metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.150D10, C10.282- 1+4, C10.1322+ 1+4, (C4×D20)⋊47C2, C42.C2⋊6D5, C4⋊C4.207D10, C42⋊2D5⋊9C2, D20⋊8C4⋊36C2, D10⋊Q8⋊35C2, D10⋊2Q8⋊37C2, C4⋊D20.13C2, D10.17(C4○D4), (C2×C20).188C23, (C2×C10).236C24, (C4×C20).196C22, D10.13D4⋊34C2, C2.57(D4⋊8D10), Dic5.Q8⋊33C2, (C2×D20).170C22, C4⋊Dic5.314C22, C22.257(C23×D5), D10⋊C4.10C22, C5⋊8(C22.33C24), (C4×Dic5).151C22, (C2×Dic5).268C23, C10.D4.52C22, (C22×D5).102C23, C2.29(Q8.10D10), (C2×Dic10).186C22, (D5×C4⋊C4)⋊36C2, C2.87(D5×C4○D4), C4⋊C4⋊D5⋊34C2, (C5×C42.C2)⋊9C2, C10.198(C2×C4○D4), (C2×C4×D5).135C22, (C2×C4).80(C22×D5), (C5×C4⋊C4).191C22, SmallGroup(320,1364)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.150D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 830 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, D10, D10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C22.33C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4×D20, C42⋊2D5, Dic5.Q8, D5×C4⋊C4, D20⋊8C4, D10.13D4, C4⋊D20, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42.C2, C42.150D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.33C24, C23×D5, Q8.10D10, D5×C4○D4, D4⋊8D10, C42.150D10
►On 160 points
Generators in S160
(1 144 80 37)(2 155 61 28)(3 146 62 39)(4 157 63 30)(5 148 64 21)(6 159 65 32)(7 150 66 23)(8 141 67 34)(9 152 68 25)(10 143 69 36)(11 154 70 27)(12 145 71 38)(13 156 72 29)(14 147 73 40)(15 158 74 31)(16 149 75 22)(17 160 76 33)(18 151 77 24)(19 142 78 35)(20 153 79 26)(41 100 109 137)(42 91 110 128)(43 82 111 139)(44 93 112 130)(45 84 113 121)(46 95 114 132)(47 86 115 123)(48 97 116 134)(49 88 117 125)(50 99 118 136)(51 90 119 127)(52 81 120 138)(53 92 101 129)(54 83 102 140)(55 94 103 131)(56 85 104 122)(57 96 105 133)(58 87 106 124)(59 98 107 135)(60 89 108 126)
(1 48 70 106)(2 107 71 49)(3 50 72 108)(4 109 73 51)(5 52 74 110)(6 111 75 53)(7 54 76 112)(8 113 77 55)(9 56 78 114)(10 115 79 57)(11 58 80 116)(12 117 61 59)(13 60 62 118)(14 119 63 41)(15 42 64 120)(16 101 65 43)(17 44 66 102)(18 103 67 45)(19 46 68 104)(20 105 69 47)(21 138 158 91)(22 92 159 139)(23 140 160 93)(24 94 141 121)(25 122 142 95)(26 96 143 123)(27 124 144 97)(28 98 145 125)(29 126 146 99)(30 100 147 127)(31 128 148 81)(32 82 149 129)(33 130 150 83)(34 84 151 131)(35 132 152 85)(36 86 153 133)(37 134 154 87)(38 88 155 135)(39 136 156 89)(40 90 157 137)
(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)
(1 96 11 86)(2 85 12 95)(3 94 13 84)(4 83 14 93)(5 92 15 82)(6 81 16 91)(7 90 17 100)(8 99 18 89)(9 88 19 98)(10 97 20 87)(21 43 31 53)(22 52 32 42)(23 41 33 51)(24 50 34 60)(25 59 35 49)(26 48 36 58)(27 57 37 47)(28 46 38 56)(29 55 39 45)(30 44 40 54)(61 122 71 132)(62 131 72 121)(63 140 73 130)(64 129 74 139)(65 138 75 128)(66 127 76 137)(67 136 77 126)(68 125 78 135)(69 134 79 124)(70 123 80 133)(101 148 111 158)(102 157 112 147)(103 146 113 156)(104 155 114 145)(105 144 115 154)(106 153 116 143)(107 142 117 152)(108 151 118 141)(109 160 119 150)(110 149 120 159)
G:=sub<Sym(160)| (1,144,80,37)(2,155,61,28)(3,146,62,39)(4,157,63,30)(5,148,64,21)(6,159,65,32)(7,150,66,23)(8,141,67,34)(9,152,68,25)(10,143,69,36)(11,154,70,27)(12,145,71,38)(13,156,72,29)(14,147,73,40)(15,158,74,31)(16,149,75,22)(17,160,76,33)(18,151,77,24)(19,142,78,35)(20,153,79,26)(41,100,109,137)(42,91,110,128)(43,82,111,139)(44,93,112,130)(45,84,113,121)(46,95,114,132)(47,86,115,123)(48,97,116,134)(49,88,117,125)(50,99,118,136)(51,90,119,127)(52,81,120,138)(53,92,101,129)(54,83,102,140)(55,94,103,131)(56,85,104,122)(57,96,105,133)(58,87,106,124)(59,98,107,135)(60,89,108,126), (1,48,70,106)(2,107,71,49)(3,50,72,108)(4,109,73,51)(5,52,74,110)(6,111,75,53)(7,54,76,112)(8,113,77,55)(9,56,78,114)(10,115,79,57)(11,58,80,116)(12,117,61,59)(13,60,62,118)(14,119,63,41)(15,42,64,120)(16,101,65,43)(17,44,66,102)(18,103,67,45)(19,46,68,104)(20,105,69,47)(21,138,158,91)(22,92,159,139)(23,140,160,93)(24,94,141,121)(25,122,142,95)(26,96,143,123)(27,124,144,97)(28,98,145,125)(29,126,146,99)(30,100,147,127)(31,128,148,81)(32,82,149,129)(33,130,150,83)(34,84,151,131)(35,132,152,85)(36,86,153,133)(37,134,154,87)(38,88,155,135)(39,136,156,89)(40,90,157,137), (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), (1,96,11,86)(2,85,12,95)(3,94,13,84)(4,83,14,93)(5,92,15,82)(6,81,16,91)(7,90,17,100)(8,99,18,89)(9,88,19,98)(10,97,20,87)(21,43,31,53)(22,52,32,42)(23,41,33,51)(24,50,34,60)(25,59,35,49)(26,48,36,58)(27,57,37,47)(28,46,38,56)(29,55,39,45)(30,44,40,54)(61,122,71,132)(62,131,72,121)(63,140,73,130)(64,129,74,139)(65,138,75,128)(66,127,76,137)(67,136,77,126)(68,125,78,135)(69,134,79,124)(70,123,80,133)(101,148,111,158)(102,157,112,147)(103,146,113,156)(104,155,114,145)(105,144,115,154)(106,153,116,143)(107,142,117,152)(108,151,118,141)(109,160,119,150)(110,149,120,159)>;
G:=Group( (1,144,80,37)(2,155,61,28)(3,146,62,39)(4,157,63,30)(5,148,64,21)(6,159,65,32)(7,150,66,23)(8,141,67,34)(9,152,68,25)(10,143,69,36)(11,154,70,27)(12,145,71,38)(13,156,72,29)(14,147,73,40)(15,158,74,31)(16,149,75,22)(17,160,76,33)(18,151,77,24)(19,142,78,35)(20,153,79,26)(41,100,109,137)(42,91,110,128)(43,82,111,139)(44,93,112,130)(45,84,113,121)(46,95,114,132)(47,86,115,123)(48,97,116,134)(49,88,117,125)(50,99,118,136)(51,90,119,127)(52,81,120,138)(53,92,101,129)(54,83,102,140)(55,94,103,131)(56,85,104,122)(57,96,105,133)(58,87,106,124)(59,98,107,135)(60,89,108,126), (1,48,70,106)(2,107,71,49)(3,50,72,108)(4,109,73,51)(5,52,74,110)(6,111,75,53)(7,54,76,112)(8,113,77,55)(9,56,78,114)(10,115,79,57)(11,58,80,116)(12,117,61,59)(13,60,62,118)(14,119,63,41)(15,42,64,120)(16,101,65,43)(17,44,66,102)(18,103,67,45)(19,46,68,104)(20,105,69,47)(21,138,158,91)(22,92,159,139)(23,140,160,93)(24,94,141,121)(25,122,142,95)(26,96,143,123)(27,124,144,97)(28,98,145,125)(29,126,146,99)(30,100,147,127)(31,128,148,81)(32,82,149,129)(33,130,150,83)(34,84,151,131)(35,132,152,85)(36,86,153,133)(37,134,154,87)(38,88,155,135)(39,136,156,89)(40,90,157,137), (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), (1,96,11,86)(2,85,12,95)(3,94,13,84)(4,83,14,93)(5,92,15,82)(6,81,16,91)(7,90,17,100)(8,99,18,89)(9,88,19,98)(10,97,20,87)(21,43,31,53)(22,52,32,42)(23,41,33,51)(24,50,34,60)(25,59,35,49)(26,48,36,58)(27,57,37,47)(28,46,38,56)(29,55,39,45)(30,44,40,54)(61,122,71,132)(62,131,72,121)(63,140,73,130)(64,129,74,139)(65,138,75,128)(66,127,76,137)(67,136,77,126)(68,125,78,135)(69,134,79,124)(70,123,80,133)(101,148,111,158)(102,157,112,147)(103,146,113,156)(104,155,114,145)(105,144,115,154)(106,153,116,143)(107,142,117,152)(108,151,118,141)(109,160,119,150)(110,149,120,159) );
G=PermutationGroup([[(1,144,80,37),(2,155,61,28),(3,146,62,39),(4,157,63,30),(5,148,64,21),(6,159,65,32),(7,150,66,23),(8,141,67,34),(9,152,68,25),(10,143,69,36),(11,154,70,27),(12,145,71,38),(13,156,72,29),(14,147,73,40),(15,158,74,31),(16,149,75,22),(17,160,76,33),(18,151,77,24),(19,142,78,35),(20,153,79,26),(41,100,109,137),(42,91,110,128),(43,82,111,139),(44,93,112,130),(45,84,113,121),(46,95,114,132),(47,86,115,123),(48,97,116,134),(49,88,117,125),(50,99,118,136),(51,90,119,127),(52,81,120,138),(53,92,101,129),(54,83,102,140),(55,94,103,131),(56,85,104,122),(57,96,105,133),(58,87,106,124),(59,98,107,135),(60,89,108,126)], [(1,48,70,106),(2,107,71,49),(3,50,72,108),(4,109,73,51),(5,52,74,110),(6,111,75,53),(7,54,76,112),(8,113,77,55),(9,56,78,114),(10,115,79,57),(11,58,80,116),(12,117,61,59),(13,60,62,118),(14,119,63,41),(15,42,64,120),(16,101,65,43),(17,44,66,102),(18,103,67,45),(19,46,68,104),(20,105,69,47),(21,138,158,91),(22,92,159,139),(23,140,160,93),(24,94,141,121),(25,122,142,95),(26,96,143,123),(27,124,144,97),(28,98,145,125),(29,126,146,99),(30,100,147,127),(31,128,148,81),(32,82,149,129),(33,130,150,83),(34,84,151,131),(35,132,152,85),(36,86,153,133),(37,134,154,87),(38,88,155,135),(39,136,156,89),(40,90,157,137)], [(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)], [(1,96,11,86),(2,85,12,95),(3,94,13,84),(4,83,14,93),(5,92,15,82),(6,81,16,91),(7,90,17,100),(8,99,18,89),(9,88,19,98),(10,97,20,87),(21,43,31,53),(22,52,32,42),(23,41,33,51),(24,50,34,60),(25,59,35,49),(26,48,36,58),(27,57,37,47),(28,46,38,56),(29,55,39,45),(30,44,40,54),(61,122,71,132),(62,131,72,121),(63,140,73,130),(64,129,74,139),(65,138,75,128),(66,127,76,137),(67,136,77,126),(68,125,78,135),(69,134,79,124),(70,123,80,133),(101,148,111,158),(102,157,112,147),(103,146,113,156),(104,155,114,145),(105,144,115,154),(106,153,116,143),(107,142,117,152),(108,151,118,141),(109,160,119,150),(110,149,120,159)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ 1+4 | 2- 1+4 | Q8.10D10 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.150D10 | C4×D20 | C42⋊2D5 | Dic5.Q8 | D5×C4⋊C4 | D20⋊8C4 | D10.13D4 | C4⋊D20 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | D10 | C42 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 12 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.150D10 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 23 | 9 | 2 |
| 0 | 0 | 0 | 0 | 0 | 39 | 0 | 32 |
| 1 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 9 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 32 | 39 |
| 0 | 0 | 0 | 0 | 0 | 9 | 32 | 40 |
| 0 | 0 | 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 | 1 | 32 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,37,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,18,9,23,39,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,32],[1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,37,10,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,18,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,40],[40,34,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,37,10,0,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,18,9,23,39,0,0,0,0,32,32,0,1,0,0,0,0,39,40,0,32] >;
C42.150D10 in GAP, Magma, Sage, TeX
C_4^2._{150}D_{10} % in TeX
G:=Group("C4^2.150D10"); // GroupNames label
G:=SmallGroup(320,1364);
// by ID
G=gap.SmallGroup(320,1364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations