direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊2D4, C24.39D10, C20⋊7(C2×D4), D10⋊4(C2×D4), (C2×C20)⋊12D4, (C2×D4)⋊36D10, (C22×D4)⋊6D5, C10⋊4(C4⋊D4), (C22×D5)⋊12D4, (D4×C10)⋊43C22, C4⋊Dic5⋊77C22, C22.147(D4×D5), (C2×C10).295C24, (C2×C20).542C23, (C22×C4).379D10, C10.142(C22×D4), C23.D5⋊61C22, (C23×C10).76C22, C22.308(C23×D5), C23.337(C22×D5), C22.79(D4⋊2D5), (C22×C20).275C22, (C22×C10).419C23, (C2×Dic5).152C23, (C23×D5).125C22, (C22×D5).249C23, (C22×Dic5).163C22, (D4×C2×C10)⋊4C2, C4⋊3(C2×C5⋊D4), C5⋊5(C2×C4⋊D4), C2.102(C2×D4×D5), (D5×C22×C4)⋊6C2, (C2×C4×D5)⋊57C22, (C2×C4)⋊13(C5⋊D4), (C2×C4⋊Dic5)⋊45C2, C10.105(C2×C4○D4), C2.69(C2×D4⋊2D5), (C2×C10).580(C2×D4), (C2×C5⋊D4)⋊44C22, (C22×C5⋊D4)⋊13C2, (C2×C23.D5)⋊28C2, C2.15(C22×C5⋊D4), (C2×C4).625(C22×D5), C22.110(C2×C5⋊D4), (C2×C10).177(C4○D4), SmallGroup(320,1472)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1486 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×36], C5, C2×C4 [×6], C2×C4 [×20], D4 [×24], C23, C23 [×4], C23 [×22], D5 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24, Dic5 [×6], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×16], C2×C20 [×6], C5×D4 [×8], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C4⋊D4, C4⋊Dic5 [×4], C23.D5 [×8], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5, C23×C10 [×2], C2×C4⋊Dic5, C20⋊2D4 [×8], C2×C23.D5 [×2], D5×C22×C4, C22×C5⋊D4 [×2], D4×C2×C10, C2×C20⋊2D4
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, C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊D4, D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×6], C23×D5, C20⋊2D4 [×4], C2×D4×D5, C2×D4⋊2D5, C22×C5⋊D4, C2×C20⋊2D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >
►On 160 points
Generators in S160
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 141)(130 142)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 151)(140 152)
(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 67 122 113)(2 66 123 112)(3 65 124 111)(4 64 125 110)(5 63 126 109)(6 62 127 108)(7 61 128 107)(8 80 129 106)(9 79 130 105)(10 78 131 104)(11 77 132 103)(12 76 133 102)(13 75 134 101)(14 74 135 120)(15 73 136 119)(16 72 137 118)(17 71 138 117)(18 70 139 116)(19 69 140 115)(20 68 121 114)(21 43 160 95)(22 42 141 94)(23 41 142 93)(24 60 143 92)(25 59 144 91)(26 58 145 90)(27 57 146 89)(28 56 147 88)(29 55 148 87)(30 54 149 86)(31 53 150 85)(32 52 151 84)(33 51 152 83)(34 50 153 82)(35 49 154 81)(36 48 155 100)(37 47 156 99)(38 46 157 98)(39 45 158 97)(40 44 159 96)
(1 25)(2 34)(3 23)(4 32)(5 21)(6 30)(7 39)(8 28)(9 37)(10 26)(11 35)(12 24)(13 33)(14 22)(15 31)(16 40)(17 29)(18 38)(19 27)(20 36)(41 111)(42 120)(43 109)(44 118)(45 107)(46 116)(47 105)(48 114)(49 103)(50 112)(51 101)(52 110)(53 119)(54 108)(55 117)(56 106)(57 115)(58 104)(59 113)(60 102)(61 97)(62 86)(63 95)(64 84)(65 93)(66 82)(67 91)(68 100)(69 89)(70 98)(71 87)(72 96)(73 85)(74 94)(75 83)(76 92)(77 81)(78 90)(79 99)(80 88)(121 155)(122 144)(123 153)(124 142)(125 151)(126 160)(127 149)(128 158)(129 147)(130 156)(131 145)(132 154)(133 143)(134 152)(135 141)(136 150)(137 159)(138 148)(139 157)(140 146)
G:=sub<Sym(160)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,151)(140,152), (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), (1,67,122,113)(2,66,123,112)(3,65,124,111)(4,64,125,110)(5,63,126,109)(6,62,127,108)(7,61,128,107)(8,80,129,106)(9,79,130,105)(10,78,131,104)(11,77,132,103)(12,76,133,102)(13,75,134,101)(14,74,135,120)(15,73,136,119)(16,72,137,118)(17,71,138,117)(18,70,139,116)(19,69,140,115)(20,68,121,114)(21,43,160,95)(22,42,141,94)(23,41,142,93)(24,60,143,92)(25,59,144,91)(26,58,145,90)(27,57,146,89)(28,56,147,88)(29,55,148,87)(30,54,149,86)(31,53,150,85)(32,52,151,84)(33,51,152,83)(34,50,153,82)(35,49,154,81)(36,48,155,100)(37,47,156,99)(38,46,157,98)(39,45,158,97)(40,44,159,96), (1,25)(2,34)(3,23)(4,32)(5,21)(6,30)(7,39)(8,28)(9,37)(10,26)(11,35)(12,24)(13,33)(14,22)(15,31)(16,40)(17,29)(18,38)(19,27)(20,36)(41,111)(42,120)(43,109)(44,118)(45,107)(46,116)(47,105)(48,114)(49,103)(50,112)(51,101)(52,110)(53,119)(54,108)(55,117)(56,106)(57,115)(58,104)(59,113)(60,102)(61,97)(62,86)(63,95)(64,84)(65,93)(66,82)(67,91)(68,100)(69,89)(70,98)(71,87)(72,96)(73,85)(74,94)(75,83)(76,92)(77,81)(78,90)(79,99)(80,88)(121,155)(122,144)(123,153)(124,142)(125,151)(126,160)(127,149)(128,158)(129,147)(130,156)(131,145)(132,154)(133,143)(134,152)(135,141)(136,150)(137,159)(138,148)(139,157)(140,146)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,151)(140,152), (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), (1,67,122,113)(2,66,123,112)(3,65,124,111)(4,64,125,110)(5,63,126,109)(6,62,127,108)(7,61,128,107)(8,80,129,106)(9,79,130,105)(10,78,131,104)(11,77,132,103)(12,76,133,102)(13,75,134,101)(14,74,135,120)(15,73,136,119)(16,72,137,118)(17,71,138,117)(18,70,139,116)(19,69,140,115)(20,68,121,114)(21,43,160,95)(22,42,141,94)(23,41,142,93)(24,60,143,92)(25,59,144,91)(26,58,145,90)(27,57,146,89)(28,56,147,88)(29,55,148,87)(30,54,149,86)(31,53,150,85)(32,52,151,84)(33,51,152,83)(34,50,153,82)(35,49,154,81)(36,48,155,100)(37,47,156,99)(38,46,157,98)(39,45,158,97)(40,44,159,96), (1,25)(2,34)(3,23)(4,32)(5,21)(6,30)(7,39)(8,28)(9,37)(10,26)(11,35)(12,24)(13,33)(14,22)(15,31)(16,40)(17,29)(18,38)(19,27)(20,36)(41,111)(42,120)(43,109)(44,118)(45,107)(46,116)(47,105)(48,114)(49,103)(50,112)(51,101)(52,110)(53,119)(54,108)(55,117)(56,106)(57,115)(58,104)(59,113)(60,102)(61,97)(62,86)(63,95)(64,84)(65,93)(66,82)(67,91)(68,100)(69,89)(70,98)(71,87)(72,96)(73,85)(74,94)(75,83)(76,92)(77,81)(78,90)(79,99)(80,88)(121,155)(122,144)(123,153)(124,142)(125,151)(126,160)(127,149)(128,158)(129,147)(130,156)(131,145)(132,154)(133,143)(134,152)(135,141)(136,150)(137,159)(138,148)(139,157)(140,146) );
G=PermutationGroup([(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(41,79),(42,80),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,111),(100,112),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,141),(130,142),(131,143),(132,144),(133,145),(134,146),(135,147),(136,148),(137,149),(138,150),(139,151),(140,152)], [(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)], [(1,67,122,113),(2,66,123,112),(3,65,124,111),(4,64,125,110),(5,63,126,109),(6,62,127,108),(7,61,128,107),(8,80,129,106),(9,79,130,105),(10,78,131,104),(11,77,132,103),(12,76,133,102),(13,75,134,101),(14,74,135,120),(15,73,136,119),(16,72,137,118),(17,71,138,117),(18,70,139,116),(19,69,140,115),(20,68,121,114),(21,43,160,95),(22,42,141,94),(23,41,142,93),(24,60,143,92),(25,59,144,91),(26,58,145,90),(27,57,146,89),(28,56,147,88),(29,55,148,87),(30,54,149,86),(31,53,150,85),(32,52,151,84),(33,51,152,83),(34,50,153,82),(35,49,154,81),(36,48,155,100),(37,47,156,99),(38,46,157,98),(39,45,158,97),(40,44,159,96)], [(1,25),(2,34),(3,23),(4,32),(5,21),(6,30),(7,39),(8,28),(9,37),(10,26),(11,35),(12,24),(13,33),(14,22),(15,31),(16,40),(17,29),(18,38),(19,27),(20,36),(41,111),(42,120),(43,109),(44,118),(45,107),(46,116),(47,105),(48,114),(49,103),(50,112),(51,101),(52,110),(53,119),(54,108),(55,117),(56,106),(57,115),(58,104),(59,113),(60,102),(61,97),(62,86),(63,95),(64,84),(65,93),(66,82),(67,91),(68,100),(69,89),(70,98),(71,87),(72,96),(73,85),(74,94),(75,83),(76,92),(77,81),(78,90),(79,99),(80,88),(121,155),(122,144),(123,153),(124,142),(125,151),(126,160),(127,149),(128,158),(129,147),(130,156),(131,145),(132,154),(133,143),(134,152),(135,141),(136,150),(137,159),(138,148),(139,157),(140,146)])
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 | 2 | 0 | 0 |
| 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 24 | 38 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 0 | 1 | 34 |
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],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,7,1,0,0,0,40,0],[1,0,0,0,0,0,40,0,0,0,0,2,1,0,0,0,0,0,3,24,0,0,0,3,38],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,34,34] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C2×C20⋊2D4 | C2×C4⋊Dic5 | C20⋊2D4 | C2×C23.D5 | D5×C22×C4 | C22×C5⋊D4 | D4×C2×C10 | C2×C20 | C22×D5 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 4 | 4 | 2 | 4 | 2 | 8 | 4 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes_2D_4 % in TeX
G:=Group("C2xC20:2D4"); // GroupNames label
G:=SmallGroup(320,1472);
// by ID
G=gap.SmallGroup(320,1472);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,12550]);
// 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^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀