metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊7D4, (C5×Q8)⋊6D4, C4.64(D4×D5), C20⋊2D4⋊7C2, C5⋊6(D4⋊D4), Q8⋊3(C5⋊D4), C20.49(C2×D4), (C2×SD16)⋊13D5, (C2×D4).74D10, (C2×C8).148D10, D20⋊5C4⋊36C2, D10⋊1C8⋊34C2, C10.59C22≀C2, (C10×SD16)⋊22C2, C10.64(C4○D8), Q8⋊Dic5⋊30C2, (C2×Q8).117D10, (C22×D5).45D4, C22.269(D4×D5), C2.29(D40⋊C2), C10.79(C8⋊C22), (C2×C40).295C22, (C2×C20).449C23, (C2×Dic5).240D4, (D4×C10).98C22, (Q8×C10).78C22, C2.27(C23⋊D10), (C2×D20).125C22, C4⋊Dic5.176C22, C2.30(SD16⋊3D5), (C2×D4⋊D5)⋊20C2, C4.44(C2×C5⋊D4), (C2×Q8⋊2D5)⋊2C2, (C2×C4×D5).53C22, (C2×C10).361(C2×D4), (C2×C4).538(C22×D5), (C2×C5⋊2C8).158C22, SmallGroup(320,799)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊7D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, dad=a11, cbc-1=a3b, dbd=a5b, dcd=c-1 >
Subgroups: 750 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×D5, C22×C10, D4⋊D4, C2×C5⋊2C8, C4⋊Dic5, D4⋊D5, C23.D5, C2×C40, C5×SD16, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, C2×C5⋊D4, D4×C10, Q8×C10, D10⋊1C8, D20⋊5C4, Q8⋊Dic5, C2×D4⋊D5, C20⋊2D4, C10×SD16, C2×Q8⋊2D5, D20⋊7D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4⋊D4, D4×D5, C2×C5⋊D4, D40⋊C2, SD16⋊3D5, C23⋊D10, D20⋊7D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 100)(95 99)(96 98)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)(141 149)(142 148)(143 147)(144 146)(150 160)(151 159)(152 158)(153 157)(154 156)
(1 126 57 110)(2 125 58 109)(3 124 59 108)(4 123 60 107)(5 122 41 106)(6 121 42 105)(7 140 43 104)(8 139 44 103)(9 138 45 102)(10 137 46 101)(11 136 47 120)(12 135 48 119)(13 134 49 118)(14 133 50 117)(15 132 51 116)(16 131 52 115)(17 130 53 114)(18 129 54 113)(19 128 55 112)(20 127 56 111)(21 84 74 152)(22 83 75 151)(23 82 76 150)(24 81 77 149)(25 100 78 148)(26 99 79 147)(27 98 80 146)(28 97 61 145)(29 96 62 144)(30 95 63 143)(31 94 64 142)(32 93 65 141)(33 92 66 160)(34 91 67 159)(35 90 68 158)(36 89 69 157)(37 88 70 156)(38 87 71 155)(39 86 72 154)(40 85 73 153)
(1 148)(2 159)(3 150)(4 141)(5 152)(6 143)(7 154)(8 145)(9 156)(10 147)(11 158)(12 149)(13 160)(14 151)(15 142)(16 153)(17 144)(18 155)(19 146)(20 157)(21 106)(22 117)(23 108)(24 119)(25 110)(26 101)(27 112)(28 103)(29 114)(30 105)(31 116)(32 107)(33 118)(34 109)(35 120)(36 111)(37 102)(38 113)(39 104)(40 115)(41 84)(42 95)(43 86)(44 97)(45 88)(46 99)(47 90)(48 81)(49 92)(50 83)(51 94)(52 85)(53 96)(54 87)(55 98)(56 89)(57 100)(58 91)(59 82)(60 93)(61 139)(62 130)(63 121)(64 132)(65 123)(66 134)(67 125)(68 136)(69 127)(70 138)(71 129)(72 140)(73 131)(74 122)(75 133)(76 124)(77 135)(78 126)(79 137)(80 128)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,126,57,110)(2,125,58,109)(3,124,59,108)(4,123,60,107)(5,122,41,106)(6,121,42,105)(7,140,43,104)(8,139,44,103)(9,138,45,102)(10,137,46,101)(11,136,47,120)(12,135,48,119)(13,134,49,118)(14,133,50,117)(15,132,51,116)(16,131,52,115)(17,130,53,114)(18,129,54,113)(19,128,55,112)(20,127,56,111)(21,84,74,152)(22,83,75,151)(23,82,76,150)(24,81,77,149)(25,100,78,148)(26,99,79,147)(27,98,80,146)(28,97,61,145)(29,96,62,144)(30,95,63,143)(31,94,64,142)(32,93,65,141)(33,92,66,160)(34,91,67,159)(35,90,68,158)(36,89,69,157)(37,88,70,156)(38,87,71,155)(39,86,72,154)(40,85,73,153), (1,148)(2,159)(3,150)(4,141)(5,152)(6,143)(7,154)(8,145)(9,156)(10,147)(11,158)(12,149)(13,160)(14,151)(15,142)(16,153)(17,144)(18,155)(19,146)(20,157)(21,106)(22,117)(23,108)(24,119)(25,110)(26,101)(27,112)(28,103)(29,114)(30,105)(31,116)(32,107)(33,118)(34,109)(35,120)(36,111)(37,102)(38,113)(39,104)(40,115)(41,84)(42,95)(43,86)(44,97)(45,88)(46,99)(47,90)(48,81)(49,92)(50,83)(51,94)(52,85)(53,96)(54,87)(55,98)(56,89)(57,100)(58,91)(59,82)(60,93)(61,139)(62,130)(63,121)(64,132)(65,123)(66,134)(67,125)(68,136)(69,127)(70,138)(71,129)(72,140)(73,131)(74,122)(75,133)(76,124)(77,135)(78,126)(79,137)(80,128)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,126,57,110)(2,125,58,109)(3,124,59,108)(4,123,60,107)(5,122,41,106)(6,121,42,105)(7,140,43,104)(8,139,44,103)(9,138,45,102)(10,137,46,101)(11,136,47,120)(12,135,48,119)(13,134,49,118)(14,133,50,117)(15,132,51,116)(16,131,52,115)(17,130,53,114)(18,129,54,113)(19,128,55,112)(20,127,56,111)(21,84,74,152)(22,83,75,151)(23,82,76,150)(24,81,77,149)(25,100,78,148)(26,99,79,147)(27,98,80,146)(28,97,61,145)(29,96,62,144)(30,95,63,143)(31,94,64,142)(32,93,65,141)(33,92,66,160)(34,91,67,159)(35,90,68,158)(36,89,69,157)(37,88,70,156)(38,87,71,155)(39,86,72,154)(40,85,73,153), (1,148)(2,159)(3,150)(4,141)(5,152)(6,143)(7,154)(8,145)(9,156)(10,147)(11,158)(12,149)(13,160)(14,151)(15,142)(16,153)(17,144)(18,155)(19,146)(20,157)(21,106)(22,117)(23,108)(24,119)(25,110)(26,101)(27,112)(28,103)(29,114)(30,105)(31,116)(32,107)(33,118)(34,109)(35,120)(36,111)(37,102)(38,113)(39,104)(40,115)(41,84)(42,95)(43,86)(44,97)(45,88)(46,99)(47,90)(48,81)(49,92)(50,83)(51,94)(52,85)(53,96)(54,87)(55,98)(56,89)(57,100)(58,91)(59,82)(60,93)(61,139)(62,130)(63,121)(64,132)(65,123)(66,134)(67,125)(68,136)(69,127)(70,138)(71,129)(72,140)(73,131)(74,122)(75,133)(76,124)(77,135)(78,126)(79,137)(80,128) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,100),(95,99),(96,98),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,140),(137,139),(141,149),(142,148),(143,147),(144,146),(150,160),(151,159),(152,158),(153,157),(154,156)], [(1,126,57,110),(2,125,58,109),(3,124,59,108),(4,123,60,107),(5,122,41,106),(6,121,42,105),(7,140,43,104),(8,139,44,103),(9,138,45,102),(10,137,46,101),(11,136,47,120),(12,135,48,119),(13,134,49,118),(14,133,50,117),(15,132,51,116),(16,131,52,115),(17,130,53,114),(18,129,54,113),(19,128,55,112),(20,127,56,111),(21,84,74,152),(22,83,75,151),(23,82,76,150),(24,81,77,149),(25,100,78,148),(26,99,79,147),(27,98,80,146),(28,97,61,145),(29,96,62,144),(30,95,63,143),(31,94,64,142),(32,93,65,141),(33,92,66,160),(34,91,67,159),(35,90,68,158),(36,89,69,157),(37,88,70,156),(38,87,71,155),(39,86,72,154),(40,85,73,153)], [(1,148),(2,159),(3,150),(4,141),(5,152),(6,143),(7,154),(8,145),(9,156),(10,147),(11,158),(12,149),(13,160),(14,151),(15,142),(16,153),(17,144),(18,155),(19,146),(20,157),(21,106),(22,117),(23,108),(24,119),(25,110),(26,101),(27,112),(28,103),(29,114),(30,105),(31,116),(32,107),(33,118),(34,109),(35,120),(36,111),(37,102),(38,113),(39,104),(40,115),(41,84),(42,95),(43,86),(44,97),(45,88),(46,99),(47,90),(48,81),(49,92),(50,83),(51,94),(52,85),(53,96),(54,87),(55,98),(56,89),(57,100),(58,91),(59,82),(60,93),(61,139),(62,130),(63,121),(64,132),(65,123),(66,134),(67,125),(68,136),(69,127),(70,138),(71,129),(72,140),(73,131),(74,122),(75,133),(76,124),(77,135),(78,126),(79,137),(80,128)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 | SD16⋊3D5 |
| kernel | D20⋊7D4 | D10⋊1C8 | D20⋊5C4 | Q8⋊Dic5 | C2×D4⋊D5 | C20⋊2D4 | C10×SD16 | C2×Q8⋊2D5 | D20 | C2×Dic5 | C5×Q8 | C22×D5 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C10 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20⋊7D4 ►in GL4(𝔽41) generated by
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 8 | 1 |
| 1 | 2 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 40 |
| 30 | 30 | 0 | 0 |
| 26 | 11 | 0 | 0 |
| 0 | 0 | 38 | 20 |
| 0 | 0 | 20 | 3 |
| 0 | 17 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 17 | 35 |
| 0 | 0 | 7 | 24 |
G:=sub<GL(4,GF(41))| [1,40,0,0,2,40,0,0,0,0,34,8,0,0,40,1],[1,0,0,0,2,40,0,0,0,0,1,0,0,0,1,40],[30,26,0,0,30,11,0,0,0,0,38,20,0,0,20,3],[0,29,0,0,17,0,0,0,0,0,17,7,0,0,35,24] >;
D20⋊7D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_7D_4 % in TeX
G:=Group("D20:7D4"); // GroupNames label
G:=SmallGroup(320,799);
// by ID
G=gap.SmallGroup(320,799);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^11,c*b*c^-1=a^3*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations