metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊22D4, C10.182- (1+4), C22⋊Q8⋊8D5, C5⋊5(D4⋊6D4), C4.112(D4×D5), C4⋊C4.189D10, D10.44(C2×D4), C20.235(C2×D4), D20⋊8C4⋊26C2, D10⋊D4⋊25C2, D10⋊2Q8⋊25C2, Dic5⋊5(C4○D4), (C2×Q8).126D10, C22⋊C4.16D10, C10.77(C22×D4), Dic5⋊Q8⋊14C2, (C2×C10).175C24, (C2×C20).503C23, (C22×C4).237D10, D10.13D4⋊18C2, D10.12D4⋊25C2, (C2×D20).155C22, C4⋊Dic5.215C22, (Q8×C10).107C22, C22.196(C23×D5), C23.119(C22×D5), D10⋊C4.23C22, (C22×C10).203C23, (C22×C20).255C22, (C2×Dic5).244C23, (C4×Dic5).113C22, C10.D4.27C22, (C22×D5).207C23, C23.D5.116C22, C2.19(Q8.10D10), (C2×Dic10).302C22, C2.50(C2×D4×D5), (D5×C4⋊C4)⋊26C2, (C4×C5⋊D4)⋊23C2, C2.49(D5×C4○D4), (C2×C4○D20)⋊24C2, (C2×Q8⋊2D5)⋊8C2, (C5×C22⋊Q8)⋊11C2, C10.161(C2×C4○D4), (C2×C4×D5).104C22, (C2×C4).48(C22×D5), (C5×C4⋊C4).158C22, (C2×C5⋊D4).131C22, (C5×C22⋊C4).30C22, SmallGroup(320,1303)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1078 in 292 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×14], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×4], C23, C23 [×3], D5 [×5], C10 [×3], C10, C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×7], C2×D4 [×6], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×5], D10 [×4], D10 [×7], C2×C10, C2×C10 [×3], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×2], C4×D5 [×14], D20 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10, D4⋊6D4, C4×Dic5, C10.D4, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5 [×6], C2×D20, C2×D20 [×2], C4○D20 [×4], Q8⋊2D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10.12D4 [×2], D10⋊D4 [×2], D5×C4⋊C4 [×2], D20⋊8C4, D10.13D4 [×2], D10⋊2Q8, C4×C5⋊D4, Dic5⋊Q8, C5×C22⋊Q8, C2×C4○D20, C2×Q8⋊2D5, D20⋊22D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- (1+4), C22×D5 [×7], D4⋊6D4, D4×D5 [×2], C23×D5, C2×D4×D5, Q8.10D10, D5×C4○D4, D20⋊22D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a9, cbc-1=a8b, dbd=a18b, dcd=c-1 >
►On 160 points
Generators in S160
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 70)(21 119)(22 118)(23 117)(24 116)(25 115)(26 114)(27 113)(28 112)(29 111)(30 110)(31 109)(32 108)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 120)(41 135)(42 134)(43 133)(44 132)(45 131)(46 130)(47 129)(48 128)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 140)(57 139)(58 138)(59 137)(60 136)(81 145)(82 144)(83 143)(84 142)(85 141)(86 160)(87 159)(88 158)(89 157)(90 156)(91 155)(92 154)(93 153)(94 152)(95 151)(96 150)(97 149)(98 148)(99 147)(100 146)
(1 144 117 138)(2 153 118 127)(3 142 119 136)(4 151 120 125)(5 160 101 134)(6 149 102 123)(7 158 103 132)(8 147 104 121)(9 156 105 130)(10 145 106 139)(11 154 107 128)(12 143 108 137)(13 152 109 126)(14 141 110 135)(15 150 111 124)(16 159 112 133)(17 148 113 122)(18 157 114 131)(19 146 115 140)(20 155 116 129)(21 52 67 96)(22 41 68 85)(23 50 69 94)(24 59 70 83)(25 48 71 92)(26 57 72 81)(27 46 73 90)(28 55 74 99)(29 44 75 88)(30 53 76 97)(31 42 77 86)(32 51 78 95)(33 60 79 84)(34 49 80 93)(35 58 61 82)(36 47 62 91)(37 56 63 100)(38 45 64 89)(39 54 65 98)(40 43 66 87)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 27)(22 36)(23 25)(24 34)(26 32)(28 30)(29 39)(31 37)(33 35)(38 40)(41 91)(42 100)(43 89)(44 98)(45 87)(46 96)(47 85)(48 94)(49 83)(50 92)(51 81)(52 90)(53 99)(54 88)(55 97)(56 86)(57 95)(58 84)(59 93)(60 82)(61 79)(62 68)(63 77)(64 66)(65 75)(67 73)(69 71)(70 80)(72 78)(74 76)(101 113)(103 111)(104 120)(105 109)(106 118)(108 116)(110 114)(115 119)(121 151)(122 160)(123 149)(124 158)(125 147)(126 156)(127 145)(128 154)(129 143)(130 152)(131 141)(132 150)(133 159)(134 148)(135 157)(136 146)(137 155)(138 144)(139 153)(140 142)
G:=sub<Sym(160)| (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,112)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,120)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,140)(57,139)(58,138)(59,137)(60,136)(81,145)(82,144)(83,143)(84,142)(85,141)(86,160)(87,159)(88,158)(89,157)(90,156)(91,155)(92,154)(93,153)(94,152)(95,151)(96,150)(97,149)(98,148)(99,147)(100,146), (1,144,117,138)(2,153,118,127)(3,142,119,136)(4,151,120,125)(5,160,101,134)(6,149,102,123)(7,158,103,132)(8,147,104,121)(9,156,105,130)(10,145,106,139)(11,154,107,128)(12,143,108,137)(13,152,109,126)(14,141,110,135)(15,150,111,124)(16,159,112,133)(17,148,113,122)(18,157,114,131)(19,146,115,140)(20,155,116,129)(21,52,67,96)(22,41,68,85)(23,50,69,94)(24,59,70,83)(25,48,71,92)(26,57,72,81)(27,46,73,90)(28,55,74,99)(29,44,75,88)(30,53,76,97)(31,42,77,86)(32,51,78,95)(33,60,79,84)(34,49,80,93)(35,58,61,82)(36,47,62,91)(37,56,63,100)(38,45,64,89)(39,54,65,98)(40,43,66,87), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,27)(22,36)(23,25)(24,34)(26,32)(28,30)(29,39)(31,37)(33,35)(38,40)(41,91)(42,100)(43,89)(44,98)(45,87)(46,96)(47,85)(48,94)(49,83)(50,92)(51,81)(52,90)(53,99)(54,88)(55,97)(56,86)(57,95)(58,84)(59,93)(60,82)(61,79)(62,68)(63,77)(64,66)(65,75)(67,73)(69,71)(70,80)(72,78)(74,76)(101,113)(103,111)(104,120)(105,109)(106,118)(108,116)(110,114)(115,119)(121,151)(122,160)(123,149)(124,158)(125,147)(126,156)(127,145)(128,154)(129,143)(130,152)(131,141)(132,150)(133,159)(134,148)(135,157)(136,146)(137,155)(138,144)(139,153)(140,142)>;
G:=Group( (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,112)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,120)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,140)(57,139)(58,138)(59,137)(60,136)(81,145)(82,144)(83,143)(84,142)(85,141)(86,160)(87,159)(88,158)(89,157)(90,156)(91,155)(92,154)(93,153)(94,152)(95,151)(96,150)(97,149)(98,148)(99,147)(100,146), (1,144,117,138)(2,153,118,127)(3,142,119,136)(4,151,120,125)(5,160,101,134)(6,149,102,123)(7,158,103,132)(8,147,104,121)(9,156,105,130)(10,145,106,139)(11,154,107,128)(12,143,108,137)(13,152,109,126)(14,141,110,135)(15,150,111,124)(16,159,112,133)(17,148,113,122)(18,157,114,131)(19,146,115,140)(20,155,116,129)(21,52,67,96)(22,41,68,85)(23,50,69,94)(24,59,70,83)(25,48,71,92)(26,57,72,81)(27,46,73,90)(28,55,74,99)(29,44,75,88)(30,53,76,97)(31,42,77,86)(32,51,78,95)(33,60,79,84)(34,49,80,93)(35,58,61,82)(36,47,62,91)(37,56,63,100)(38,45,64,89)(39,54,65,98)(40,43,66,87), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,27)(22,36)(23,25)(24,34)(26,32)(28,30)(29,39)(31,37)(33,35)(38,40)(41,91)(42,100)(43,89)(44,98)(45,87)(46,96)(47,85)(48,94)(49,83)(50,92)(51,81)(52,90)(53,99)(54,88)(55,97)(56,86)(57,95)(58,84)(59,93)(60,82)(61,79)(62,68)(63,77)(64,66)(65,75)(67,73)(69,71)(70,80)(72,78)(74,76)(101,113)(103,111)(104,120)(105,109)(106,118)(108,116)(110,114)(115,119)(121,151)(122,160)(123,149)(124,158)(125,147)(126,156)(127,145)(128,154)(129,143)(130,152)(131,141)(132,150)(133,159)(134,148)(135,157)(136,146)(137,155)(138,144)(139,153)(140,142) );
G=PermutationGroup([(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,80),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,73),(18,72),(19,71),(20,70),(21,119),(22,118),(23,117),(24,116),(25,115),(26,114),(27,113),(28,112),(29,111),(30,110),(31,109),(32,108),(33,107),(34,106),(35,105),(36,104),(37,103),(38,102),(39,101),(40,120),(41,135),(42,134),(43,133),(44,132),(45,131),(46,130),(47,129),(48,128),(49,127),(50,126),(51,125),(52,124),(53,123),(54,122),(55,121),(56,140),(57,139),(58,138),(59,137),(60,136),(81,145),(82,144),(83,143),(84,142),(85,141),(86,160),(87,159),(88,158),(89,157),(90,156),(91,155),(92,154),(93,153),(94,152),(95,151),(96,150),(97,149),(98,148),(99,147),(100,146)], [(1,144,117,138),(2,153,118,127),(3,142,119,136),(4,151,120,125),(5,160,101,134),(6,149,102,123),(7,158,103,132),(8,147,104,121),(9,156,105,130),(10,145,106,139),(11,154,107,128),(12,143,108,137),(13,152,109,126),(14,141,110,135),(15,150,111,124),(16,159,112,133),(17,148,113,122),(18,157,114,131),(19,146,115,140),(20,155,116,129),(21,52,67,96),(22,41,68,85),(23,50,69,94),(24,59,70,83),(25,48,71,92),(26,57,72,81),(27,46,73,90),(28,55,74,99),(29,44,75,88),(30,53,76,97),(31,42,77,86),(32,51,78,95),(33,60,79,84),(34,49,80,93),(35,58,61,82),(36,47,62,91),(37,56,63,100),(38,45,64,89),(39,54,65,98),(40,43,66,87)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,27),(22,36),(23,25),(24,34),(26,32),(28,30),(29,39),(31,37),(33,35),(38,40),(41,91),(42,100),(43,89),(44,98),(45,87),(46,96),(47,85),(48,94),(49,83),(50,92),(51,81),(52,90),(53,99),(54,88),(55,97),(56,86),(57,95),(58,84),(59,93),(60,82),(61,79),(62,68),(63,77),(64,66),(65,75),(67,73),(69,71),(70,80),(72,78),(74,76),(101,113),(103,111),(104,120),(105,109),(106,118),(108,116),(110,114),(115,119),(121,151),(122,160),(123,149),(124,158),(125,147),(126,156),(127,145),(128,154),(129,143),(130,152),(131,141),(132,150),(133,159),(134,148),(135,157),(136,146),(137,155),(138,144),(139,153),(140,142)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 37 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 6 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 37 | 0 | 0 | 0 | 0 |
| 26 | 29 | 0 | 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 | 6 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 5 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,37,9,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,4,0,0,0,0,6,10,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,26,0,0,0,0,37,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
53 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8.10D10 | D5×C4○D4 |
| kernel | D20⋊22D4 | D10.12D4 | D10⋊D4 | D5×C4⋊C4 | D20⋊8C4 | D10.13D4 | D10⋊2Q8 | C4×C5⋊D4 | Dic5⋊Q8 | C5×C22⋊Q8 | C2×C4○D20 | C2×Q8⋊2D5 | D20 | C22⋊Q8 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{22}D_4 % in TeX
G:=Group("D20:22D4"); // GroupNames label
G:=SmallGroup(320,1303);
// by ID
G=gap.SmallGroup(320,1303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^9,c*b*c^-1=a^8*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀