metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊1D20, C20⋊13SD16, C42.57D10, (C4×Q8)⋊3D5, (C5×Q8)⋊8D4, C4⋊3(Q8⋊D5), (Q8×C20)⋊3C2, C20⋊3C8⋊26C2, C5⋊3(C4⋊SD16), C20.20(C2×D4), (C2×C20).66D4, C4.16(C2×D20), C4⋊C4.253D10, C20⋊4D4.7C2, D20⋊6C4⋊32C2, C4.12(C4○D20), C20.60(C4○D4), (C4×C20).98C22, (C2×Q8).160D10, C10.70(C2×SD16), C2.15(C20⋊7D4), C10.67(C4⋊D4), (C2×C20).347C23, (C2×D20).99C22, C2.10(D4⋊D10), C10.112(C8⋊C22), (Q8×C10).195C22, (C2×Q8⋊D5)⋊7C2, C2.7(C2×Q8⋊D5), (C2×C10).478(C2×D4), (C2×C4).249(C5⋊D4), (C5×C4⋊C4).284C22, (C2×C4).447(C22×D5), C22.155(C2×C5⋊D4), (C2×C5⋊2C8).101C22, SmallGroup(320,654)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D20
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Subgroups: 646 in 128 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C5⋊2C8, D20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C4⋊SD16, C2×C5⋊2C8, Q8⋊D5, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×D20, Q8×C10, C20⋊3C8, D20⋊6C4, C20⋊4D4, C2×Q8⋊D5, Q8×C20, Q8⋊D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, D20, C5⋊D4, C22×D5, C4⋊SD16, Q8⋊D5, C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, C2×Q8⋊D5, D4⋊D10, Q8⋊D20
►On 160 points
Generators in S160
(1 134 148 98)(2 135 149 99)(3 136 150 100)(4 137 151 81)(5 138 152 82)(6 139 153 83)(7 140 154 84)(8 121 155 85)(9 122 156 86)(10 123 157 87)(11 124 158 88)(12 125 159 89)(13 126 160 90)(14 127 141 91)(15 128 142 92)(16 129 143 93)(17 130 144 94)(18 131 145 95)(19 132 146 96)(20 133 147 97)(21 53 72 108)(22 54 73 109)(23 55 74 110)(24 56 75 111)(25 57 76 112)(26 58 77 113)(27 59 78 114)(28 60 79 115)(29 41 80 116)(30 42 61 117)(31 43 62 118)(32 44 63 119)(33 45 64 120)(34 46 65 101)(35 47 66 102)(36 48 67 103)(37 49 68 104)(38 50 69 105)(39 51 70 106)(40 52 71 107)
(1 29 148 80)(2 30 149 61)(3 31 150 62)(4 32 151 63)(5 33 152 64)(6 34 153 65)(7 35 154 66)(8 36 155 67)(9 37 156 68)(10 38 157 69)(11 39 158 70)(12 40 159 71)(13 21 160 72)(14 22 141 73)(15 23 142 74)(16 24 143 75)(17 25 144 76)(18 26 145 77)(19 27 146 78)(20 28 147 79)(41 134 116 98)(42 135 117 99)(43 136 118 100)(44 137 119 81)(45 138 120 82)(46 139 101 83)(47 140 102 84)(48 121 103 85)(49 122 104 86)(50 123 105 87)(51 124 106 88)(52 125 107 89)(53 126 108 90)(54 127 109 91)(55 128 110 92)(56 129 111 93)(57 130 112 94)(58 131 113 95)(59 132 114 96)(60 133 115 97)
(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 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(81 130)(82 129)(83 128)(84 127)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 140)(92 139)(93 138)(94 137)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 160)(156 159)(157 158)
G:=sub<Sym(160)| (1,134,148,98)(2,135,149,99)(3,136,150,100)(4,137,151,81)(5,138,152,82)(6,139,153,83)(7,140,154,84)(8,121,155,85)(9,122,156,86)(10,123,157,87)(11,124,158,88)(12,125,159,89)(13,126,160,90)(14,127,141,91)(15,128,142,92)(16,129,143,93)(17,130,144,94)(18,131,145,95)(19,132,146,96)(20,133,147,97)(21,53,72,108)(22,54,73,109)(23,55,74,110)(24,56,75,111)(25,57,76,112)(26,58,77,113)(27,59,78,114)(28,60,79,115)(29,41,80,116)(30,42,61,117)(31,43,62,118)(32,44,63,119)(33,45,64,120)(34,46,65,101)(35,47,66,102)(36,48,67,103)(37,49,68,104)(38,50,69,105)(39,51,70,106)(40,52,71,107), (1,29,148,80)(2,30,149,61)(3,31,150,62)(4,32,151,63)(5,33,152,64)(6,34,153,65)(7,35,154,66)(8,36,155,67)(9,37,156,68)(10,38,157,69)(11,39,158,70)(12,40,159,71)(13,21,160,72)(14,22,141,73)(15,23,142,74)(16,24,143,75)(17,25,144,76)(18,26,145,77)(19,27,146,78)(20,28,147,79)(41,134,116,98)(42,135,117,99)(43,136,118,100)(44,137,119,81)(45,138,120,82)(46,139,101,83)(47,140,102,84)(48,121,103,85)(49,122,104,86)(50,123,105,87)(51,124,106,88)(52,125,107,89)(53,126,108,90)(54,127,109,91)(55,128,110,92)(56,129,111,93)(57,130,112,94)(58,131,113,95)(59,132,114,96)(60,133,115,97), (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)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)>;
G:=Group( (1,134,148,98)(2,135,149,99)(3,136,150,100)(4,137,151,81)(5,138,152,82)(6,139,153,83)(7,140,154,84)(8,121,155,85)(9,122,156,86)(10,123,157,87)(11,124,158,88)(12,125,159,89)(13,126,160,90)(14,127,141,91)(15,128,142,92)(16,129,143,93)(17,130,144,94)(18,131,145,95)(19,132,146,96)(20,133,147,97)(21,53,72,108)(22,54,73,109)(23,55,74,110)(24,56,75,111)(25,57,76,112)(26,58,77,113)(27,59,78,114)(28,60,79,115)(29,41,80,116)(30,42,61,117)(31,43,62,118)(32,44,63,119)(33,45,64,120)(34,46,65,101)(35,47,66,102)(36,48,67,103)(37,49,68,104)(38,50,69,105)(39,51,70,106)(40,52,71,107), (1,29,148,80)(2,30,149,61)(3,31,150,62)(4,32,151,63)(5,33,152,64)(6,34,153,65)(7,35,154,66)(8,36,155,67)(9,37,156,68)(10,38,157,69)(11,39,158,70)(12,40,159,71)(13,21,160,72)(14,22,141,73)(15,23,142,74)(16,24,143,75)(17,25,144,76)(18,26,145,77)(19,27,146,78)(20,28,147,79)(41,134,116,98)(42,135,117,99)(43,136,118,100)(44,137,119,81)(45,138,120,82)(46,139,101,83)(47,140,102,84)(48,121,103,85)(49,122,104,86)(50,123,105,87)(51,124,106,88)(52,125,107,89)(53,126,108,90)(54,127,109,91)(55,128,110,92)(56,129,111,93)(57,130,112,94)(58,131,113,95)(59,132,114,96)(60,133,115,97), (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)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158) );
G=PermutationGroup([[(1,134,148,98),(2,135,149,99),(3,136,150,100),(4,137,151,81),(5,138,152,82),(6,139,153,83),(7,140,154,84),(8,121,155,85),(9,122,156,86),(10,123,157,87),(11,124,158,88),(12,125,159,89),(13,126,160,90),(14,127,141,91),(15,128,142,92),(16,129,143,93),(17,130,144,94),(18,131,145,95),(19,132,146,96),(20,133,147,97),(21,53,72,108),(22,54,73,109),(23,55,74,110),(24,56,75,111),(25,57,76,112),(26,58,77,113),(27,59,78,114),(28,60,79,115),(29,41,80,116),(30,42,61,117),(31,43,62,118),(32,44,63,119),(33,45,64,120),(34,46,65,101),(35,47,66,102),(36,48,67,103),(37,49,68,104),(38,50,69,105),(39,51,70,106),(40,52,71,107)], [(1,29,148,80),(2,30,149,61),(3,31,150,62),(4,32,151,63),(5,33,152,64),(6,34,153,65),(7,35,154,66),(8,36,155,67),(9,37,156,68),(10,38,157,69),(11,39,158,70),(12,40,159,71),(13,21,160,72),(14,22,141,73),(15,23,142,74),(16,24,143,75),(17,25,144,76),(18,26,145,77),(19,27,146,78),(20,28,147,79),(41,134,116,98),(42,135,117,99),(43,136,118,100),(44,137,119,81),(45,138,120,82),(46,139,101,83),(47,140,102,84),(48,121,103,85),(49,122,104,86),(50,123,105,87),(51,124,106,88),(52,125,107,89),(53,126,108,90),(54,127,109,91),(55,128,110,92),(56,129,111,93),(57,130,112,94),(58,131,113,95),(59,132,114,96),(60,133,115,97)], [(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),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(61,114),(62,113),(63,112),(64,111),(65,110),(66,109),(67,108),(68,107),(69,106),(70,105),(71,104),(72,103),(73,102),(74,101),(75,120),(76,119),(77,118),(78,117),(79,116),(80,115),(81,130),(82,129),(83,128),(84,127),(85,126),(86,125),(87,124),(88,123),(89,122),(90,121),(91,140),(92,139),(93,138),(94,137),(95,136),(96,135),(97,134),(98,133),(99,132),(100,131),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148),(155,160),(156,159),(157,158)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | C8⋊C22 | Q8⋊D5 | D4⋊D10 |
| kernel | Q8⋊D20 | C20⋊3C8 | D20⋊6C4 | C20⋊4D4 | C2×Q8⋊D5 | Q8×C20 | C2×C20 | C5×Q8 | C4×Q8 | C20 | C20 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of Q8⋊D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 30 |
| 0 | 0 | 26 | 30 |
| 28 | 39 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 28 | 39 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 40 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,11,26,0,0,30,30],[28,2,0,0,39,16,0,0,0,0,40,0,0,0,0,40],[28,2,0,0,39,13,0,0,0,0,40,40,0,0,0,1] >;
Q8⋊D20 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{20} % in TeX
G:=Group("Q8:D20"); // GroupNames label
G:=SmallGroup(320,654);
// by ID
G=gap.SmallGroup(320,654);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations