metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D20, C42.112D10, C10.612- (1+4), (C4×D4)⋊17D5, (C5×D4)⋊11D4, (D4×C20)⋊19C2, (C4×D20)⋊31C2, C5⋊3(D4⋊6D4), C4.23(C2×D20), C20.55(C2×D4), C20⋊14(C4○D4), C20⋊7D4⋊10C2, C4⋊5(D4⋊2D5), C4⋊C4.284D10, C20⋊2Q8⋊25C2, C22.2(C2×D20), D10⋊2Q8⋊15C2, (C2×D4).249D10, (C2×C10).99C24, C2.19(C22×D20), C10.17(C22×D4), (C4×C20).155C22, (C2×C20).160C23, C22⋊C4.113D10, (C22×C4).211D10, C22.D20⋊6C2, C4⋊Dic5.39C22, (D4×C10).260C22, (C2×D20).220C22, D10⋊C4.5C22, (C22×C20).81C22, (C22×D5).34C23, C22.124(C23×D5), C23.173(C22×D5), (C22×C10).169C23, (C2×Dic5).216C23, C2.18(D4.10D10), (C2×Dic10).150C22, (C22×Dic5).97C22, (C2×C10).2(C2×D4), (C2×D4⋊2D5)⋊4C2, (C2×C4⋊Dic5)⋊25C2, C10.74(C2×C4○D4), (C2×C4×D5).74C22, C2.22(C2×D4⋊2D5), (C5×C4⋊C4).329C22, (C2×C4).732(C22×D5), (C2×C5⋊D4).15C22, (C5×C22⋊C4).106C22, SmallGroup(320,1227)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1030 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic5 [×6], C20 [×4], C20 [×3], D10 [×6], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×8], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5 [×2], C22×C10 [×2], D4⋊6D4, C4⋊Dic5, C4⋊Dic5 [×8], D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D4⋊2D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20⋊2Q8, C4×D20, C22.D20 [×4], D10⋊2Q8 [×2], C2×C4⋊Dic5 [×2], C20⋊7D4 [×2], D4×C20, C2×D4⋊2D5 [×2], D4⋊6D20
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), D20 [×4], C22×D5 [×7], D4⋊6D4, C2×D20 [×6], D4⋊2D5 [×2], C23×D5, C22×D20, C2×D4⋊2D5, D4.10D10, D4⋊6D20
Generators and relations
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
►On 160 points
Generators in S160
(1 139 33 88)(2 140 34 89)(3 121 35 90)(4 122 36 91)(5 123 37 92)(6 124 38 93)(7 125 39 94)(8 126 40 95)(9 127 21 96)(10 128 22 97)(11 129 23 98)(12 130 24 99)(13 131 25 100)(14 132 26 81)(15 133 27 82)(16 134 28 83)(17 135 29 84)(18 136 30 85)(19 137 31 86)(20 138 32 87)(41 154 75 114)(42 155 76 115)(43 156 77 116)(44 157 78 117)(45 158 79 118)(46 159 80 119)(47 160 61 120)(48 141 62 101)(49 142 63 102)(50 143 64 103)(51 144 65 104)(52 145 66 105)(53 146 67 106)(54 147 68 107)(55 148 69 108)(56 149 70 109)(57 150 71 110)(58 151 72 111)(59 152 73 112)(60 153 74 113)
(1 157)(2 158)(3 159)(4 160)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 101)(38 102)(39 103)(40 104)(41 136)(42 137)(43 138)(44 139)(45 140)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 100)(96 99)(97 98)(101 152)(102 151)(103 150)(104 149)(105 148)(106 147)(107 146)(108 145)(109 144)(110 143)(111 142)(112 141)(113 160)(114 159)(115 158)(116 157)(117 156)(118 155)(119 154)(120 153)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 140)(138 139)
G:=sub<Sym(160)| (1,139,33,88)(2,140,34,89)(3,121,35,90)(4,122,36,91)(5,123,37,92)(6,124,38,93)(7,125,39,94)(8,126,40,95)(9,127,21,96)(10,128,22,97)(11,129,23,98)(12,130,24,99)(13,131,25,100)(14,132,26,81)(15,133,27,82)(16,134,28,83)(17,135,29,84)(18,136,30,85)(19,137,31,86)(20,138,32,87)(41,154,75,114)(42,155,76,115)(43,156,77,116)(44,157,78,117)(45,158,79,118)(46,159,80,119)(47,160,61,120)(48,141,62,101)(49,142,63,102)(50,143,64,103)(51,144,65,104)(52,145,66,105)(53,146,67,106)(54,147,68,107)(55,148,69,108)(56,149,70,109)(57,150,71,110)(58,151,72,111)(59,152,73,112)(60,153,74,113), (1,157)(2,158)(3,159)(4,160)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,101)(38,102)(39,103)(40,104)(41,136)(42,137)(43,138)(44,139)(45,140)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(95,100)(96,99)(97,98)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(137,140)(138,139)>;
G:=Group( (1,139,33,88)(2,140,34,89)(3,121,35,90)(4,122,36,91)(5,123,37,92)(6,124,38,93)(7,125,39,94)(8,126,40,95)(9,127,21,96)(10,128,22,97)(11,129,23,98)(12,130,24,99)(13,131,25,100)(14,132,26,81)(15,133,27,82)(16,134,28,83)(17,135,29,84)(18,136,30,85)(19,137,31,86)(20,138,32,87)(41,154,75,114)(42,155,76,115)(43,156,77,116)(44,157,78,117)(45,158,79,118)(46,159,80,119)(47,160,61,120)(48,141,62,101)(49,142,63,102)(50,143,64,103)(51,144,65,104)(52,145,66,105)(53,146,67,106)(54,147,68,107)(55,148,69,108)(56,149,70,109)(57,150,71,110)(58,151,72,111)(59,152,73,112)(60,153,74,113), (1,157)(2,158)(3,159)(4,160)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,101)(38,102)(39,103)(40,104)(41,136)(42,137)(43,138)(44,139)(45,140)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(95,100)(96,99)(97,98)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(137,140)(138,139) );
G=PermutationGroup([(1,139,33,88),(2,140,34,89),(3,121,35,90),(4,122,36,91),(5,123,37,92),(6,124,38,93),(7,125,39,94),(8,126,40,95),(9,127,21,96),(10,128,22,97),(11,129,23,98),(12,130,24,99),(13,131,25,100),(14,132,26,81),(15,133,27,82),(16,134,28,83),(17,135,29,84),(18,136,30,85),(19,137,31,86),(20,138,32,87),(41,154,75,114),(42,155,76,115),(43,156,77,116),(44,157,78,117),(45,158,79,118),(46,159,80,119),(47,160,61,120),(48,141,62,101),(49,142,63,102),(50,143,64,103),(51,144,65,104),(52,145,66,105),(53,146,67,106),(54,147,68,107),(55,148,69,108),(56,149,70,109),(57,150,71,110),(58,151,72,111),(59,152,73,112),(60,153,74,113)], [(1,157),(2,158),(3,159),(4,160),(5,141),(6,142),(7,143),(8,144),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,101),(38,102),(39,103),(40,104),(41,136),(42,137),(43,138),(44,139),(45,140),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(95,100),(96,99),(97,98),(101,152),(102,151),(103,150),(104,149),(105,148),(106,147),(107,146),(108,145),(109,144),(110,143),(111,142),(112,141),(113,160),(114,159),(115,158),(116,157),(117,156),(118,155),(119,154),(120,153),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(137,140),(138,139)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 32 | 0 |
| 27 | 2 | 0 | 0 |
| 25 | 11 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 30 | 2 | 0 | 0 |
| 22 | 11 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,9,0],[27,25,0,0,2,11,0,0,0,0,40,0,0,0,0,40],[30,22,0,0,2,11,0,0,0,0,40,0,0,0,0,1] >;
65 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 | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | D20 | 2- (1+4) | D4⋊2D5 | D4.10D10 |
| kernel | D4⋊6D20 | C20⋊2Q8 | C4×D20 | C22.D20 | D10⋊2Q8 | C2×C4⋊Dic5 | C20⋊7D4 | D4×C20 | C2×D4⋊2D5 | C5×D4 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_4\rtimes_6D_{20} % in TeX
G:=Group("D4:6D20"); // GroupNames label
G:=SmallGroup(320,1227);
// by ID
G=gap.SmallGroup(320,1227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀