metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.3C8, C40.66D4, C8.31D20, Dic10.3C8, (C2×C80)⋊2C2, (C2×C16)⋊2D5, C4.8(C8×D5), C5⋊3(D4.C8), C20.52(C2×C8), C4○D20.5C4, C20.4C8⋊9C2, (C2×C8).322D10, C8.45(C5⋊D4), C4.Dic5.5C4, C2.9(D10⋊1C8), C10.23(C22⋊C8), (C2×C40).402C22, D20.3C4.3C2, C22.2(C8⋊D5), (C2×C10).27M4(2), C4.41(D10⋊C4), C20.103(C22⋊C4), (C2×C4).97(C4×D5), (C2×C20).404(C2×C4), SmallGroup(320,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.3C8
G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, ac=ca, cbc-1=a15b >
Smallest permutation representation of D20.3C8
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 149)(142 148)(143 147)(144 146)(150 160)(151 159)(152 158)(153 157)(154 156)
(1 143 102 136 29 48 92 74 11 153 112 126 39 58 82 64)(2 144 103 137 30 49 93 75 12 154 113 127 40 59 83 65)(3 145 104 138 31 50 94 76 13 155 114 128 21 60 84 66)(4 146 105 139 32 51 95 77 14 156 115 129 22 41 85 67)(5 147 106 140 33 52 96 78 15 157 116 130 23 42 86 68)(6 148 107 121 34 53 97 79 16 158 117 131 24 43 87 69)(7 149 108 122 35 54 98 80 17 159 118 132 25 44 88 70)(8 150 109 123 36 55 99 61 18 160 119 133 26 45 89 71)(9 151 110 124 37 56 100 62 19 141 120 134 27 46 90 72)(10 152 111 125 38 57 81 63 20 142 101 135 28 47 91 73)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,143,102,136,29,48,92,74,11,153,112,126,39,58,82,64)(2,144,103,137,30,49,93,75,12,154,113,127,40,59,83,65)(3,145,104,138,31,50,94,76,13,155,114,128,21,60,84,66)(4,146,105,139,32,51,95,77,14,156,115,129,22,41,85,67)(5,147,106,140,33,52,96,78,15,157,116,130,23,42,86,68)(6,148,107,121,34,53,97,79,16,158,117,131,24,43,87,69)(7,149,108,122,35,54,98,80,17,159,118,132,25,44,88,70)(8,150,109,123,36,55,99,61,18,160,119,133,26,45,89,71)(9,151,110,124,37,56,100,62,19,141,120,134,27,46,90,72)(10,152,111,125,38,57,81,63,20,142,101,135,28,47,91,73)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,143,102,136,29,48,92,74,11,153,112,126,39,58,82,64)(2,144,103,137,30,49,93,75,12,154,113,127,40,59,83,65)(3,145,104,138,31,50,94,76,13,155,114,128,21,60,84,66)(4,146,105,139,32,51,95,77,14,156,115,129,22,41,85,67)(5,147,106,140,33,52,96,78,15,157,116,130,23,42,86,68)(6,148,107,121,34,53,97,79,16,158,117,131,24,43,87,69)(7,149,108,122,35,54,98,80,17,159,118,132,25,44,88,70)(8,150,109,123,36,55,99,61,18,160,119,133,26,45,89,71)(9,151,110,124,37,56,100,62,19,141,120,134,27,46,90,72)(10,152,111,125,38,57,81,63,20,142,101,135,28,47,91,73) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,100),(94,99),(95,98),(96,97),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117),(121,125),(122,124),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(141,149),(142,148),(143,147),(144,146),(150,160),(151,159),(152,158),(153,157),(154,156)], [(1,143,102,136,29,48,92,74,11,153,112,126,39,58,82,64),(2,144,103,137,30,49,93,75,12,154,113,127,40,59,83,65),(3,145,104,138,31,50,94,76,13,155,114,128,21,60,84,66),(4,146,105,139,32,51,95,77,14,156,115,129,22,41,85,67),(5,147,106,140,33,52,96,78,15,157,116,130,23,42,86,68),(6,148,107,121,34,53,97,79,16,158,117,131,24,43,87,69),(7,149,108,122,35,54,98,80,17,159,118,132,25,44,88,70),(8,150,109,123,36,55,99,61,18,160,119,133,26,45,89,71),(9,151,110,124,37,56,100,62,19,141,120,134,27,46,90,72),(10,152,111,125,38,57,81,63,20,142,101,135,28,47,91,73)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 2 | 20 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D5 | M4(2) | D10 | D20 | C5⋊D4 | C4×D5 | D4.C8 | C8×D5 | C8⋊D5 | D20.3C8 |
| kernel | D20.3C8 | C20.4C8 | C2×C80 | D20.3C4 | C4.Dic5 | C4○D20 | Dic10 | D20 | C40 | C2×C16 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 32 |
Matrix representation of D20.3C8 ►in GL2(𝔽241) generated by
| 156 | 200 |
| 41 | 119 |
| 0 | 240 |
| 240 | 0 |
| 161 | 71 |
| 170 | 84 |
G:=sub<GL(2,GF(241))| [156,41,200,119],[0,240,240,0],[161,170,71,84] >;
D20.3C8 in GAP, Magma, Sage, TeX
D_{20}._3C_8 % in TeX
G:=Group("D20.3C8"); // GroupNames label
G:=SmallGroup(320,66);
// by ID
G=gap.SmallGroup(320,66);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,1123,136,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations
Export