metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊5D20, C42.128D10, C10.112- 1+4, (C4×Q8)⋊9D5, (C5×Q8)⋊10D4, (C4×D20)⋊38C2, (Q8×C20)⋊11C2, C5⋊2(Q8⋊5D4), C20.57(C2×D4), C4.25(C2×D20), C4⋊D20⋊17C2, C4⋊C4.295D10, D10⋊13(C4○D4), D10⋊2Q8⋊18C2, C4.D20⋊20C2, (C2×Q8).204D10, C2.21(C22×D20), C10.19(C22×D4), (C4×C20).172C22, (C2×C20).498C23, (C2×C10).120C24, D10⋊C4.6C22, (C2×D20).224C22, C4⋊Dic5.306C22, (Q8×C10).220C22, (C2×Dic5).54C23, (C22×D5).45C23, C22.141(C23×D5), C2.12(Q8.10D10), (C2×Dic10).155C22, (C2×Q8×D5)⋊4C2, C2.29(D5×C4○D4), (C2×Q8⋊2D5)⋊3C2, (C2×C4×D5).81C22, C10.145(C2×C4○D4), (C5×C4⋊C4).348C22, (C2×C4).168(C22×D5), SmallGroup(320,1248)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊5D20
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1126 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, Q8⋊5D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×D5, Q8⋊2D5, Q8×C10, C4×D20, C4.D20, C4⋊D20, D10⋊2Q8, Q8×C20, C2×Q8×D5, C2×Q8⋊2D5, Q8⋊5D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, D20, C22×D5, Q8⋊5D4, C2×D20, C23×D5, C22×D20, Q8.10D10, D5×C4○D4, Q8⋊5D20
►On 160 points
Generators in S160
(1 55 79 39)(2 56 80 40)(3 57 61 21)(4 58 62 22)(5 59 63 23)(6 60 64 24)(7 41 65 25)(8 42 66 26)(9 43 67 27)(10 44 68 28)(11 45 69 29)(12 46 70 30)(13 47 71 31)(14 48 72 32)(15 49 73 33)(16 50 74 34)(17 51 75 35)(18 52 76 36)(19 53 77 37)(20 54 78 38)(81 160 134 108)(82 141 135 109)(83 142 136 110)(84 143 137 111)(85 144 138 112)(86 145 139 113)(87 146 140 114)(88 147 121 115)(89 148 122 116)(90 149 123 117)(91 150 124 118)(92 151 125 119)(93 152 126 120)(94 153 127 101)(95 154 128 102)(96 155 129 103)(97 156 130 104)(98 157 131 105)(99 158 132 106)(100 159 133 107)
(1 93 79 126)(2 127 80 94)(3 95 61 128)(4 129 62 96)(5 97 63 130)(6 131 64 98)(7 99 65 132)(8 133 66 100)(9 81 67 134)(10 135 68 82)(11 83 69 136)(12 137 70 84)(13 85 71 138)(14 139 72 86)(15 87 73 140)(16 121 74 88)(17 89 75 122)(18 123 76 90)(19 91 77 124)(20 125 78 92)(21 154 57 102)(22 103 58 155)(23 156 59 104)(24 105 60 157)(25 158 41 106)(26 107 42 159)(27 160 43 108)(28 109 44 141)(29 142 45 110)(30 111 46 143)(31 144 47 112)(32 113 48 145)(33 146 49 114)(34 115 50 147)(35 148 51 116)(36 117 52 149)(37 150 53 118)(38 119 54 151)(39 152 55 120)(40 101 56 153)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 122)(82 121)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 155)(102 154)(103 153)(104 152)(105 151)(106 150)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(113 143)(114 142)(115 141)(116 160)(117 159)(118 158)(119 157)(120 156)
G:=sub<Sym(160)| (1,55,79,39)(2,56,80,40)(3,57,61,21)(4,58,62,22)(5,59,63,23)(6,60,64,24)(7,41,65,25)(8,42,66,26)(9,43,67,27)(10,44,68,28)(11,45,69,29)(12,46,70,30)(13,47,71,31)(14,48,72,32)(15,49,73,33)(16,50,74,34)(17,51,75,35)(18,52,76,36)(19,53,77,37)(20,54,78,38)(81,160,134,108)(82,141,135,109)(83,142,136,110)(84,143,137,111)(85,144,138,112)(86,145,139,113)(87,146,140,114)(88,147,121,115)(89,148,122,116)(90,149,123,117)(91,150,124,118)(92,151,125,119)(93,152,126,120)(94,153,127,101)(95,154,128,102)(96,155,129,103)(97,156,130,104)(98,157,131,105)(99,158,132,106)(100,159,133,107), (1,93,79,126)(2,127,80,94)(3,95,61,128)(4,129,62,96)(5,97,63,130)(6,131,64,98)(7,99,65,132)(8,133,66,100)(9,81,67,134)(10,135,68,82)(11,83,69,136)(12,137,70,84)(13,85,71,138)(14,139,72,86)(15,87,73,140)(16,121,74,88)(17,89,75,122)(18,123,76,90)(19,91,77,124)(20,125,78,92)(21,154,57,102)(22,103,58,155)(23,156,59,104)(24,105,60,157)(25,158,41,106)(26,107,42,159)(27,160,43,108)(28,109,44,141)(29,142,45,110)(30,111,46,143)(31,144,47,112)(32,113,48,145)(33,146,49,114)(34,115,50,147)(35,148,51,116)(36,117,52,149)(37,150,53,118)(38,119,54,151)(39,152,55,120)(40,101,56,153), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)(116,160)(117,159)(118,158)(119,157)(120,156)>;
G:=Group( (1,55,79,39)(2,56,80,40)(3,57,61,21)(4,58,62,22)(5,59,63,23)(6,60,64,24)(7,41,65,25)(8,42,66,26)(9,43,67,27)(10,44,68,28)(11,45,69,29)(12,46,70,30)(13,47,71,31)(14,48,72,32)(15,49,73,33)(16,50,74,34)(17,51,75,35)(18,52,76,36)(19,53,77,37)(20,54,78,38)(81,160,134,108)(82,141,135,109)(83,142,136,110)(84,143,137,111)(85,144,138,112)(86,145,139,113)(87,146,140,114)(88,147,121,115)(89,148,122,116)(90,149,123,117)(91,150,124,118)(92,151,125,119)(93,152,126,120)(94,153,127,101)(95,154,128,102)(96,155,129,103)(97,156,130,104)(98,157,131,105)(99,158,132,106)(100,159,133,107), (1,93,79,126)(2,127,80,94)(3,95,61,128)(4,129,62,96)(5,97,63,130)(6,131,64,98)(7,99,65,132)(8,133,66,100)(9,81,67,134)(10,135,68,82)(11,83,69,136)(12,137,70,84)(13,85,71,138)(14,139,72,86)(15,87,73,140)(16,121,74,88)(17,89,75,122)(18,123,76,90)(19,91,77,124)(20,125,78,92)(21,154,57,102)(22,103,58,155)(23,156,59,104)(24,105,60,157)(25,158,41,106)(26,107,42,159)(27,160,43,108)(28,109,44,141)(29,142,45,110)(30,111,46,143)(31,144,47,112)(32,113,48,145)(33,146,49,114)(34,115,50,147)(35,148,51,116)(36,117,52,149)(37,150,53,118)(38,119,54,151)(39,152,55,120)(40,101,56,153), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)(116,160)(117,159)(118,158)(119,157)(120,156) );
G=PermutationGroup([[(1,55,79,39),(2,56,80,40),(3,57,61,21),(4,58,62,22),(5,59,63,23),(6,60,64,24),(7,41,65,25),(8,42,66,26),(9,43,67,27),(10,44,68,28),(11,45,69,29),(12,46,70,30),(13,47,71,31),(14,48,72,32),(15,49,73,33),(16,50,74,34),(17,51,75,35),(18,52,76,36),(19,53,77,37),(20,54,78,38),(81,160,134,108),(82,141,135,109),(83,142,136,110),(84,143,137,111),(85,144,138,112),(86,145,139,113),(87,146,140,114),(88,147,121,115),(89,148,122,116),(90,149,123,117),(91,150,124,118),(92,151,125,119),(93,152,126,120),(94,153,127,101),(95,154,128,102),(96,155,129,103),(97,156,130,104),(98,157,131,105),(99,158,132,106),(100,159,133,107)], [(1,93,79,126),(2,127,80,94),(3,95,61,128),(4,129,62,96),(5,97,63,130),(6,131,64,98),(7,99,65,132),(8,133,66,100),(9,81,67,134),(10,135,68,82),(11,83,69,136),(12,137,70,84),(13,85,71,138),(14,139,72,86),(15,87,73,140),(16,121,74,88),(17,89,75,122),(18,123,76,90),(19,91,77,124),(20,125,78,92),(21,154,57,102),(22,103,58,155),(23,156,59,104),(24,105,60,157),(25,158,41,106),(26,107,42,159),(27,160,43,108),(28,109,44,141),(29,142,45,110),(30,111,46,143),(31,144,47,112),(32,113,48,145),(33,146,49,114),(34,115,50,147),(35,148,51,116),(36,117,52,149),(37,150,53,118),(38,119,54,151),(39,152,55,120),(40,101,56,153)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,122),(82,121),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123),(101,155),(102,154),(103,153),(104,152),(105,151),(106,150),(107,149),(108,148),(109,147),(110,146),(111,145),(112,144),(113,143),(114,142),(115,141),(116,160),(117,159),(118,158),(119,157),(120,156)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D20 | 2- 1+4 | Q8.10D10 | D5×C4○D4 |
| kernel | Q8⋊5D20 | C4×D20 | C4.D20 | C4⋊D20 | D10⋊2Q8 | Q8×C20 | C2×Q8×D5 | C2×Q8⋊2D5 | C5×Q8 | C4×Q8 | D10 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of Q8⋊5D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 28 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 37 |
| 0 | 0 | 0 | 0 | 26 | 12 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,28,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,26,0,0,0,0,37,12],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,35,0,0,0,0,0,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,1,35,0,0,0,0,0,40] >;
Q8⋊5D20 in GAP, Magma, Sage, TeX
Q_8\rtimes_5D_{20} % in TeX
G:=Group("Q8:5D20"); // GroupNames label
G:=SmallGroup(320,1248);
// by ID
G=gap.SmallGroup(320,1248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations