metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊10Q8, C42.130D10, C10.1112+ 1+4, (C4×Q8)⋊12D5, C4.50(Q8×D5), (Q8×C20)⋊14C2, C5⋊4(D4⋊3Q8), C4⋊C4.326D10, D10⋊3Q8⋊8C2, C20⋊2Q8⋊28C2, (C4×D20).21C2, D10.20(C2×Q8), C20.108(C2×Q8), D10⋊Q8⋊11C2, C4.67(C4○D20), (C2×Q8).178D10, C20.6Q8⋊18C2, C20.118(C4○D4), C10.31(C22×Q8), (C2×C10).123C24, (C4×C20).175C22, (C2×C20).590C23, C2.23(D4⋊8D10), (C2×D20).297C22, C4⋊Dic5.202C22, (Q8×C10).223C22, (C2×Dic5).55C23, C22.144(C23×D5), (C2×Dic10).33C22, C10.D4.69C22, (C22×D5).190C23, D10⋊C4.103C22, (D5×C4⋊C4)⋊18C2, C2.14(C2×Q8×D5), C2.62(C2×C4○D20), C10.55(C2×C4○D4), (C2×C4×D5).83C22, (C5×C4⋊C4).351C22, (C2×C4).169(C22×D5), SmallGroup(320,1251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊10Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd-1=c-1 >
Subgroups: 790 in 228 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4⋊3Q8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C20⋊2Q8, C20.6Q8, C4×D20, C4×D20, D5×C4⋊C4, D10⋊Q8, D10⋊3Q8, Q8×C20, D20⋊10Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊3Q8, C4○D20, Q8×D5, C23×D5, C2×C4○D20, C2×Q8×D5, D4⋊8D10, D20⋊10Q8
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 91)(82 90)(83 89)(84 88)(85 87)(92 100)(93 99)(94 98)(95 97)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(116 120)(117 119)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(138 140)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)
(1 106 147 33)(2 107 148 34)(3 108 149 35)(4 109 150 36)(5 110 151 37)(6 111 152 38)(7 112 153 39)(8 113 154 40)(9 114 155 21)(10 115 156 22)(11 116 157 23)(12 117 158 24)(13 118 159 25)(14 119 160 26)(15 120 141 27)(16 101 142 28)(17 102 143 29)(18 103 144 30)(19 104 145 31)(20 105 146 32)(41 78 121 93)(42 79 122 94)(43 80 123 95)(44 61 124 96)(45 62 125 97)(46 63 126 98)(47 64 127 99)(48 65 128 100)(49 66 129 81)(50 67 130 82)(51 68 131 83)(52 69 132 84)(53 70 133 85)(54 71 134 86)(55 72 135 87)(56 73 136 88)(57 74 137 89)(58 75 138 90)(59 76 139 91)(60 77 140 92)
(1 89 147 74)(2 90 148 75)(3 91 149 76)(4 92 150 77)(5 93 151 78)(6 94 152 79)(7 95 153 80)(8 96 154 61)(9 97 155 62)(10 98 156 63)(11 99 157 64)(12 100 158 65)(13 81 159 66)(14 82 160 67)(15 83 141 68)(16 84 142 69)(17 85 143 70)(18 86 144 71)(19 87 145 72)(20 88 146 73)(21 45 114 125)(22 46 115 126)(23 47 116 127)(24 48 117 128)(25 49 118 129)(26 50 119 130)(27 51 120 131)(28 52 101 132)(29 53 102 133)(30 54 103 134)(31 55 104 135)(32 56 105 136)(33 57 106 137)(34 58 107 138)(35 59 108 139)(36 60 109 140)(37 41 110 121)(38 42 111 122)(39 43 112 123)(40 44 113 124)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(116,120)(117,119)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,140)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155), (1,106,147,33)(2,107,148,34)(3,108,149,35)(4,109,150,36)(5,110,151,37)(6,111,152,38)(7,112,153,39)(8,113,154,40)(9,114,155,21)(10,115,156,22)(11,116,157,23)(12,117,158,24)(13,118,159,25)(14,119,160,26)(15,120,141,27)(16,101,142,28)(17,102,143,29)(18,103,144,30)(19,104,145,31)(20,105,146,32)(41,78,121,93)(42,79,122,94)(43,80,123,95)(44,61,124,96)(45,62,125,97)(46,63,126,98)(47,64,127,99)(48,65,128,100)(49,66,129,81)(50,67,130,82)(51,68,131,83)(52,69,132,84)(53,70,133,85)(54,71,134,86)(55,72,135,87)(56,73,136,88)(57,74,137,89)(58,75,138,90)(59,76,139,91)(60,77,140,92), (1,89,147,74)(2,90,148,75)(3,91,149,76)(4,92,150,77)(5,93,151,78)(6,94,152,79)(7,95,153,80)(8,96,154,61)(9,97,155,62)(10,98,156,63)(11,99,157,64)(12,100,158,65)(13,81,159,66)(14,82,160,67)(15,83,141,68)(16,84,142,69)(17,85,143,70)(18,86,144,71)(19,87,145,72)(20,88,146,73)(21,45,114,125)(22,46,115,126)(23,47,116,127)(24,48,117,128)(25,49,118,129)(26,50,119,130)(27,51,120,131)(28,52,101,132)(29,53,102,133)(30,54,103,134)(31,55,104,135)(32,56,105,136)(33,57,106,137)(34,58,107,138)(35,59,108,139)(36,60,109,140)(37,41,110,121)(38,42,111,122)(39,43,112,123)(40,44,113,124)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(116,120)(117,119)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,140)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155), (1,106,147,33)(2,107,148,34)(3,108,149,35)(4,109,150,36)(5,110,151,37)(6,111,152,38)(7,112,153,39)(8,113,154,40)(9,114,155,21)(10,115,156,22)(11,116,157,23)(12,117,158,24)(13,118,159,25)(14,119,160,26)(15,120,141,27)(16,101,142,28)(17,102,143,29)(18,103,144,30)(19,104,145,31)(20,105,146,32)(41,78,121,93)(42,79,122,94)(43,80,123,95)(44,61,124,96)(45,62,125,97)(46,63,126,98)(47,64,127,99)(48,65,128,100)(49,66,129,81)(50,67,130,82)(51,68,131,83)(52,69,132,84)(53,70,133,85)(54,71,134,86)(55,72,135,87)(56,73,136,88)(57,74,137,89)(58,75,138,90)(59,76,139,91)(60,77,140,92), (1,89,147,74)(2,90,148,75)(3,91,149,76)(4,92,150,77)(5,93,151,78)(6,94,152,79)(7,95,153,80)(8,96,154,61)(9,97,155,62)(10,98,156,63)(11,99,157,64)(12,100,158,65)(13,81,159,66)(14,82,160,67)(15,83,141,68)(16,84,142,69)(17,85,143,70)(18,86,144,71)(19,87,145,72)(20,88,146,73)(21,45,114,125)(22,46,115,126)(23,47,116,127)(24,48,117,128)(25,49,118,129)(26,50,119,130)(27,51,120,131)(28,52,101,132)(29,53,102,133)(30,54,103,134)(31,55,104,135)(32,56,105,136)(33,57,106,137)(34,58,107,138)(35,59,108,139)(36,60,109,140)(37,41,110,121)(38,42,111,122)(39,43,112,123)(40,44,113,124) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,91),(82,90),(83,89),(84,88),(85,87),(92,100),(93,99),(94,98),(95,97),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109),(116,120),(117,119),(121,137),(122,136),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(138,140),(141,147),(142,146),(143,145),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155)], [(1,106,147,33),(2,107,148,34),(3,108,149,35),(4,109,150,36),(5,110,151,37),(6,111,152,38),(7,112,153,39),(8,113,154,40),(9,114,155,21),(10,115,156,22),(11,116,157,23),(12,117,158,24),(13,118,159,25),(14,119,160,26),(15,120,141,27),(16,101,142,28),(17,102,143,29),(18,103,144,30),(19,104,145,31),(20,105,146,32),(41,78,121,93),(42,79,122,94),(43,80,123,95),(44,61,124,96),(45,62,125,97),(46,63,126,98),(47,64,127,99),(48,65,128,100),(49,66,129,81),(50,67,130,82),(51,68,131,83),(52,69,132,84),(53,70,133,85),(54,71,134,86),(55,72,135,87),(56,73,136,88),(57,74,137,89),(58,75,138,90),(59,76,139,91),(60,77,140,92)], [(1,89,147,74),(2,90,148,75),(3,91,149,76),(4,92,150,77),(5,93,151,78),(6,94,152,79),(7,95,153,80),(8,96,154,61),(9,97,155,62),(10,98,156,63),(11,99,157,64),(12,100,158,65),(13,81,159,66),(14,82,160,67),(15,83,141,68),(16,84,142,69),(17,85,143,70),(18,86,144,71),(19,87,145,72),(20,88,146,73),(21,45,114,125),(22,46,115,126),(23,47,116,127),(24,48,117,128),(25,49,118,129),(26,50,119,130),(27,51,120,131),(28,52,101,132),(29,53,102,133),(30,54,103,134),(31,55,104,135),(32,56,105,136),(33,57,106,137),(34,58,107,138),(35,59,108,139),(36,60,109,140),(37,41,110,121),(38,42,111,122),(39,43,112,123),(40,44,113,124)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 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 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | Q8×D5 | D4⋊8D10 |
| kernel | D20⋊10Q8 | C20⋊2Q8 | C20.6Q8 | C4×D20 | D5×C4⋊C4 | D10⋊Q8 | D10⋊3Q8 | Q8×C20 | D20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 2 | 4 | 2 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D20⋊10Q8 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 | 3 | 20 |
| 0 | 0 | 0 | 0 | 20 | 38 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,20,0,0,0,0,20,38] >;
D20⋊10Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{10}Q_8 % in TeX
G:=Group("D20:10Q8"); // GroupNames label
G:=SmallGroup(320,1251);
// by ID
G=gap.SmallGroup(320,1251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations