metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.6C8, C16.18D10, C40.65C23, C80.28C22, Dic10.6C8, (C2×C16)⋊9D5, (C2×C80)⋊16C2, (D5×C16)⋊6C2, C5⋊3(D4○C16), C80⋊C2⋊7C2, C8.23(C4×D5), C4.10(C8×D5), C5⋊D4.3C8, C20.54(C2×C8), D10.2(C2×C8), C8⋊D5.5C4, C22.2(C8×D5), C40.103(C2×C4), C4○D20.10C4, (C2×C8).325D10, Dic5.5(C2×C8), C20.4C8⋊15C2, C8.59(C22×D5), C10.38(C22×C8), (C8×D5).46C22, C4.Dic5.11C4, (C2×C40).410C22, C20.189(C22×C4), C5⋊2C16.11C22, D20.3C4.6C2, C2.15(D5×C2×C8), C4.104(C2×C4×D5), (C2×C10).44(C2×C8), C5⋊2C8.29(C2×C4), (C4×D5).57(C2×C4), (C2×C4).105(C4×D5), (C2×C20).405(C2×C4), SmallGroup(320,528)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.6C8
G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, ac=ca, bc=cb >
Subgroups: 214 in 84 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C16, C2×C8, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C2×C16, C2×C16, M5(2), C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, D4○C16, C5⋊2C16, C80, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D5×C16, C80⋊C2, C20.4C8, C2×C80, D20.3C4, D20.6C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, D10, C22×C8, C4×D5, C22×D5, D4○C16, C8×D5, C2×C4×D5, D5×C2×C8, D20.6C8
►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 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 149 35 117 62 84 134 50 11 159 25 107 72 94 124 60)(2 150 36 118 63 85 135 51 12 160 26 108 73 95 125 41)(3 151 37 119 64 86 136 52 13 141 27 109 74 96 126 42)(4 152 38 120 65 87 137 53 14 142 28 110 75 97 127 43)(5 153 39 101 66 88 138 54 15 143 29 111 76 98 128 44)(6 154 40 102 67 89 139 55 16 144 30 112 77 99 129 45)(7 155 21 103 68 90 140 56 17 145 31 113 78 100 130 46)(8 156 22 104 69 91 121 57 18 146 32 114 79 81 131 47)(9 157 23 105 70 92 122 58 19 147 33 115 80 82 132 48)(10 158 24 106 71 93 123 59 20 148 34 116 61 83 133 49)
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)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,149,35,117,62,84,134,50,11,159,25,107,72,94,124,60)(2,150,36,118,63,85,135,51,12,160,26,108,73,95,125,41)(3,151,37,119,64,86,136,52,13,141,27,109,74,96,126,42)(4,152,38,120,65,87,137,53,14,142,28,110,75,97,127,43)(5,153,39,101,66,88,138,54,15,143,29,111,76,98,128,44)(6,154,40,102,67,89,139,55,16,144,30,112,77,99,129,45)(7,155,21,103,68,90,140,56,17,145,31,113,78,100,130,46)(8,156,22,104,69,91,121,57,18,146,32,114,79,81,131,47)(9,157,23,105,70,92,122,58,19,147,33,115,80,82,132,48)(10,158,24,106,71,93,123,59,20,148,34,116,61,83,133,49)>;
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)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,149,35,117,62,84,134,50,11,159,25,107,72,94,124,60)(2,150,36,118,63,85,135,51,12,160,26,108,73,95,125,41)(3,151,37,119,64,86,136,52,13,141,27,109,74,96,126,42)(4,152,38,120,65,87,137,53,14,142,28,110,75,97,127,43)(5,153,39,101,66,88,138,54,15,143,29,111,76,98,128,44)(6,154,40,102,67,89,139,55,16,144,30,112,77,99,129,45)(7,155,21,103,68,90,140,56,17,145,31,113,78,100,130,46)(8,156,22,104,69,91,121,57,18,146,32,114,79,81,131,47)(9,157,23,105,70,92,122,58,19,147,33,115,80,82,132,48)(10,158,24,106,71,93,123,59,20,148,34,116,61,83,133,49) );
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),(21,23),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,80),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(101,107),(102,106),(103,105),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(141,151),(142,150),(143,149),(144,148),(145,147),(152,160),(153,159),(154,158),(155,157)], [(1,149,35,117,62,84,134,50,11,159,25,107,72,94,124,60),(2,150,36,118,63,85,135,51,12,160,26,108,73,95,125,41),(3,151,37,119,64,86,136,52,13,141,27,109,74,96,126,42),(4,152,38,120,65,87,137,53,14,142,28,110,75,97,127,43),(5,153,39,101,66,88,138,54,15,143,29,111,76,98,128,44),(6,154,40,102,67,89,139,55,16,144,30,112,77,99,129,45),(7,155,21,103,68,90,140,56,17,145,31,113,78,100,130,46),(8,156,22,104,69,91,121,57,18,146,32,114,79,81,131,47),(9,157,23,105,70,92,122,58,19,147,33,115,80,82,132,48),(10,158,24,106,71,93,123,59,20,148,34,116,61,83,133,49)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | ··· | 10F | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 16M | ··· | 16T | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D5 | D10 | D10 | C4×D5 | C4×D5 | D4○C16 | C8×D5 | C8×D5 | D20.6C8 |
| kernel | D20.6C8 | D5×C16 | C80⋊C2 | C20.4C8 | C2×C80 | D20.3C4 | C8⋊D5 | C4.Dic5 | C4○D20 | Dic10 | D20 | C5⋊D4 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 32 |
Matrix representation of D20.6C8 ►in GL2(𝔽241) generated by
| 156 | 200 |
| 41 | 119 |
| 0 | 240 |
| 240 | 0 |
| 44 | 0 |
| 0 | 44 |
G:=sub<GL(2,GF(241))| [156,41,200,119],[0,240,240,0],[44,0,0,44] >;
D20.6C8 in GAP, Magma, Sage, TeX
D_{20}._6C_8 % in TeX
G:=Group("D20.6C8"); // GroupNames label
G:=SmallGroup(320,528);
// by ID
G=gap.SmallGroup(320,528);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,58,80,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations