metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.1D20, C42.58D10, (C4×Q8)⋊4D5, (Q8×C20)⋊4C2, C20⋊3C8⋊27C2, C4.17(C2×D20), (C2×C20).67D4, C20.21(C2×D4), (C5×Q8).18D4, C4⋊C4.254D10, C5⋊4(Q8.D4), C4.13(C4○D20), C20.61(C4○D4), C10.93(C4○D8), C10.Q16⋊32C2, (C4×C20).99C22, (C2×Q8).161D10, C4.D20.8C2, D20⋊6C4.11C2, C2.16(C20⋊7D4), C10.68(C4⋊D4), (C2×C20).348C23, C2.9(C20.C23), (C2×D20).100C22, C10.88(C8.C22), (Q8×C10).196C22, C2.13(D4.8D10), (C2×Dic10).105C22, (C2×C5⋊Q16)⋊7C2, (C2×Q8⋊D5).5C2, (C2×C10).479(C2×D4), (C2×C4).222(C5⋊D4), (C5×C4⋊C4).285C22, (C2×C4).448(C22×D5), C22.156(C2×C5⋊D4), (C2×C5⋊2C8).102C22, SmallGroup(320,655)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.1D20
G = < a,b,c,d | a4=c20=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 454 in 112 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C5⋊2C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, Q8.D4, C2×C5⋊2C8, D10⋊C4, Q8⋊D5, C5⋊Q16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, Q8×C10, C20⋊3C8, D20⋊6C4, C10.Q16, C4.D20, C2×Q8⋊D5, C2×C5⋊Q16, Q8×C20, Q8.1D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, D20, C5⋊D4, C22×D5, Q8.D4, C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, C20.C23, D4.8D10, Q8.1D20
►On 160 points
Generators in S160
(1 81 69 141)(2 82 70 142)(3 83 71 143)(4 84 72 144)(5 85 73 145)(6 86 74 146)(7 87 75 147)(8 88 76 148)(9 89 77 149)(10 90 78 150)(11 91 79 151)(12 92 80 152)(13 93 61 153)(14 94 62 154)(15 95 63 155)(16 96 64 156)(17 97 65 157)(18 98 66 158)(19 99 67 159)(20 100 68 160)(21 55 131 117)(22 56 132 118)(23 57 133 119)(24 58 134 120)(25 59 135 101)(26 60 136 102)(27 41 137 103)(28 42 138 104)(29 43 139 105)(30 44 140 106)(31 45 121 107)(32 46 122 108)(33 47 123 109)(34 48 124 110)(35 49 125 111)(36 50 126 112)(37 51 127 113)(38 52 128 114)(39 53 129 115)(40 54 130 116)
(1 47 69 109)(2 48 70 110)(3 49 71 111)(4 50 72 112)(5 51 73 113)(6 52 74 114)(7 53 75 115)(8 54 76 116)(9 55 77 117)(10 56 78 118)(11 57 79 119)(12 58 80 120)(13 59 61 101)(14 60 62 102)(15 41 63 103)(16 42 64 104)(17 43 65 105)(18 44 66 106)(19 45 67 107)(20 46 68 108)(21 149 131 89)(22 150 132 90)(23 151 133 91)(24 152 134 92)(25 153 135 93)(26 154 136 94)(27 155 137 95)(28 156 138 96)(29 157 139 97)(30 158 140 98)(31 159 121 99)(32 160 122 100)(33 141 123 81)(34 142 124 82)(35 143 125 83)(36 144 126 84)(37 145 127 85)(38 146 128 86)(39 147 129 87)(40 148 130 88)
(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 68 69 20)(2 19 70 67)(3 66 71 18)(4 17 72 65)(5 64 73 16)(6 15 74 63)(7 62 75 14)(8 13 76 61)(9 80 77 12)(10 11 78 79)(21 58 131 120)(22 119 132 57)(23 56 133 118)(24 117 134 55)(25 54 135 116)(26 115 136 53)(27 52 137 114)(28 113 138 51)(29 50 139 112)(30 111 140 49)(31 48 121 110)(32 109 122 47)(33 46 123 108)(34 107 124 45)(35 44 125 106)(36 105 126 43)(37 42 127 104)(38 103 128 41)(39 60 129 102)(40 101 130 59)(81 100 141 160)(82 159 142 99)(83 98 143 158)(84 157 144 97)(85 96 145 156)(86 155 146 95)(87 94 147 154)(88 153 148 93)(89 92 149 152)(90 151 150 91)
G:=sub<Sym(160)| (1,81,69,141)(2,82,70,142)(3,83,71,143)(4,84,72,144)(5,85,73,145)(6,86,74,146)(7,87,75,147)(8,88,76,148)(9,89,77,149)(10,90,78,150)(11,91,79,151)(12,92,80,152)(13,93,61,153)(14,94,62,154)(15,95,63,155)(16,96,64,156)(17,97,65,157)(18,98,66,158)(19,99,67,159)(20,100,68,160)(21,55,131,117)(22,56,132,118)(23,57,133,119)(24,58,134,120)(25,59,135,101)(26,60,136,102)(27,41,137,103)(28,42,138,104)(29,43,139,105)(30,44,140,106)(31,45,121,107)(32,46,122,108)(33,47,123,109)(34,48,124,110)(35,49,125,111)(36,50,126,112)(37,51,127,113)(38,52,128,114)(39,53,129,115)(40,54,130,116), (1,47,69,109)(2,48,70,110)(3,49,71,111)(4,50,72,112)(5,51,73,113)(6,52,74,114)(7,53,75,115)(8,54,76,116)(9,55,77,117)(10,56,78,118)(11,57,79,119)(12,58,80,120)(13,59,61,101)(14,60,62,102)(15,41,63,103)(16,42,64,104)(17,43,65,105)(18,44,66,106)(19,45,67,107)(20,46,68,108)(21,149,131,89)(22,150,132,90)(23,151,133,91)(24,152,134,92)(25,153,135,93)(26,154,136,94)(27,155,137,95)(28,156,138,96)(29,157,139,97)(30,158,140,98)(31,159,121,99)(32,160,122,100)(33,141,123,81)(34,142,124,82)(35,143,125,83)(36,144,126,84)(37,145,127,85)(38,146,128,86)(39,147,129,87)(40,148,130,88), (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), (1,68,69,20)(2,19,70,67)(3,66,71,18)(4,17,72,65)(5,64,73,16)(6,15,74,63)(7,62,75,14)(8,13,76,61)(9,80,77,12)(10,11,78,79)(21,58,131,120)(22,119,132,57)(23,56,133,118)(24,117,134,55)(25,54,135,116)(26,115,136,53)(27,52,137,114)(28,113,138,51)(29,50,139,112)(30,111,140,49)(31,48,121,110)(32,109,122,47)(33,46,123,108)(34,107,124,45)(35,44,125,106)(36,105,126,43)(37,42,127,104)(38,103,128,41)(39,60,129,102)(40,101,130,59)(81,100,141,160)(82,159,142,99)(83,98,143,158)(84,157,144,97)(85,96,145,156)(86,155,146,95)(87,94,147,154)(88,153,148,93)(89,92,149,152)(90,151,150,91)>;
G:=Group( (1,81,69,141)(2,82,70,142)(3,83,71,143)(4,84,72,144)(5,85,73,145)(6,86,74,146)(7,87,75,147)(8,88,76,148)(9,89,77,149)(10,90,78,150)(11,91,79,151)(12,92,80,152)(13,93,61,153)(14,94,62,154)(15,95,63,155)(16,96,64,156)(17,97,65,157)(18,98,66,158)(19,99,67,159)(20,100,68,160)(21,55,131,117)(22,56,132,118)(23,57,133,119)(24,58,134,120)(25,59,135,101)(26,60,136,102)(27,41,137,103)(28,42,138,104)(29,43,139,105)(30,44,140,106)(31,45,121,107)(32,46,122,108)(33,47,123,109)(34,48,124,110)(35,49,125,111)(36,50,126,112)(37,51,127,113)(38,52,128,114)(39,53,129,115)(40,54,130,116), (1,47,69,109)(2,48,70,110)(3,49,71,111)(4,50,72,112)(5,51,73,113)(6,52,74,114)(7,53,75,115)(8,54,76,116)(9,55,77,117)(10,56,78,118)(11,57,79,119)(12,58,80,120)(13,59,61,101)(14,60,62,102)(15,41,63,103)(16,42,64,104)(17,43,65,105)(18,44,66,106)(19,45,67,107)(20,46,68,108)(21,149,131,89)(22,150,132,90)(23,151,133,91)(24,152,134,92)(25,153,135,93)(26,154,136,94)(27,155,137,95)(28,156,138,96)(29,157,139,97)(30,158,140,98)(31,159,121,99)(32,160,122,100)(33,141,123,81)(34,142,124,82)(35,143,125,83)(36,144,126,84)(37,145,127,85)(38,146,128,86)(39,147,129,87)(40,148,130,88), (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), (1,68,69,20)(2,19,70,67)(3,66,71,18)(4,17,72,65)(5,64,73,16)(6,15,74,63)(7,62,75,14)(8,13,76,61)(9,80,77,12)(10,11,78,79)(21,58,131,120)(22,119,132,57)(23,56,133,118)(24,117,134,55)(25,54,135,116)(26,115,136,53)(27,52,137,114)(28,113,138,51)(29,50,139,112)(30,111,140,49)(31,48,121,110)(32,109,122,47)(33,46,123,108)(34,107,124,45)(35,44,125,106)(36,105,126,43)(37,42,127,104)(38,103,128,41)(39,60,129,102)(40,101,130,59)(81,100,141,160)(82,159,142,99)(83,98,143,158)(84,157,144,97)(85,96,145,156)(86,155,146,95)(87,94,147,154)(88,153,148,93)(89,92,149,152)(90,151,150,91) );
G=PermutationGroup([[(1,81,69,141),(2,82,70,142),(3,83,71,143),(4,84,72,144),(5,85,73,145),(6,86,74,146),(7,87,75,147),(8,88,76,148),(9,89,77,149),(10,90,78,150),(11,91,79,151),(12,92,80,152),(13,93,61,153),(14,94,62,154),(15,95,63,155),(16,96,64,156),(17,97,65,157),(18,98,66,158),(19,99,67,159),(20,100,68,160),(21,55,131,117),(22,56,132,118),(23,57,133,119),(24,58,134,120),(25,59,135,101),(26,60,136,102),(27,41,137,103),(28,42,138,104),(29,43,139,105),(30,44,140,106),(31,45,121,107),(32,46,122,108),(33,47,123,109),(34,48,124,110),(35,49,125,111),(36,50,126,112),(37,51,127,113),(38,52,128,114),(39,53,129,115),(40,54,130,116)], [(1,47,69,109),(2,48,70,110),(3,49,71,111),(4,50,72,112),(5,51,73,113),(6,52,74,114),(7,53,75,115),(8,54,76,116),(9,55,77,117),(10,56,78,118),(11,57,79,119),(12,58,80,120),(13,59,61,101),(14,60,62,102),(15,41,63,103),(16,42,64,104),(17,43,65,105),(18,44,66,106),(19,45,67,107),(20,46,68,108),(21,149,131,89),(22,150,132,90),(23,151,133,91),(24,152,134,92),(25,153,135,93),(26,154,136,94),(27,155,137,95),(28,156,138,96),(29,157,139,97),(30,158,140,98),(31,159,121,99),(32,160,122,100),(33,141,123,81),(34,142,124,82),(35,143,125,83),(36,144,126,84),(37,145,127,85),(38,146,128,86),(39,147,129,87),(40,148,130,88)], [(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)], [(1,68,69,20),(2,19,70,67),(3,66,71,18),(4,17,72,65),(5,64,73,16),(6,15,74,63),(7,62,75,14),(8,13,76,61),(9,80,77,12),(10,11,78,79),(21,58,131,120),(22,119,132,57),(23,56,133,118),(24,117,134,55),(25,54,135,116),(26,115,136,53),(27,52,137,114),(28,113,138,51),(29,50,139,112),(30,111,140,49),(31,48,121,110),(32,109,122,47),(33,46,123,108),(34,107,124,45),(35,44,125,106),(36,105,126,43),(37,42,127,104),(38,103,128,41),(39,60,129,102),(40,101,130,59),(81,100,141,160),(82,159,142,99),(83,98,143,158),(84,157,144,97),(85,96,145,156),(86,155,146,95),(87,94,147,154),(88,153,148,93),(89,92,149,152),(90,151,150,91)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | C4○D20 | C8.C22 | C20.C23 | D4.8D10 |
| kernel | Q8.1D20 | C20⋊3C8 | D20⋊6C4 | C10.Q16 | C4.D20 | C2×Q8⋊D5 | C2×C5⋊Q16 | Q8×C20 | C2×C20 | C5×Q8 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C2×C4 | Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of Q8.1D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 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 | 11 | 30 |
| 0 | 0 | 0 | 0 | 26 | 30 |
| 7 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,26,0,0,0,0,30,30],[7,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,9,0,0,0,0,0,32] >;
Q8.1D20 in GAP, Magma, Sage, TeX
Q_8._1D_{20} % in TeX
G:=Group("Q8.1D20"); // GroupNames label
G:=SmallGroup(320,655);
// by ID
G=gap.SmallGroup(320,655);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations