metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.122D10, C10.72- 1+4, (C4×Q8)⋊5D5, (Q8×C20)⋊5C2, C4⋊C4.291D10, D10⋊Q8⋊10C2, (C4×Dic10)⋊36C2, Dic5⋊Q8⋊9C2, C4.18(C4○D20), C42⋊2D5⋊17C2, C42⋊D5⋊33C2, (C2×Q8).176D10, Dic5⋊3Q8⋊17C2, D20⋊8C4.10C2, C20.116(C4○D4), (C2×C20).621C23, (C2×C10).112C24, (C4×C20).238C22, C4.D20.12C2, C20.23D4.9C2, Dic5.37(C4○D4), (C2×D20).146C22, C4⋊Dic5.303C22, (Q8×C10).212C22, (C4×Dic5).89C22, (C22×D5).44C23, C22.137(C23×D5), D10⋊C4.68C22, C5⋊3(C22.50C24), (C2×Dic5).221C23, C10.D4.68C22, C2.10(Q8.10D10), (C2×Dic10).153C22, C2.27(D5×C4○D4), C4⋊C4⋊D5⋊10C2, C10.53(C2×C4○D4), C2.60(C2×C4○D20), (C2×C4×D5).257C22, (C5×C4⋊C4).340C22, (C2×C4).653(C22×D5), SmallGroup(320,1240)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.122D10
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=a2b-1, dcd=c9 >
Subgroups: 694 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.50C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C4×Dic10, C42⋊D5, C4.D20, C42⋊2D5, Dic5⋊3Q8, D20⋊8C4, D10⋊Q8, C4⋊C4⋊D5, Dic5⋊Q8, C20.23D4, Q8×C20, C42.122D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, C4○D20, C23×D5, C2×C4○D20, Q8.10D10, D5×C4○D4, C42.122D10
►On 160 points
Generators in S160
(1 94 150 133)(2 95 151 134)(3 96 152 135)(4 97 153 136)(5 98 154 137)(6 99 155 138)(7 100 156 139)(8 81 157 140)(9 82 158 121)(10 83 159 122)(11 84 160 123)(12 85 141 124)(13 86 142 125)(14 87 143 126)(15 88 144 127)(16 89 145 128)(17 90 146 129)(18 91 147 130)(19 92 148 131)(20 93 149 132)(21 118 80 47)(22 119 61 48)(23 120 62 49)(24 101 63 50)(25 102 64 51)(26 103 65 52)(27 104 66 53)(28 105 67 54)(29 106 68 55)(30 107 69 56)(31 108 70 57)(32 109 71 58)(33 110 72 59)(34 111 73 60)(35 112 74 41)(36 113 75 42)(37 114 76 43)(38 115 77 44)(39 116 78 45)(40 117 79 46)
(1 22 11 32)(2 72 12 62)(3 24 13 34)(4 74 14 64)(5 26 15 36)(6 76 16 66)(7 28 17 38)(8 78 18 68)(9 30 19 40)(10 80 20 70)(21 149 31 159)(23 151 33 141)(25 153 35 143)(27 155 37 145)(29 157 39 147)(41 87 51 97)(42 137 52 127)(43 89 53 99)(44 139 54 129)(45 91 55 81)(46 121 56 131)(47 93 57 83)(48 123 58 133)(49 95 59 85)(50 125 60 135)(61 160 71 150)(63 142 73 152)(65 144 75 154)(67 146 77 156)(69 148 79 158)(82 107 92 117)(84 109 94 119)(86 111 96 101)(88 113 98 103)(90 115 100 105)(102 136 112 126)(104 138 114 128)(106 140 116 130)(108 122 118 132)(110 124 120 134)
(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 21)(2 30)(3 39)(4 28)(5 37)(6 26)(7 35)(8 24)(9 33)(10 22)(11 31)(12 40)(13 29)(14 38)(15 27)(16 36)(17 25)(18 34)(19 23)(20 32)(41 129)(42 138)(43 127)(44 136)(45 125)(46 134)(47 123)(48 132)(49 121)(50 130)(51 139)(52 128)(53 137)(54 126)(55 135)(56 124)(57 133)(58 122)(59 131)(60 140)(61 159)(62 148)(63 157)(64 146)(65 155)(66 144)(67 153)(68 142)(69 151)(70 160)(71 149)(72 158)(73 147)(74 156)(75 145)(76 154)(77 143)(78 152)(79 141)(80 150)(81 111)(82 120)(83 109)(84 118)(85 107)(86 116)(87 105)(88 114)(89 103)(90 112)(91 101)(92 110)(93 119)(94 108)(95 117)(96 106)(97 115)(98 104)(99 113)(100 102)
G:=sub<Sym(160)| (1,94,150,133)(2,95,151,134)(3,96,152,135)(4,97,153,136)(5,98,154,137)(6,99,155,138)(7,100,156,139)(8,81,157,140)(9,82,158,121)(10,83,159,122)(11,84,160,123)(12,85,141,124)(13,86,142,125)(14,87,143,126)(15,88,144,127)(16,89,145,128)(17,90,146,129)(18,91,147,130)(19,92,148,131)(20,93,149,132)(21,118,80,47)(22,119,61,48)(23,120,62,49)(24,101,63,50)(25,102,64,51)(26,103,65,52)(27,104,66,53)(28,105,67,54)(29,106,68,55)(30,107,69,56)(31,108,70,57)(32,109,71,58)(33,110,72,59)(34,111,73,60)(35,112,74,41)(36,113,75,42)(37,114,76,43)(38,115,77,44)(39,116,78,45)(40,117,79,46), (1,22,11,32)(2,72,12,62)(3,24,13,34)(4,74,14,64)(5,26,15,36)(6,76,16,66)(7,28,17,38)(8,78,18,68)(9,30,19,40)(10,80,20,70)(21,149,31,159)(23,151,33,141)(25,153,35,143)(27,155,37,145)(29,157,39,147)(41,87,51,97)(42,137,52,127)(43,89,53,99)(44,139,54,129)(45,91,55,81)(46,121,56,131)(47,93,57,83)(48,123,58,133)(49,95,59,85)(50,125,60,135)(61,160,71,150)(63,142,73,152)(65,144,75,154)(67,146,77,156)(69,148,79,158)(82,107,92,117)(84,109,94,119)(86,111,96,101)(88,113,98,103)(90,115,100,105)(102,136,112,126)(104,138,114,128)(106,140,116,130)(108,122,118,132)(110,124,120,134), (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), (1,21)(2,30)(3,39)(4,28)(5,37)(6,26)(7,35)(8,24)(9,33)(10,22)(11,31)(12,40)(13,29)(14,38)(15,27)(16,36)(17,25)(18,34)(19,23)(20,32)(41,129)(42,138)(43,127)(44,136)(45,125)(46,134)(47,123)(48,132)(49,121)(50,130)(51,139)(52,128)(53,137)(54,126)(55,135)(56,124)(57,133)(58,122)(59,131)(60,140)(61,159)(62,148)(63,157)(64,146)(65,155)(66,144)(67,153)(68,142)(69,151)(70,160)(71,149)(72,158)(73,147)(74,156)(75,145)(76,154)(77,143)(78,152)(79,141)(80,150)(81,111)(82,120)(83,109)(84,118)(85,107)(86,116)(87,105)(88,114)(89,103)(90,112)(91,101)(92,110)(93,119)(94,108)(95,117)(96,106)(97,115)(98,104)(99,113)(100,102)>;
G:=Group( (1,94,150,133)(2,95,151,134)(3,96,152,135)(4,97,153,136)(5,98,154,137)(6,99,155,138)(7,100,156,139)(8,81,157,140)(9,82,158,121)(10,83,159,122)(11,84,160,123)(12,85,141,124)(13,86,142,125)(14,87,143,126)(15,88,144,127)(16,89,145,128)(17,90,146,129)(18,91,147,130)(19,92,148,131)(20,93,149,132)(21,118,80,47)(22,119,61,48)(23,120,62,49)(24,101,63,50)(25,102,64,51)(26,103,65,52)(27,104,66,53)(28,105,67,54)(29,106,68,55)(30,107,69,56)(31,108,70,57)(32,109,71,58)(33,110,72,59)(34,111,73,60)(35,112,74,41)(36,113,75,42)(37,114,76,43)(38,115,77,44)(39,116,78,45)(40,117,79,46), (1,22,11,32)(2,72,12,62)(3,24,13,34)(4,74,14,64)(5,26,15,36)(6,76,16,66)(7,28,17,38)(8,78,18,68)(9,30,19,40)(10,80,20,70)(21,149,31,159)(23,151,33,141)(25,153,35,143)(27,155,37,145)(29,157,39,147)(41,87,51,97)(42,137,52,127)(43,89,53,99)(44,139,54,129)(45,91,55,81)(46,121,56,131)(47,93,57,83)(48,123,58,133)(49,95,59,85)(50,125,60,135)(61,160,71,150)(63,142,73,152)(65,144,75,154)(67,146,77,156)(69,148,79,158)(82,107,92,117)(84,109,94,119)(86,111,96,101)(88,113,98,103)(90,115,100,105)(102,136,112,126)(104,138,114,128)(106,140,116,130)(108,122,118,132)(110,124,120,134), (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), (1,21)(2,30)(3,39)(4,28)(5,37)(6,26)(7,35)(8,24)(9,33)(10,22)(11,31)(12,40)(13,29)(14,38)(15,27)(16,36)(17,25)(18,34)(19,23)(20,32)(41,129)(42,138)(43,127)(44,136)(45,125)(46,134)(47,123)(48,132)(49,121)(50,130)(51,139)(52,128)(53,137)(54,126)(55,135)(56,124)(57,133)(58,122)(59,131)(60,140)(61,159)(62,148)(63,157)(64,146)(65,155)(66,144)(67,153)(68,142)(69,151)(70,160)(71,149)(72,158)(73,147)(74,156)(75,145)(76,154)(77,143)(78,152)(79,141)(80,150)(81,111)(82,120)(83,109)(84,118)(85,107)(86,116)(87,105)(88,114)(89,103)(90,112)(91,101)(92,110)(93,119)(94,108)(95,117)(96,106)(97,115)(98,104)(99,113)(100,102) );
G=PermutationGroup([[(1,94,150,133),(2,95,151,134),(3,96,152,135),(4,97,153,136),(5,98,154,137),(6,99,155,138),(7,100,156,139),(8,81,157,140),(9,82,158,121),(10,83,159,122),(11,84,160,123),(12,85,141,124),(13,86,142,125),(14,87,143,126),(15,88,144,127),(16,89,145,128),(17,90,146,129),(18,91,147,130),(19,92,148,131),(20,93,149,132),(21,118,80,47),(22,119,61,48),(23,120,62,49),(24,101,63,50),(25,102,64,51),(26,103,65,52),(27,104,66,53),(28,105,67,54),(29,106,68,55),(30,107,69,56),(31,108,70,57),(32,109,71,58),(33,110,72,59),(34,111,73,60),(35,112,74,41),(36,113,75,42),(37,114,76,43),(38,115,77,44),(39,116,78,45),(40,117,79,46)], [(1,22,11,32),(2,72,12,62),(3,24,13,34),(4,74,14,64),(5,26,15,36),(6,76,16,66),(7,28,17,38),(8,78,18,68),(9,30,19,40),(10,80,20,70),(21,149,31,159),(23,151,33,141),(25,153,35,143),(27,155,37,145),(29,157,39,147),(41,87,51,97),(42,137,52,127),(43,89,53,99),(44,139,54,129),(45,91,55,81),(46,121,56,131),(47,93,57,83),(48,123,58,133),(49,95,59,85),(50,125,60,135),(61,160,71,150),(63,142,73,152),(65,144,75,154),(67,146,77,156),(69,148,79,158),(82,107,92,117),(84,109,94,119),(86,111,96,101),(88,113,98,103),(90,115,100,105),(102,136,112,126),(104,138,114,128),(106,140,116,130),(108,122,118,132),(110,124,120,134)], [(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)], [(1,21),(2,30),(3,39),(4,28),(5,37),(6,26),(7,35),(8,24),(9,33),(10,22),(11,31),(12,40),(13,29),(14,38),(15,27),(16,36),(17,25),(18,34),(19,23),(20,32),(41,129),(42,138),(43,127),(44,136),(45,125),(46,134),(47,123),(48,132),(49,121),(50,130),(51,139),(52,128),(53,137),(54,126),(55,135),(56,124),(57,133),(58,122),(59,131),(60,140),(61,159),(62,148),(63,157),(64,146),(65,155),(66,144),(67,153),(68,142),(69,151),(70,160),(71,149),(72,158),(73,147),(74,156),(75,145),(76,154),(77,143),(78,152),(79,141),(80,150),(81,111),(82,120),(83,109),(84,118),(85,107),(86,116),(87,105),(88,114),(89,103),(90,112),(91,101),(92,110),(93,119),(94,108),(95,117),(96,106),(97,115),(98,104),(99,113),(100,102)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | Q8.10D10 | D5×C4○D4 |
| kernel | C42.122D10 | C4×Dic10 | C42⋊D5 | C4.D20 | C42⋊2D5 | Dic5⋊3Q8 | D20⋊8C4 | D10⋊Q8 | C4⋊C4⋊D5 | Dic5⋊Q8 | C20.23D4 | Q8×C20 | C4×Q8 | Dic5 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.122D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 5 |
| 0 | 0 | 0 | 0 | 38 | 37 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 36 |
| 0 | 0 | 0 | 0 | 3 | 4 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,18,0,0,0,0,0,1],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,4,38,0,0,0,0,5,37],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,37,3,0,0,0,0,36,4] >;
C42.122D10 in GAP, Magma, Sage, TeX
C_4^2._{122}D_{10} % in TeX
G:=Group("C4^2.122D10"); // GroupNames label
G:=SmallGroup(320,1240);
// by ID
G=gap.SmallGroup(320,1240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,232,758,100,794,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^9>;
// generators/relations