metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6D20, C42.129D10, C10.1102+ 1+4, (C4×Q8)⋊10D5, (C5×Q8)⋊11D4, (C4×D20)⋊39C2, (Q8×C20)⋊12C2, C5⋊2(Q8⋊6D4), C4.26(C2×D20), C20.58(C2×D4), C20⋊17(C4○D4), C20⋊4D4⋊13C2, C4⋊D20⋊18C2, C4⋊C4.296D10, C4⋊3(Q8⋊2D5), (C2×Q8).205D10, C2.22(C22×D20), C10.20(C22×D4), (C2×C10).121C24, (C2×C20).170C23, (C4×C20).173C22, (C2×D20).30C22, C2.22(D4⋊8D10), C4⋊Dic5.399C22, (Q8×C10).221C22, (C22×D5).46C23, C22.142(C23×D5), (C2×Dic5).225C23, D10⋊C4.101C22, (C2×Q8⋊2D5)⋊4C2, (C2×C4×D5).82C22, C10.112(C2×C4○D4), C2.11(C2×Q8⋊2D5), (C5×C4⋊C4).349C22, (C2×C4).734(C22×D5), SmallGroup(320,1249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊6D20
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1366 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, Q8⋊6D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8⋊2D5, Q8×C10, C4×D20, C20⋊4D4, C4⋊D20, Q8×C20, C2×Q8⋊2D5, Q8⋊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, Q8⋊6D4, C2×D20, Q8⋊2D5, C23×D5, C22×D20, C2×Q8⋊2D5, D4⋊8D10, Q8⋊6D20
►On 160 points
Generators in S160
(1 151 134 37)(2 152 135 38)(3 153 136 39)(4 154 137 40)(5 155 138 21)(6 156 139 22)(7 157 140 23)(8 158 121 24)(9 159 122 25)(10 160 123 26)(11 141 124 27)(12 142 125 28)(13 143 126 29)(14 144 127 30)(15 145 128 31)(16 146 129 32)(17 147 130 33)(18 148 131 34)(19 149 132 35)(20 150 133 36)(41 95 115 76)(42 96 116 77)(43 97 117 78)(44 98 118 79)(45 99 119 80)(46 100 120 61)(47 81 101 62)(48 82 102 63)(49 83 103 64)(50 84 104 65)(51 85 105 66)(52 86 106 67)(53 87 107 68)(54 88 108 69)(55 89 109 70)(56 90 110 71)(57 91 111 72)(58 92 112 73)(59 93 113 74)(60 94 114 75)
(1 98 134 79)(2 99 135 80)(3 100 136 61)(4 81 137 62)(5 82 138 63)(6 83 139 64)(7 84 140 65)(8 85 121 66)(9 86 122 67)(10 87 123 68)(11 88 124 69)(12 89 125 70)(13 90 126 71)(14 91 127 72)(15 92 128 73)(16 93 129 74)(17 94 130 75)(18 95 131 76)(19 96 132 77)(20 97 133 78)(21 102 155 48)(22 103 156 49)(23 104 157 50)(24 105 158 51)(25 106 159 52)(26 107 160 53)(27 108 141 54)(28 109 142 55)(29 110 143 56)(30 111 144 57)(31 112 145 58)(32 113 146 59)(33 114 147 60)(34 115 148 41)(35 116 149 42)(36 117 150 43)(37 118 151 44)(38 119 152 45)(39 120 153 46)(40 101 154 47)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(81 104)(82 103)(83 102)(84 101)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 160)(135 159)(136 158)(137 157)(138 156)(139 155)(140 154)
G:=sub<Sym(160)| (1,151,134,37)(2,152,135,38)(3,153,136,39)(4,154,137,40)(5,155,138,21)(6,156,139,22)(7,157,140,23)(8,158,121,24)(9,159,122,25)(10,160,123,26)(11,141,124,27)(12,142,125,28)(13,143,126,29)(14,144,127,30)(15,145,128,31)(16,146,129,32)(17,147,130,33)(18,148,131,34)(19,149,132,35)(20,150,133,36)(41,95,115,76)(42,96,116,77)(43,97,117,78)(44,98,118,79)(45,99,119,80)(46,100,120,61)(47,81,101,62)(48,82,102,63)(49,83,103,64)(50,84,104,65)(51,85,105,66)(52,86,106,67)(53,87,107,68)(54,88,108,69)(55,89,109,70)(56,90,110,71)(57,91,111,72)(58,92,112,73)(59,93,113,74)(60,94,114,75), (1,98,134,79)(2,99,135,80)(3,100,136,61)(4,81,137,62)(5,82,138,63)(6,83,139,64)(7,84,140,65)(8,85,121,66)(9,86,122,67)(10,87,123,68)(11,88,124,69)(12,89,125,70)(13,90,126,71)(14,91,127,72)(15,92,128,73)(16,93,129,74)(17,94,130,75)(18,95,131,76)(19,96,132,77)(20,97,133,78)(21,102,155,48)(22,103,156,49)(23,104,157,50)(24,105,158,51)(25,106,159,52)(26,107,160,53)(27,108,141,54)(28,109,142,55)(29,110,143,56)(30,111,144,57)(31,112,145,58)(32,113,146,59)(33,114,147,60)(34,115,148,41)(35,116,149,42)(36,117,150,43)(37,118,151,44)(38,119,152,45)(39,120,153,46)(40,101,154,47), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(81,104)(82,103)(83,102)(84,101)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,160)(135,159)(136,158)(137,157)(138,156)(139,155)(140,154)>;
G:=Group( (1,151,134,37)(2,152,135,38)(3,153,136,39)(4,154,137,40)(5,155,138,21)(6,156,139,22)(7,157,140,23)(8,158,121,24)(9,159,122,25)(10,160,123,26)(11,141,124,27)(12,142,125,28)(13,143,126,29)(14,144,127,30)(15,145,128,31)(16,146,129,32)(17,147,130,33)(18,148,131,34)(19,149,132,35)(20,150,133,36)(41,95,115,76)(42,96,116,77)(43,97,117,78)(44,98,118,79)(45,99,119,80)(46,100,120,61)(47,81,101,62)(48,82,102,63)(49,83,103,64)(50,84,104,65)(51,85,105,66)(52,86,106,67)(53,87,107,68)(54,88,108,69)(55,89,109,70)(56,90,110,71)(57,91,111,72)(58,92,112,73)(59,93,113,74)(60,94,114,75), (1,98,134,79)(2,99,135,80)(3,100,136,61)(4,81,137,62)(5,82,138,63)(6,83,139,64)(7,84,140,65)(8,85,121,66)(9,86,122,67)(10,87,123,68)(11,88,124,69)(12,89,125,70)(13,90,126,71)(14,91,127,72)(15,92,128,73)(16,93,129,74)(17,94,130,75)(18,95,131,76)(19,96,132,77)(20,97,133,78)(21,102,155,48)(22,103,156,49)(23,104,157,50)(24,105,158,51)(25,106,159,52)(26,107,160,53)(27,108,141,54)(28,109,142,55)(29,110,143,56)(30,111,144,57)(31,112,145,58)(32,113,146,59)(33,114,147,60)(34,115,148,41)(35,116,149,42)(36,117,150,43)(37,118,151,44)(38,119,152,45)(39,120,153,46)(40,101,154,47), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(81,104)(82,103)(83,102)(84,101)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,160)(135,159)(136,158)(137,157)(138,156)(139,155)(140,154) );
G=PermutationGroup([[(1,151,134,37),(2,152,135,38),(3,153,136,39),(4,154,137,40),(5,155,138,21),(6,156,139,22),(7,157,140,23),(8,158,121,24),(9,159,122,25),(10,160,123,26),(11,141,124,27),(12,142,125,28),(13,143,126,29),(14,144,127,30),(15,145,128,31),(16,146,129,32),(17,147,130,33),(18,148,131,34),(19,149,132,35),(20,150,133,36),(41,95,115,76),(42,96,116,77),(43,97,117,78),(44,98,118,79),(45,99,119,80),(46,100,120,61),(47,81,101,62),(48,82,102,63),(49,83,103,64),(50,84,104,65),(51,85,105,66),(52,86,106,67),(53,87,107,68),(54,88,108,69),(55,89,109,70),(56,90,110,71),(57,91,111,72),(58,92,112,73),(59,93,113,74),(60,94,114,75)], [(1,98,134,79),(2,99,135,80),(3,100,136,61),(4,81,137,62),(5,82,138,63),(6,83,139,64),(7,84,140,65),(8,85,121,66),(9,86,122,67),(10,87,123,68),(11,88,124,69),(12,89,125,70),(13,90,126,71),(14,91,127,72),(15,92,128,73),(16,93,129,74),(17,94,130,75),(18,95,131,76),(19,96,132,77),(20,97,133,78),(21,102,155,48),(22,103,156,49),(23,104,157,50),(24,105,158,51),(25,106,159,52),(26,107,160,53),(27,108,141,54),(28,109,142,55),(29,110,143,56),(30,111,144,57),(31,112,145,58),(32,113,146,59),(33,114,147,60),(34,115,148,41),(35,116,149,42),(36,117,150,43),(37,118,151,44),(38,119,152,45),(39,120,153,46),(40,101,154,47)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(81,104),(82,103),(83,102),(84,101),(85,120),(86,119),(87,118),(88,117),(89,116),(90,115),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(99,106),(100,105),(121,153),(122,152),(123,151),(124,150),(125,149),(126,148),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,141),(134,160),(135,159),(136,158),(137,157),(138,156),(139,155),(140,154)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D20 | 2+ 1+4 | Q8⋊2D5 | D4⋊8D10 |
| kernel | Q8⋊6D20 | C4×D20 | C20⋊4D4 | C4⋊D20 | Q8×C20 | C2×Q8⋊2D5 | C5×Q8 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of Q8⋊6D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 21 |
| 0 | 0 | 37 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 25 |
| 0 | 0 | 0 | 9 |
| 27 | 30 | 0 | 0 |
| 11 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 27 | 0 | 0 |
| 32 | 11 | 0 | 0 |
| 0 | 0 | 40 | 21 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,37,0,0,21,1],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,25,9],[27,11,0,0,30,32,0,0,0,0,1,0,0,0,0,1],[30,32,0,0,27,11,0,0,0,0,40,0,0,0,21,1] >;
Q8⋊6D20 in GAP, Magma, Sage, TeX
Q_8\rtimes_6D_{20} % in TeX
G:=Group("Q8:6D20"); // GroupNames label
G:=SmallGroup(320,1249);
// by ID
G=gap.SmallGroup(320,1249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,184,675,80,12550]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations