direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊7D4, C23⋊4D20, C24.71D10, C20⋊15(C2×D4), (C2×C20)⋊37D4, (C23×C4)⋊5D5, (C23×C20)⋊8C2, C22⋊2(C2×D20), C10⋊3(C4⋊D4), (C22×C4)⋊44D10, (C22×C10)⋊15D4, (C2×D20)⋊50C22, (C22×D20)⋊12C2, C4⋊Dic5⋊64C22, C2.33(C22×D20), (C2×C10).288C24, (C2×C20).705C23, (C22×C20)⋊60C22, C10.134(C22×D4), D10⋊C4⋊42C22, C22.83(C4○D20), (C23×D5).75C22, C23.234(C22×D5), C22.303(C23×D5), (C22×C10).417C23, (C23×C10).110C22, (C2×Dic5).150C23, (C22×D5).126C23, (C22×Dic5).162C22, C4⋊4(C2×C5⋊D4), C5⋊4(C2×C4⋊D4), (C2×C10)⋊11(C2×D4), (C2×C4)⋊16(C5⋊D4), (C2×C4⋊Dic5)⋊29C2, C10.63(C2×C4○D4), C2.71(C2×C4○D20), C2.7(C22×C5⋊D4), (C2×C5⋊D4)⋊42C22, (C22×C5⋊D4)⋊11C2, (C2×D10⋊C4)⋊14C2, (C2×C4).658(C22×D5), C22.104(C2×C5⋊D4), (C2×C10).114(C4○D4), SmallGroup(320,1462)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1646 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], C5, C2×C4 [×8], C2×C4 [×18], D4 [×24], C23, C23 [×6], C23 [×20], D5 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×24], C24, C24 [×2], Dic5 [×4], C20 [×4], C20 [×2], D10 [×20], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×8], C2×C20 [×10], C22×D5 [×4], C22×D5 [×12], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C4⋊D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×D5 [×2], C23×C10, C2×C4⋊Dic5, C2×D10⋊C4 [×2], C20⋊7D4 [×8], C22×D20, C22×C5⋊D4 [×2], C23×C20, C2×C20⋊7D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D20 [×4], C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊D4, C2×D20 [×6], C4○D20 [×2], C2×C5⋊D4 [×6], C23×D5, C20⋊7D4 [×4], C22×D20, C2×C4○D20, C22×C5⋊D4, C2×C20⋊7D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
►On 160 points
Generators in S160
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 121)(15 122)(16 123)(17 124)(18 125)(19 126)(20 127)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 81)(37 82)(38 83)(39 84)(40 85)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 116)(62 117)(63 118)(64 119)(65 120)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)
(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 53 22 64)(2 52 23 63)(3 51 24 62)(4 50 25 61)(5 49 26 80)(6 48 27 79)(7 47 28 78)(8 46 29 77)(9 45 30 76)(10 44 31 75)(11 43 32 74)(12 42 33 73)(13 41 34 72)(14 60 35 71)(15 59 36 70)(16 58 37 69)(17 57 38 68)(18 56 39 67)(19 55 40 66)(20 54 21 65)(81 105 122 148)(82 104 123 147)(83 103 124 146)(84 102 125 145)(85 101 126 144)(86 120 127 143)(87 119 128 142)(88 118 129 141)(89 117 130 160)(90 116 131 159)(91 115 132 158)(92 114 133 157)(93 113 134 156)(94 112 135 155)(95 111 136 154)(96 110 137 153)(97 109 138 152)(98 108 139 151)(99 107 140 150)(100 106 121 149)
(1 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 100)(19 99)(20 98)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 140)(41 144)(42 143)(43 142)(44 141)(45 160)(46 159)(47 158)(48 157)(49 156)(50 155)(51 154)(52 153)(53 152)(54 151)(55 150)(56 149)(57 148)(58 147)(59 146)(60 145)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)
G:=sub<Sym(160)| (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,81)(37,82)(38,83)(39,84)(40,85)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,116)(62,117)(63,118)(64,119)(65,120)(66,101)(67,102)(68,103)(69,104)(70,105)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,113)(79,114)(80,115), (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), (1,53,22,64)(2,52,23,63)(3,51,24,62)(4,50,25,61)(5,49,26,80)(6,48,27,79)(7,47,28,78)(8,46,29,77)(9,45,30,76)(10,44,31,75)(11,43,32,74)(12,42,33,73)(13,41,34,72)(14,60,35,71)(15,59,36,70)(16,58,37,69)(17,57,38,68)(18,56,39,67)(19,55,40,66)(20,54,21,65)(81,105,122,148)(82,104,123,147)(83,103,124,146)(84,102,125,145)(85,101,126,144)(86,120,127,143)(87,119,128,142)(88,118,129,141)(89,117,130,160)(90,116,131,159)(91,115,132,158)(92,114,133,157)(93,113,134,156)(94,112,135,155)(95,111,136,154)(96,110,137,153)(97,109,138,152)(98,108,139,151)(99,107,140,150)(100,106,121,149), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,100)(19,99)(20,98)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,140)(41,144)(42,143)(43,142)(44,141)(45,160)(46,159)(47,158)(48,157)(49,156)(50,155)(51,154)(52,153)(53,152)(54,151)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)>;
G:=Group( (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,81)(37,82)(38,83)(39,84)(40,85)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,116)(62,117)(63,118)(64,119)(65,120)(66,101)(67,102)(68,103)(69,104)(70,105)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,113)(79,114)(80,115), (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), (1,53,22,64)(2,52,23,63)(3,51,24,62)(4,50,25,61)(5,49,26,80)(6,48,27,79)(7,47,28,78)(8,46,29,77)(9,45,30,76)(10,44,31,75)(11,43,32,74)(12,42,33,73)(13,41,34,72)(14,60,35,71)(15,59,36,70)(16,58,37,69)(17,57,38,68)(18,56,39,67)(19,55,40,66)(20,54,21,65)(81,105,122,148)(82,104,123,147)(83,103,124,146)(84,102,125,145)(85,101,126,144)(86,120,127,143)(87,119,128,142)(88,118,129,141)(89,117,130,160)(90,116,131,159)(91,115,132,158)(92,114,133,157)(93,113,134,156)(94,112,135,155)(95,111,136,154)(96,110,137,153)(97,109,138,152)(98,108,139,151)(99,107,140,150)(100,106,121,149), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,100)(19,99)(20,98)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,140)(41,144)(42,143)(43,142)(44,141)(45,160)(46,159)(47,158)(48,157)(49,156)(50,155)(51,154)(52,153)(53,152)(54,151)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113) );
G=PermutationGroup([(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,136),(10,137),(11,138),(12,139),(13,140),(14,121),(15,122),(16,123),(17,124),(18,125),(19,126),(20,127),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,81),(37,82),(38,83),(39,84),(40,85),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,116),(62,117),(63,118),(64,119),(65,120),(66,101),(67,102),(68,103),(69,104),(70,105),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,113),(79,114),(80,115)], [(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)], [(1,53,22,64),(2,52,23,63),(3,51,24,62),(4,50,25,61),(5,49,26,80),(6,48,27,79),(7,47,28,78),(8,46,29,77),(9,45,30,76),(10,44,31,75),(11,43,32,74),(12,42,33,73),(13,41,34,72),(14,60,35,71),(15,59,36,70),(16,58,37,69),(17,57,38,68),(18,56,39,67),(19,55,40,66),(20,54,21,65),(81,105,122,148),(82,104,123,147),(83,103,124,146),(84,102,125,145),(85,101,126,144),(86,120,127,143),(87,119,128,142),(88,118,129,141),(89,117,130,160),(90,116,131,159),(91,115,132,158),(92,114,133,157),(93,113,134,156),(94,112,135,155),(95,111,136,154),(96,110,137,153),(97,109,138,152),(98,108,139,151),(99,107,140,150),(100,106,121,149)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,81),(18,100),(19,99),(20,98),(21,139),(22,138),(23,137),(24,136),(25,135),(26,134),(27,133),(28,132),(29,131),(30,130),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(37,123),(38,122),(39,121),(40,140),(41,144),(42,143),(43,142),(44,141),(45,160),(46,159),(47,158),(48,157),(49,156),(50,155),(51,154),(52,153),(53,152),(54,151),(55,150),(56,149),(57,148),(58,147),(59,146),(60,145),(61,112),(62,111),(63,110),(64,109),(65,108),(66,107),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 8 | 0 | 0 |
| 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 0 | 39 | 25 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 38 | 20 | 0 | 0 |
| 0 | 20 | 3 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 35 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 33 | 0 | 0 |
| 0 | 6 | 34 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 35 | 1 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,40,0,0,0,8,7,0,0,0,0,0,13,39,0,0,0,2,25],[40,0,0,0,0,0,38,20,0,0,0,20,3,0,0,0,0,0,40,35,0,0,0,0,1],[1,0,0,0,0,0,7,6,0,0,0,33,34,0,0,0,0,0,40,35,0,0,0,0,1] >;
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 |
| kernel | C2×C20⋊7D4 | C2×C4⋊Dic5 | C2×D10⋊C4 | C20⋊7D4 | C22×D20 | C22×C5⋊D4 | C23×C20 | C2×C20 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 4 | 4 | 2 | 4 | 12 | 2 | 16 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes_7D_4 % in TeX
G:=Group("C2xC20:7D4"); // GroupNames label
G:=SmallGroup(320,1462);
// by ID
G=gap.SmallGroup(320,1462);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,12550]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀