direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C2×C42, C20⋊9(C22×C4), C10⋊2(C2×C42), C5⋊2(C22×C42), (C4×C20)⋊54C22, (C2×C10).15C24, C10.23(C23×C4), Dic5⋊9(C22×C4), (C2×C20).874C23, (C4×Dic5)⋊83C22, D10.50(C22×C4), (C22×C4).466D10, C22.12(C23×D5), C23.312(C22×D5), (C22×C20).563C22, (C22×C10).377C23, (C2×Dic5).368C23, (C23×D5).143C22, (C22×D5).289C23, (C22×Dic5).287C22, (C2×C4×C20)⋊16C2, (C2×C20)⋊44(C2×C4), C2.1(D5×C22×C4), (C2×C4×Dic5)⋊39C2, C22.67(C2×C4×D5), (D5×C22×C4).38C2, (C2×Dic5)⋊38(C2×C4), (C2×C4×D5).421C22, (C2×C4).816(C22×D5), (C2×C10).246(C22×C4), (C22×D5).140(C2×C4), SmallGroup(320,1143)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C42 |
Generators and relations for D5×C2×C42
G = < a,b,c,d,e | a2=b4=c4=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: 1278 in 498 conjugacy classes, 303 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C42, C42, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C42, C2×C42, C23×C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, C22×C42, C4×Dic5, C4×C20, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D5×C42, C2×C4×Dic5, C2×C4×C20, D5×C22×C4, D5×C2×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, C24, D10, C2×C42, C23×C4, C4×D5, C22×D5, C22×C42, C2×C4×D5, C23×D5, D5×C42, D5×C22×C4, D5×C2×C42
►On 160 points
Generators in S160
(1 89)(2 90)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 74 19 69)(2 75 20 70)(3 71 16 66)(4 72 17 67)(5 73 18 68)(6 76 11 61)(7 77 12 62)(8 78 13 63)(9 79 14 64)(10 80 15 65)(21 41 36 56)(22 42 37 57)(23 43 38 58)(24 44 39 59)(25 45 40 60)(26 46 31 51)(27 47 32 52)(28 48 33 53)(29 49 34 54)(30 50 35 55)(81 151 96 146)(82 152 97 147)(83 153 98 148)(84 154 99 149)(85 155 100 150)(86 156 91 141)(87 157 92 142)(88 158 93 143)(89 159 94 144)(90 160 95 145)(101 121 116 136)(102 122 117 137)(103 123 118 138)(104 124 119 139)(105 125 120 140)(106 126 111 131)(107 127 112 132)(108 128 113 133)(109 129 114 134)(110 130 115 135)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)(81 111 91 101)(82 112 92 102)(83 113 93 103)(84 114 94 104)(85 115 95 105)(86 116 96 106)(87 117 97 107)(88 118 98 108)(89 119 99 109)(90 120 100 110)(121 151 131 141)(122 152 132 142)(123 153 133 143)(124 154 134 144)(125 155 135 145)(126 156 136 146)(127 157 137 147)(128 158 138 148)(129 159 139 149)(130 160 140 150)
(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)(151 152 153 154 155)(156 157 158 159 160)
(1 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)(81 91)(82 95)(83 94)(84 93)(85 92)(86 96)(87 100)(88 99)(89 98)(90 97)(101 111)(102 115)(103 114)(104 113)(105 112)(106 116)(107 120)(108 119)(109 118)(110 117)(121 131)(122 135)(123 134)(124 133)(125 132)(126 136)(127 140)(128 139)(129 138)(130 137)(141 151)(142 155)(143 154)(144 153)(145 152)(146 156)(147 160)(148 159)(149 158)(150 157)
G:=sub<Sym(160)| (1,89)(2,90)(3,86)(4,87)(5,88)(6,81)(7,82)(8,83)(9,84)(10,85)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (1,74,19,69)(2,75,20,70)(3,71,16,66)(4,72,17,67)(5,73,18,68)(6,76,11,61)(7,77,12,62)(8,78,13,63)(9,79,14,64)(10,80,15,65)(21,41,36,56)(22,42,37,57)(23,43,38,58)(24,44,39,59)(25,45,40,60)(26,46,31,51)(27,47,32,52)(28,48,33,53)(29,49,34,54)(30,50,35,55)(81,151,96,146)(82,152,97,147)(83,153,98,148)(84,154,99,149)(85,155,100,150)(86,156,91,141)(87,157,92,142)(88,158,93,143)(89,159,94,144)(90,160,95,145)(101,121,116,136)(102,122,117,137)(103,123,118,138)(104,124,119,139)(105,125,120,140)(106,126,111,131)(107,127,112,132)(108,128,113,133)(109,129,114,134)(110,130,115,135), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70)(81,111,91,101)(82,112,92,102)(83,113,93,103)(84,114,94,104)(85,115,95,105)(86,116,96,106)(87,117,97,107)(88,118,98,108)(89,119,99,109)(90,120,100,110)(121,151,131,141)(122,152,132,142)(123,153,133,143)(124,154,134,144)(125,155,135,145)(126,156,136,146)(127,157,137,147)(128,158,138,148)(129,159,139,149)(130,160,140,150), (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)(151,152,153,154,155)(156,157,158,159,160), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)(41,51)(42,55)(43,54)(44,53)(45,52)(46,56)(47,60)(48,59)(49,58)(50,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77)(81,91)(82,95)(83,94)(84,93)(85,92)(86,96)(87,100)(88,99)(89,98)(90,97)(101,111)(102,115)(103,114)(104,113)(105,112)(106,116)(107,120)(108,119)(109,118)(110,117)(121,131)(122,135)(123,134)(124,133)(125,132)(126,136)(127,140)(128,139)(129,138)(130,137)(141,151)(142,155)(143,154)(144,153)(145,152)(146,156)(147,160)(148,159)(149,158)(150,157)>;
G:=Group( (1,89)(2,90)(3,86)(4,87)(5,88)(6,81)(7,82)(8,83)(9,84)(10,85)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (1,74,19,69)(2,75,20,70)(3,71,16,66)(4,72,17,67)(5,73,18,68)(6,76,11,61)(7,77,12,62)(8,78,13,63)(9,79,14,64)(10,80,15,65)(21,41,36,56)(22,42,37,57)(23,43,38,58)(24,44,39,59)(25,45,40,60)(26,46,31,51)(27,47,32,52)(28,48,33,53)(29,49,34,54)(30,50,35,55)(81,151,96,146)(82,152,97,147)(83,153,98,148)(84,154,99,149)(85,155,100,150)(86,156,91,141)(87,157,92,142)(88,158,93,143)(89,159,94,144)(90,160,95,145)(101,121,116,136)(102,122,117,137)(103,123,118,138)(104,124,119,139)(105,125,120,140)(106,126,111,131)(107,127,112,132)(108,128,113,133)(109,129,114,134)(110,130,115,135), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70)(81,111,91,101)(82,112,92,102)(83,113,93,103)(84,114,94,104)(85,115,95,105)(86,116,96,106)(87,117,97,107)(88,118,98,108)(89,119,99,109)(90,120,100,110)(121,151,131,141)(122,152,132,142)(123,153,133,143)(124,154,134,144)(125,155,135,145)(126,156,136,146)(127,157,137,147)(128,158,138,148)(129,159,139,149)(130,160,140,150), (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)(151,152,153,154,155)(156,157,158,159,160), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)(41,51)(42,55)(43,54)(44,53)(45,52)(46,56)(47,60)(48,59)(49,58)(50,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77)(81,91)(82,95)(83,94)(84,93)(85,92)(86,96)(87,100)(88,99)(89,98)(90,97)(101,111)(102,115)(103,114)(104,113)(105,112)(106,116)(107,120)(108,119)(109,118)(110,117)(121,131)(122,135)(123,134)(124,133)(125,132)(126,136)(127,140)(128,139)(129,138)(130,137)(141,151)(142,155)(143,154)(144,153)(145,152)(146,156)(147,160)(148,159)(149,158)(150,157) );
G=PermutationGroup([[(1,89),(2,90),(3,86),(4,87),(5,88),(6,81),(7,82),(8,83),(9,84),(10,85),(11,96),(12,97),(13,98),(14,99),(15,100),(16,91),(17,92),(18,93),(19,94),(20,95),(21,106),(22,107),(23,108),(24,109),(25,110),(26,101),(27,102),(28,103),(29,104),(30,105),(31,116),(32,117),(33,118),(34,119),(35,120),(36,111),(37,112),(38,113),(39,114),(40,115),(41,126),(42,127),(43,128),(44,129),(45,130),(46,121),(47,122),(48,123),(49,124),(50,125),(51,136),(52,137),(53,138),(54,139),(55,140),(56,131),(57,132),(58,133),(59,134),(60,135),(61,146),(62,147),(63,148),(64,149),(65,150),(66,141),(67,142),(68,143),(69,144),(70,145),(71,156),(72,157),(73,158),(74,159),(75,160),(76,151),(77,152),(78,153),(79,154),(80,155)], [(1,74,19,69),(2,75,20,70),(3,71,16,66),(4,72,17,67),(5,73,18,68),(6,76,11,61),(7,77,12,62),(8,78,13,63),(9,79,14,64),(10,80,15,65),(21,41,36,56),(22,42,37,57),(23,43,38,58),(24,44,39,59),(25,45,40,60),(26,46,31,51),(27,47,32,52),(28,48,33,53),(29,49,34,54),(30,50,35,55),(81,151,96,146),(82,152,97,147),(83,153,98,148),(84,154,99,149),(85,155,100,150),(86,156,91,141),(87,157,92,142),(88,158,93,143),(89,159,94,144),(90,160,95,145),(101,121,116,136),(102,122,117,137),(103,123,118,138),(104,124,119,139),(105,125,120,140),(106,126,111,131),(107,127,112,132),(108,128,113,133),(109,129,114,134),(110,130,115,135)], [(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70),(81,111,91,101),(82,112,92,102),(83,113,93,103),(84,114,94,104),(85,115,95,105),(86,116,96,106),(87,117,97,107),(88,118,98,108),(89,119,99,109),(90,120,100,110),(121,151,131,141),(122,152,132,142),(123,153,133,143),(124,154,134,144),(125,155,135,145),(126,156,136,146),(127,157,137,147),(128,158,138,148),(129,159,139,149),(130,160,140,150)], [(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),(151,152,153,154,155),(156,157,158,159,160)], [(1,13),(2,12),(3,11),(4,15),(5,14),(6,16),(7,20),(8,19),(9,18),(10,17),(21,31),(22,35),(23,34),(24,33),(25,32),(26,36),(27,40),(28,39),(29,38),(30,37),(41,51),(42,55),(43,54),(44,53),(45,52),(46,56),(47,60),(48,59),(49,58),(50,57),(61,71),(62,75),(63,74),(64,73),(65,72),(66,76),(67,80),(68,79),(69,78),(70,77),(81,91),(82,95),(83,94),(84,93),(85,92),(86,96),(87,100),(88,99),(89,98),(90,97),(101,111),(102,115),(103,114),(104,113),(105,112),(106,116),(107,120),(108,119),(109,118),(110,117),(121,131),(122,135),(123,134),(124,133),(125,132),(126,136),(127,140),(128,139),(129,138),(130,137),(141,151),(142,155),(143,154),(144,153),(145,152),(146,156),(147,160),(148,159),(149,158),(150,157)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4X | 4Y | ··· | 4AV | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | C4×D5 |
| kernel | D5×C2×C42 | D5×C42 | C2×C4×Dic5 | C2×C4×C20 | D5×C22×C4 | C2×C4×D5 | C2×C42 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 8 | 3 | 1 | 3 | 48 | 2 | 8 | 6 | 48 |
Matrix representation of D5×C2×C42 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 36 |
| 0 | 0 | 40 | 35 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 40 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[40,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,40,0,0,36,35],[40,0,0,0,0,1,0,0,0,0,40,40,0,0,0,1] >;
D5×C2×C42 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_4^2
% in TeX
G:=Group("D5xC2xC4^2"); // GroupNames label
G:=SmallGroup(320,1143);
// by ID
G=gap.SmallGroup(320,1143);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=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