118,107)(23,153,75,176,119,108)(24,154,76,177,120,103)(31,111,136,172,58,42)(32,112,137,173,59,37)(33,113,138,174,60,38)(34,114,133,169,55,39)(35,109,134,170,56,40)(36,110,135,171,57,41), (1,22,37,53,160)(2,23,38,54,161)(3,24,39,49,162)(4,19,40,50,157)(5,20,41,51,158)(6,21,42,52,159)(7,13,139,105,57)(8,14,140,106,58)(9,15,141,107,59)(10,16,142,108,60)(11,17,143,103,55)(12,18,144,104,56)(25,150,117,172,63)(26,145,118,173,64)(27,146,119,174,65)(28,147,120,169,66)(29,148,115,170,61)(30,149,116,171,62)(31,163,69,125,151)(32,164,70,126,152)(33,165,71,121,153)(34,166,72,122,154)(35,167,67,123,155)(36,168,68,124,156)(43,175,137,92,127)(44,176,138,93,128)(45,177,133,94,129)(46,178,134,95,130)(47,179,135,96,131)(48,180,136,91,132)(73,111,100,90,82)(74,112,101,85,83)(75,113,102,86,84)(76,114,97,87,79)(77,109,98,88,80)(78,110,99,89,81), (1,160)(2,132)(3,158)(4,130)(5,162)(6,128)(7,154)(8,119)(9,152)(10,117)(11,156)(12,115)(13,122)(14,146)(15,126)(16,150)(17,124)(18,148)(19,95)(20,49)(21,93)(22,53)(23,91)(24,51)(25,142)(26,83)(27,140)(28,81)(29,144)(30,79)(31,113)(32,59)(33,111)(34,57)(35,109)(36,55)(38,136)(39,41)(40,134)(42,138)(43,127)(44,159)(45,131)(46,157)(47,129)(48,161)(50,178)(52,176)(54,180)(56,170)(58,174)(60,172)(61,104)(62,76)(63,108)(64,74)(65,106)(66,78)(67,80)(68,143)(69,84)(70,141)(71,82)(72,139)(73,165)(75,163)(77,167)(85,145)(86,125)(87,149)(88,123)(89,147)(90,121)(92,175)(94,179)(96,177)(97,116)(98,155)(99,120)(100,153)(101,118)(102,151)(103,168)(105,166)(107,164)(110,169)(112,173)(114,171)(133,135) );
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),(151,152,153,154,155,156),(157,158,159,160,161,162),(163,164,165,166,167,168),(169,170,171,172,173,174),(175,176,177,178,179,180)], [(1,126,83,43,145,141),(2,121,84,44,146,142),(3,122,79,45,147,143),(4,123,80,46,148,144),(5,124,81,47,149,139),(6,125,82,48,150,140),(7,51,168,99,96,62),(8,52,163,100,91,63),(9,53,164,101,92,64),(10,54,165,102,93,65),(11,49,166,97,94,66),(12,50,167,98,95,61),(13,158,68,89,131,30),(14,159,69,90,132,25),(15,160,70,85,127,26),(16,161,71,86,128,27),(17,162,72,87,129,28),(18,157,67,88,130,29),(19,155,77,178,115,104),(20,156,78,179,116,105),(21,151,73,180,117,106),(22,152,74,175,118,107),(23,153,75,176,119,108),(24,154,76,177,120,103),(31,111,136,172,58,42),(32,112,137,173,59,37),(33,113,138,174,60,38),(34,114,133,169,55,39),(35,109,134,170,56,40),(36,110,135,171,57,41)], [(1,22,37,53,160),(2,23,38,54,161),(3,24,39,49,162),(4,19,40,50,157),(5,20,41,51,158),(6,21,42,52,159),(7,13,139,105,57),(8,14,140,106,58),(9,15,141,107,59),(10,16,142,108,60),(11,17,143,103,55),(12,18,144,104,56),(25,150,117,172,63),(26,145,118,173,64),(27,146,119,174,65),(28,147,120,169,66),(29,148,115,170,61),(30,149,116,171,62),(31,163,69,125,151),(32,164,70,126,152),(33,165,71,121,153),(34,166,72,122,154),(35,167,67,123,155),(36,168,68,124,156),(43,175,137,92,127),(44,176,138,93,128),(45,177,133,94,129),(46,178,134,95,130),(47,179,135,96,131),(48,180,136,91,132),(73,111,100,90,82),(74,112,101,85,83),(75,113,102,86,84),(76,114,97,87,79),(77,109,98,88,80),(78,110,99,89,81)], [(1,160),(2,132),(3,158),(4,130),(5,162),(6,128),(7,154),(8,119),(9,152),(10,117),(11,156),(12,115),(13,122),(14,146),(15,126),(16,150),(17,124),(18,148),(19,95),(20,49),(21,93),(22,53),(23,91),(24,51),(25,142),(26,83),(27,140),(28,81),(29,144),(30,79),(31,113),(32,59),(33,111),(34,57),(35,109),(36,55),(38,136),(39,41),(40,134),(42,138),(43,127),(44,159),(45,131),(46,157),(47,129),(48,161),(50,178),(52,176),(54,180),(56,170),(58,174),(60,172),(61,104),(62,76),(63,108),(64,74),(65,106),(66,78),(67,80),(68,143),(69,84),(70,141),(71,82),(72,139),(73,165),(75,163),(77,167),(85,145),(86,125),(87,149),(88,123),(89,147),(90,121),(92,175),(94,179),(96,177),(97,116),(98,155),(99,120),(100,153),(101,118),(102,151),(103,168),(105,166),(107,164),(110,169),(112,173),(114,171),(133,135)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 5A | 5B | 6A | ··· | 6L | 10A | ··· | 10F | 15A | ··· | 15P | 30A | ··· | 30AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 90 | 2 | 2 | 2 | 2 | 90 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 |
| kernel | C62⋊D5 | C3⋊Dic15 | C2×C3⋊D15 | C6×C30 | C2×C30 | C3×C15 | C62 | C30 | C3×C6 | C15 | C2×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C62⋊D5 ►in GL4(𝔽61) generated by
| 10 | 39 | 0 | 0 |
| 14 | 24 | 0 | 0 |
| 0 | 0 | 24 | 33 |
| 0 | 0 | 28 | 36 |
| 37 | 45 | 0 | 0 |
| 49 | 25 | 0 | 0 |
| 0 | 0 | 36 | 28 |
| 0 | 0 | 33 | 24 |
| 44 | 1 | 0 | 0 |
| 16 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 17 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [10,14,0,0,39,24,0,0,0,0,24,28,0,0,33,36],[37,49,0,0,45,25,0,0,0,0,36,33,0,0,28,24],[44,16,0,0,1,60,0,0,0,0,0,60,0,0,1,17],[60,0,0,0,60,1,0,0,0,0,0,1,0,0,1,0] >;
C62⋊D5 in GAP, Magma, Sage, TeX
C_6^2\rtimes D_5
% in TeX
G:=Group("C6^2:D5"); // GroupNames label
G:=SmallGroup(360,114);
// by ID
G=gap.SmallGroup(360,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,387,1444,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^5=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations