direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.23D4, (C2×Q8)⋊29D10, (C22×Q8)⋊6D5, C20.259(C2×D4), (C2×C20).215D4, C10⋊4(C4.4D4), (Q8×C10)⋊36C22, (C2×C20).646C23, (C2×C10).306C24, (C4×Dic5)⋊69C22, (C22×D20).20C2, (C22×C4).385D10, C10.154(C22×D4), D10⋊C4⋊74C22, (C2×D20).286C22, (C23×D5).78C22, C22.317(C23×D5), C23.342(C22×D5), (C22×C10).424C23, (C22×C20).439C22, C22.40(Q8⋊2D5), (C2×Dic5).299C23, (C22×D5).133C23, (C22×Dic5).257C22, (Q8×C2×C10)⋊5C2, C5⋊5(C2×C4.4D4), (C2×C4×Dic5)⋊13C2, C4.28(C2×C5⋊D4), C10.128(C2×C4○D4), (C2×C10).589(C2×D4), C2.35(C2×Q8⋊2D5), (C2×D10⋊C4)⋊43C2, C2.27(C22×C5⋊D4), (C2×C4).157(C5⋊D4), (C2×C4).243(C22×D5), C22.117(C2×C5⋊D4), (C2×C10).201(C4○D4), SmallGroup(320,1486)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1278 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×10], C2×C4 [×12], D4 [×8], Q8 [×8], C23, C23 [×16], D5 [×4], C10, C10 [×6], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], Dic5 [×4], C20 [×4], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C5×Q8 [×8], C22×D5 [×4], C22×D5 [×12], C22×C10, C2×C4.4D4, C4×Dic5 [×4], D10⋊C4 [×16], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C23×D5 [×2], C2×C4×Dic5, C2×D10⋊C4 [×4], C20.23D4 [×8], C22×D20, Q8×C2×C10, C2×C20.23D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C4.4D4, Q8⋊2D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20.23D4 [×4], C2×Q8⋊2D5 [×2], C22×C5⋊D4, C2×C20.23D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=b10c-1 >
►On 160 points
Generators in S160
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 137)(62 138)(63 139)(64 140)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)
(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 89 67 43)(2 98 68 52)(3 87 69 41)(4 96 70 50)(5 85 71 59)(6 94 72 48)(7 83 73 57)(8 92 74 46)(9 81 75 55)(10 90 76 44)(11 99 77 53)(12 88 78 42)(13 97 79 51)(14 86 80 60)(15 95 61 49)(16 84 62 58)(17 93 63 47)(18 82 64 56)(19 91 65 45)(20 100 66 54)(21 144 134 110)(22 153 135 119)(23 142 136 108)(24 151 137 117)(25 160 138 106)(26 149 139 115)(27 158 140 104)(28 147 121 113)(29 156 122 102)(30 145 123 111)(31 154 124 120)(32 143 125 109)(33 152 126 118)(34 141 127 107)(35 150 128 116)(36 159 129 105)(37 148 130 114)(38 157 131 103)(39 146 132 112)(40 155 133 101)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 160)(49 159)(50 158)(51 157)(52 156)(53 155)(54 154)(55 153)(56 152)(57 151)(58 150)(59 149)(60 148)(61 139)(62 138)(63 137)(64 136)(65 135)(66 134)(67 133)(68 132)(69 131)(70 130)(71 129)(72 128)(73 127)(74 126)(75 125)(76 124)(77 123)(78 122)(79 121)(80 140)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(100 120)
G:=sub<Sym(160)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,137)(62,138)(63,139)(64,140)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156), (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), (1,89,67,43)(2,98,68,52)(3,87,69,41)(4,96,70,50)(5,85,71,59)(6,94,72,48)(7,83,73,57)(8,92,74,46)(9,81,75,55)(10,90,76,44)(11,99,77,53)(12,88,78,42)(13,97,79,51)(14,86,80,60)(15,95,61,49)(16,84,62,58)(17,93,63,47)(18,82,64,56)(19,91,65,45)(20,100,66,54)(21,144,134,110)(22,153,135,119)(23,142,136,108)(24,151,137,117)(25,160,138,106)(26,149,139,115)(27,158,140,104)(28,147,121,113)(29,156,122,102)(30,145,123,111)(31,154,124,120)(32,143,125,109)(33,152,126,118)(34,141,127,107)(35,150,128,116)(36,159,129,105)(37,148,130,114)(38,157,131,103)(39,146,132,112)(40,155,133,101), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,160)(49,159)(50,158)(51,157)(52,156)(53,155)(54,154)(55,153)(56,152)(57,151)(58,150)(59,149)(60,148)(61,139)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,140)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(100,120)>;
G:=Group( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,137)(62,138)(63,139)(64,140)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156), (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), (1,89,67,43)(2,98,68,52)(3,87,69,41)(4,96,70,50)(5,85,71,59)(6,94,72,48)(7,83,73,57)(8,92,74,46)(9,81,75,55)(10,90,76,44)(11,99,77,53)(12,88,78,42)(13,97,79,51)(14,86,80,60)(15,95,61,49)(16,84,62,58)(17,93,63,47)(18,82,64,56)(19,91,65,45)(20,100,66,54)(21,144,134,110)(22,153,135,119)(23,142,136,108)(24,151,137,117)(25,160,138,106)(26,149,139,115)(27,158,140,104)(28,147,121,113)(29,156,122,102)(30,145,123,111)(31,154,124,120)(32,143,125,109)(33,152,126,118)(34,141,127,107)(35,150,128,116)(36,159,129,105)(37,148,130,114)(38,157,131,103)(39,146,132,112)(40,155,133,101), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,160)(49,159)(50,158)(51,157)(52,156)(53,155)(54,154)(55,153)(56,152)(57,151)(58,150)(59,149)(60,148)(61,139)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,140)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(100,120) );
G=PermutationGroup([(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,137),(62,138),(63,139),(64,140),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(81,157),(82,158),(83,159),(84,160),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156)], [(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)], [(1,89,67,43),(2,98,68,52),(3,87,69,41),(4,96,70,50),(5,85,71,59),(6,94,72,48),(7,83,73,57),(8,92,74,46),(9,81,75,55),(10,90,76,44),(11,99,77,53),(12,88,78,42),(13,97,79,51),(14,86,80,60),(15,95,61,49),(16,84,62,58),(17,93,63,47),(18,82,64,56),(19,91,65,45),(20,100,66,54),(21,144,134,110),(22,153,135,119),(23,142,136,108),(24,151,137,117),(25,160,138,106),(26,149,139,115),(27,158,140,104),(28,147,121,113),(29,156,122,102),(30,145,123,111),(31,154,124,120),(32,143,125,109),(33,152,126,118),(34,141,127,107),(35,150,128,116),(36,159,129,105),(37,148,130,114),(38,157,131,103),(39,146,132,112),(40,155,133,101)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,147),(42,146),(43,145),(44,144),(45,143),(46,142),(47,141),(48,160),(49,159),(50,158),(51,157),(52,156),(53,155),(54,154),(55,153),(56,152),(57,151),(58,150),(59,149),(60,148),(61,139),(62,138),(63,137),(64,136),(65,135),(66,134),(67,133),(68,132),(69,131),(70,130),(71,129),(72,128),(73,127),(74,126),(75,125),(76,124),(77,123),(78,122),(79,121),(80,140),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(100,120)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 33 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 38 | 37 | 0 | 0 | 0 | 0 |
| 23 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 3 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 5 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,1,0,0,0,0,33,1,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0],[38,23,0,0,0,0,37,3,0,0,0,0,0,0,20,3,0,0,0,0,3,21,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Q8⋊2D5 |
| kernel | C2×C20.23D4 | C2×C4×Dic5 | C2×D10⋊C4 | C20.23D4 | C22×D20 | Q8×C2×C10 | C2×C20 | C22×Q8 | C2×C10 | C22×C4 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 4 | 2 | 8 | 6 | 8 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}._{23}D_4 % in TeX
G:=Group("C2xC20.23D4"); // GroupNames label
G:=SmallGroup(320,1486);
// by ID
G=gap.SmallGroup(320,1486);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=b^10*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀