metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.30D4, C20.13C24, C40.42C23, Q16.11D10, D20.8C23, Dic10.30D4, D40.13C22, Dic10.8C23, Dic20.15C22, C5⋊2(Q8○D8), (D5×Q16)⋊6C2, C4.77(D4×D5), Q8⋊D5.C22, (C10×Q16)⋊3C2, (C2×Q16)⋊12D5, C20.88(C2×D4), C5⋊D4.10D4, D40⋊7C2⋊4C2, Q8.D10⋊6C2, Q16⋊D5⋊5C2, D10.51(C2×D4), (C2×C8).105D10, C5⋊2C8.5C23, (C2×Q8).91D10, (C8×D5).7C22, (C4×D5).7C23, C8.14(C22×D5), C4.13(C23×D5), C22.22(D4×D5), D20.3C4⋊3C2, Q8.7(C22×D5), (C5×Q8).7C23, (Q8×D5).1C22, C20.C23⋊8C2, (C2×C40).35C22, Dic5.57(C2×D4), C8⋊D5.3C22, C40⋊C2.3C22, C5⋊Q16.1C22, (C2×C20).530C23, Q8.10D10⋊4C2, C4○D20.53C22, C10.114(C22×D4), Q8⋊2D5.1C22, (C5×Q16).11C22, (Q8×C10).152C22, C4.Dic5.48C22, C2.87(C2×D4×D5), (C2×C10).403(C2×D4), (C2×C4).231(C22×D5), SmallGroup(320,1438)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.30D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a10c3 >
Subgroups: 886 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C8○D4, C2×Q16, C2×Q16, C4○D8, C8.C22, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8○D8, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, Q8⋊D5, C5⋊Q16, C2×C40, C5×Q16, C4○D20, C4○D20, C4○D20, Q8×D5, Q8×D5, Q8⋊2D5, Q8⋊2D5, Q8×C10, D20.3C4, D40⋊7C2, D5×Q16, Q16⋊D5, Q8.D10, C20.C23, C10×Q16, Q8.10D10, D20.30D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, Q8○D8, D4×D5, C23×D5, C2×D4×D5, D20.30D4
►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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 123)(22 122)(23 121)(24 140)(25 139)(26 138)(27 137)(28 136)(29 135)(30 134)(31 133)(32 132)(33 131)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 104)(42 103)(43 102)(44 101)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(81 149)(82 148)(83 147)(84 146)(85 145)(86 144)(87 143)(88 142)(89 141)(90 160)(91 159)(92 158)(93 157)(94 156)(95 155)(96 154)(97 153)(98 152)(99 151)(100 150)
(1 109 151 34 11 119 141 24)(2 110 152 35 12 120 142 25)(3 111 153 36 13 101 143 26)(4 112 154 37 14 102 144 27)(5 113 155 38 15 103 145 28)(6 114 156 39 16 104 146 29)(7 115 157 40 17 105 147 30)(8 116 158 21 18 106 148 31)(9 117 159 22 19 107 149 32)(10 118 160 23 20 108 150 33)(41 84 135 68 51 94 125 78)(42 85 136 69 52 95 126 79)(43 86 137 70 53 96 127 80)(44 87 138 71 54 97 128 61)(45 88 139 72 55 98 129 62)(46 89 140 73 56 99 130 63)(47 90 121 74 57 100 131 64)(48 91 122 75 58 81 132 65)(49 92 123 76 59 82 133 66)(50 93 124 77 60 83 134 67)
(1 95 11 85)(2 86 12 96)(3 97 13 87)(4 88 14 98)(5 99 15 89)(6 90 16 100)(7 81 17 91)(8 92 18 82)(9 83 19 93)(10 94 20 84)(21 133 31 123)(22 124 32 134)(23 135 33 125)(24 126 34 136)(25 137 35 127)(26 128 36 138)(27 139 37 129)(28 130 38 140)(29 121 39 131)(30 132 40 122)(41 118 51 108)(42 109 52 119)(43 120 53 110)(44 111 54 101)(45 102 55 112)(46 113 56 103)(47 104 57 114)(48 115 58 105)(49 106 59 116)(50 117 60 107)(61 153 71 143)(62 144 72 154)(63 155 73 145)(64 146 74 156)(65 157 75 147)(66 148 76 158)(67 159 77 149)(68 150 78 160)(69 141 79 151)(70 152 80 142)
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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,123)(22,122)(23,121)(24,140)(25,139)(26,138)(27,137)(28,136)(29,135)(30,134)(31,133)(32,132)(33,131)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,104)(42,103)(43,102)(44,101)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(81,149)(82,148)(83,147)(84,146)(85,145)(86,144)(87,143)(88,142)(89,141)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150), (1,109,151,34,11,119,141,24)(2,110,152,35,12,120,142,25)(3,111,153,36,13,101,143,26)(4,112,154,37,14,102,144,27)(5,113,155,38,15,103,145,28)(6,114,156,39,16,104,146,29)(7,115,157,40,17,105,147,30)(8,116,158,21,18,106,148,31)(9,117,159,22,19,107,149,32)(10,118,160,23,20,108,150,33)(41,84,135,68,51,94,125,78)(42,85,136,69,52,95,126,79)(43,86,137,70,53,96,127,80)(44,87,138,71,54,97,128,61)(45,88,139,72,55,98,129,62)(46,89,140,73,56,99,130,63)(47,90,121,74,57,100,131,64)(48,91,122,75,58,81,132,65)(49,92,123,76,59,82,133,66)(50,93,124,77,60,83,134,67), (1,95,11,85)(2,86,12,96)(3,97,13,87)(4,88,14,98)(5,99,15,89)(6,90,16,100)(7,81,17,91)(8,92,18,82)(9,83,19,93)(10,94,20,84)(21,133,31,123)(22,124,32,134)(23,135,33,125)(24,126,34,136)(25,137,35,127)(26,128,36,138)(27,139,37,129)(28,130,38,140)(29,121,39,131)(30,132,40,122)(41,118,51,108)(42,109,52,119)(43,120,53,110)(44,111,54,101)(45,102,55,112)(46,113,56,103)(47,104,57,114)(48,115,58,105)(49,106,59,116)(50,117,60,107)(61,153,71,143)(62,144,72,154)(63,155,73,145)(64,146,74,156)(65,157,75,147)(66,148,76,158)(67,159,77,149)(68,150,78,160)(69,141,79,151)(70,152,80,142)>;
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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,123)(22,122)(23,121)(24,140)(25,139)(26,138)(27,137)(28,136)(29,135)(30,134)(31,133)(32,132)(33,131)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,104)(42,103)(43,102)(44,101)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(81,149)(82,148)(83,147)(84,146)(85,145)(86,144)(87,143)(88,142)(89,141)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150), (1,109,151,34,11,119,141,24)(2,110,152,35,12,120,142,25)(3,111,153,36,13,101,143,26)(4,112,154,37,14,102,144,27)(5,113,155,38,15,103,145,28)(6,114,156,39,16,104,146,29)(7,115,157,40,17,105,147,30)(8,116,158,21,18,106,148,31)(9,117,159,22,19,107,149,32)(10,118,160,23,20,108,150,33)(41,84,135,68,51,94,125,78)(42,85,136,69,52,95,126,79)(43,86,137,70,53,96,127,80)(44,87,138,71,54,97,128,61)(45,88,139,72,55,98,129,62)(46,89,140,73,56,99,130,63)(47,90,121,74,57,100,131,64)(48,91,122,75,58,81,132,65)(49,92,123,76,59,82,133,66)(50,93,124,77,60,83,134,67), (1,95,11,85)(2,86,12,96)(3,97,13,87)(4,88,14,98)(5,99,15,89)(6,90,16,100)(7,81,17,91)(8,92,18,82)(9,83,19,93)(10,94,20,84)(21,133,31,123)(22,124,32,134)(23,135,33,125)(24,126,34,136)(25,137,35,127)(26,128,36,138)(27,139,37,129)(28,130,38,140)(29,121,39,131)(30,132,40,122)(41,118,51,108)(42,109,52,119)(43,120,53,110)(44,111,54,101)(45,102,55,112)(46,113,56,103)(47,104,57,114)(48,115,58,105)(49,106,59,116)(50,117,60,107)(61,153,71,143)(62,144,72,154)(63,155,73,145)(64,146,74,156)(65,157,75,147)(66,148,76,158)(67,159,77,149)(68,150,78,160)(69,141,79,151)(70,152,80,142) );
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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,123),(22,122),(23,121),(24,140),(25,139),(26,138),(27,137),(28,136),(29,135),(30,134),(31,133),(32,132),(33,131),(34,130),(35,129),(36,128),(37,127),(38,126),(39,125),(40,124),(41,104),(42,103),(43,102),(44,101),(45,120),(46,119),(47,118),(48,117),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,108),(58,107),(59,106),(60,105),(81,149),(82,148),(83,147),(84,146),(85,145),(86,144),(87,143),(88,142),(89,141),(90,160),(91,159),(92,158),(93,157),(94,156),(95,155),(96,154),(97,153),(98,152),(99,151),(100,150)], [(1,109,151,34,11,119,141,24),(2,110,152,35,12,120,142,25),(3,111,153,36,13,101,143,26),(4,112,154,37,14,102,144,27),(5,113,155,38,15,103,145,28),(6,114,156,39,16,104,146,29),(7,115,157,40,17,105,147,30),(8,116,158,21,18,106,148,31),(9,117,159,22,19,107,149,32),(10,118,160,23,20,108,150,33),(41,84,135,68,51,94,125,78),(42,85,136,69,52,95,126,79),(43,86,137,70,53,96,127,80),(44,87,138,71,54,97,128,61),(45,88,139,72,55,98,129,62),(46,89,140,73,56,99,130,63),(47,90,121,74,57,100,131,64),(48,91,122,75,58,81,132,65),(49,92,123,76,59,82,133,66),(50,93,124,77,60,83,134,67)], [(1,95,11,85),(2,86,12,96),(3,97,13,87),(4,88,14,98),(5,99,15,89),(6,90,16,100),(7,81,17,91),(8,92,18,82),(9,83,19,93),(10,94,20,84),(21,133,31,123),(22,124,32,134),(23,135,33,125),(24,126,34,136),(25,137,35,127),(26,128,36,138),(27,139,37,129),(28,130,38,140),(29,121,39,131),(30,132,40,122),(41,118,51,108),(42,109,52,119),(43,120,53,110),(44,111,54,101),(45,102,55,112),(46,113,56,103),(47,104,57,114),(48,115,58,105),(49,106,59,116),(50,117,60,107),(61,153,71,143),(62,144,72,154),(63,155,73,145),(64,146,74,156),(65,157,75,147),(66,148,76,158),(67,159,77,149),(68,150,78,160),(69,141,79,151),(70,152,80,142)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | Q8○D8 | D4×D5 | D4×D5 | D20.30D4 |
| kernel | D20.30D4 | D20.3C4 | D40⋊7C2 | D5×Q16 | Q16⋊D5 | Q8.D10 | C20.C23 | C10×Q16 | Q8.10D10 | Dic10 | D20 | C5⋊D4 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D20.30D4 ►in GL4(𝔽41) generated by
| 0 | 0 | 6 | 23 |
| 0 | 0 | 18 | 21 |
| 35 | 18 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 8 | 17 | 0 | 15 |
| 24 | 33 | 15 | 0 |
| 0 | 26 | 8 | 17 |
| 26 | 0 | 24 | 33 |
| 12 | 0 | 29 | 0 |
| 0 | 12 | 0 | 29 |
| 12 | 0 | 12 | 0 |
| 0 | 12 | 0 | 12 |
| 15 | 0 | 17 | 8 |
| 0 | 15 | 33 | 24 |
| 17 | 8 | 26 | 0 |
| 33 | 24 | 0 | 26 |
G:=sub<GL(4,GF(41))| [0,0,35,23,0,0,18,20,6,18,0,0,23,21,0,0],[8,24,0,26,17,33,26,0,0,15,8,24,15,0,17,33],[12,0,12,0,0,12,0,12,29,0,12,0,0,29,0,12],[15,0,17,33,0,15,8,24,17,33,26,0,8,24,0,26] >;
D20.30D4 in GAP, Magma, Sage, TeX
D_{20}._{30}D_4 % in TeX
G:=Group("D20.30D4"); // GroupNames label
G:=SmallGroup(320,1438);
// by ID
G=gap.SmallGroup(320,1438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^3>;
// generators/relations