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, (C2×D20).220C22, D10⋊C4.5C22, (D4×C10).260C22, (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
Generators and relations for D4⋊6D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1030 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, D4⋊6D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C20⋊2Q8, C4×D20, C22.D20, D10⋊2Q8, C2×C4⋊Dic5, C20⋊7D4, D4×C20, C2×D4⋊2D5, D4⋊6D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, D20, C22×D5, D4⋊6D4, C2×D20, D4⋊2D5, C23×D5, C22×D20, C2×D4⋊2D5, D4.10D10, D4⋊6D20
►On 160 points
Generators in S160
(1 90 120 157)(2 91 101 158)(3 92 102 159)(4 93 103 160)(5 94 104 141)(6 95 105 142)(7 96 106 143)(8 97 107 144)(9 98 108 145)(10 99 109 146)(11 100 110 147)(12 81 111 148)(13 82 112 149)(14 83 113 150)(15 84 114 151)(16 85 115 152)(17 86 116 153)(18 87 117 154)(19 88 118 155)(20 89 119 156)(21 78 44 133)(22 79 45 134)(23 80 46 135)(24 61 47 136)(25 62 48 137)(26 63 49 138)(27 64 50 139)(28 65 51 140)(29 66 52 121)(30 67 53 122)(31 68 54 123)(32 69 55 124)(33 70 56 125)(34 71 57 126)(35 72 58 127)(36 73 59 128)(37 74 60 129)(38 75 41 130)(39 76 42 131)(40 77 43 132)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 21)(41 116)(42 117)(43 118)(44 119)(45 120)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 159)(62 160)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 121)(98 122)(99 123)(100 124)
(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 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(61 131)(62 130)(63 129)(64 128)(65 127)(66 126)(67 125)(68 124)(69 123)(70 122)(71 121)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 132)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 120)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 160)(154 159)(155 158)(156 157)
G:=sub<Sym(160)| (1,90,120,157)(2,91,101,158)(3,92,102,159)(4,93,103,160)(5,94,104,141)(6,95,105,142)(7,96,106,143)(8,97,107,144)(9,98,108,145)(10,99,109,146)(11,100,110,147)(12,81,111,148)(13,82,112,149)(14,83,113,150)(15,84,114,151)(16,85,115,152)(17,86,116,153)(18,87,117,154)(19,88,118,155)(20,89,119,156)(21,78,44,133)(22,79,45,134)(23,80,46,135)(24,61,47,136)(25,62,48,137)(26,63,49,138)(27,64,50,139)(28,65,51,140)(29,66,52,121)(30,67,53,122)(31,68,54,123)(32,69,55,124)(33,70,56,125)(34,71,57,126)(35,72,58,127)(36,73,59,128)(37,74,60,129)(38,75,41,130)(39,76,42,131)(40,77,43,132), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,21)(41,116)(42,117)(43,118)(44,119)(45,120)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,159)(62,160)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,121)(98,122)(99,123)(100,124), (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)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(61,131)(62,130)(63,129)(64,128)(65,127)(66,126)(67,125)(68,124)(69,123)(70,122)(71,121)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,132)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,120)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(153,160)(154,159)(155,158)(156,157)>;
G:=Group( (1,90,120,157)(2,91,101,158)(3,92,102,159)(4,93,103,160)(5,94,104,141)(6,95,105,142)(7,96,106,143)(8,97,107,144)(9,98,108,145)(10,99,109,146)(11,100,110,147)(12,81,111,148)(13,82,112,149)(14,83,113,150)(15,84,114,151)(16,85,115,152)(17,86,116,153)(18,87,117,154)(19,88,118,155)(20,89,119,156)(21,78,44,133)(22,79,45,134)(23,80,46,135)(24,61,47,136)(25,62,48,137)(26,63,49,138)(27,64,50,139)(28,65,51,140)(29,66,52,121)(30,67,53,122)(31,68,54,123)(32,69,55,124)(33,70,56,125)(34,71,57,126)(35,72,58,127)(36,73,59,128)(37,74,60,129)(38,75,41,130)(39,76,42,131)(40,77,43,132), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,21)(41,116)(42,117)(43,118)(44,119)(45,120)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,159)(62,160)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,121)(98,122)(99,123)(100,124), (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)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(61,131)(62,130)(63,129)(64,128)(65,127)(66,126)(67,125)(68,124)(69,123)(70,122)(71,121)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,132)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(119,120)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(153,160)(154,159)(155,158)(156,157) );
G=PermutationGroup([[(1,90,120,157),(2,91,101,158),(3,92,102,159),(4,93,103,160),(5,94,104,141),(6,95,105,142),(7,96,106,143),(8,97,107,144),(9,98,108,145),(10,99,109,146),(11,100,110,147),(12,81,111,148),(13,82,112,149),(14,83,113,150),(15,84,114,151),(16,85,115,152),(17,86,116,153),(18,87,117,154),(19,88,118,155),(20,89,119,156),(21,78,44,133),(22,79,45,134),(23,80,46,135),(24,61,47,136),(25,62,48,137),(26,63,49,138),(27,64,50,139),(28,65,51,140),(29,66,52,121),(30,67,53,122),(31,68,54,123),(32,69,55,124),(33,70,56,125),(34,71,57,126),(35,72,58,127),(36,73,59,128),(37,74,60,129),(38,75,41,130),(39,76,42,131),(40,77,43,132)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,21),(41,116),(42,117),(43,118),(44,119),(45,120),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,159),(62,160),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156),(79,157),(80,158),(81,125),(82,126),(83,127),(84,128),(85,129),(86,130),(87,131),(88,132),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(97,121),(98,122),(99,123),(100,124)], [(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),(21,45),(22,44),(23,43),(24,42),(25,41),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(61,131),(62,130),(63,129),(64,128),(65,127),(66,126),(67,125),(68,124),(69,123),(70,122),(71,121),(72,140),(73,139),(74,138),(75,137),(76,136),(77,135),(78,134),(79,133),(80,132),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(99,100),(101,118),(102,117),(103,116),(104,115),(105,114),(106,113),(107,112),(108,111),(109,110),(119,120),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(153,160),(154,159),(155,158),(156,157)]])
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 |
Matrix representation of D4⋊6D20 ►in 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] >;
D4⋊6D20 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