metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.4C8, C40.47D4, C8.26D20, M5(2)⋊4D5, Dic10.4C8, C4.3(C8×D5), C5⋊4(D4.C8), C20.30(C2×C8), C4○D20.6C4, (C2×C8).267D10, C8.51(C5⋊D4), (C5×M5(2))⋊8C2, C4.Dic5.7C4, C10.25(C22⋊C8), (C2×C40).221C22, D20.3C4.4C2, C22.1(C8⋊D5), (C2×C10).14M4(2), C4.43(D10⋊C4), C2.11(D10⋊1C8), C20.105(C22⋊C4), (C2×C5⋊2C16)⋊13C2, (C2×C4).67(C4×D5), (C2×C20).226(C2×C4), SmallGroup(320,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.4C8
G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, cac-1=a11, cbc-1=a15b >
Smallest permutation representation of D20.4C8
►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 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)
(1 87 151 46 79 111 138 37 11 97 141 56 69 101 128 27)(2 98 152 57 80 102 139 28 12 88 142 47 70 112 129 38)(3 89 153 48 61 113 140 39 13 99 143 58 71 103 130 29)(4 100 154 59 62 104 121 30 14 90 144 49 72 114 131 40)(5 91 155 50 63 115 122 21 15 81 145 60 73 105 132 31)(6 82 156 41 64 106 123 32 16 92 146 51 74 116 133 22)(7 93 157 52 65 117 124 23 17 83 147 42 75 107 134 33)(8 84 158 43 66 108 125 34 18 94 148 53 76 118 135 24)(9 95 159 54 67 119 126 25 19 85 149 44 77 109 136 35)(10 86 160 45 68 110 127 36 20 96 150 55 78 120 137 26)
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)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151), (1,87,151,46,79,111,138,37,11,97,141,56,69,101,128,27)(2,98,152,57,80,102,139,28,12,88,142,47,70,112,129,38)(3,89,153,48,61,113,140,39,13,99,143,58,71,103,130,29)(4,100,154,59,62,104,121,30,14,90,144,49,72,114,131,40)(5,91,155,50,63,115,122,21,15,81,145,60,73,105,132,31)(6,82,156,41,64,106,123,32,16,92,146,51,74,116,133,22)(7,93,157,52,65,117,124,23,17,83,147,42,75,107,134,33)(8,84,158,43,66,108,125,34,18,94,148,53,76,118,135,24)(9,95,159,54,67,119,126,25,19,85,149,44,77,109,136,35)(10,86,160,45,68,110,127,36,20,96,150,55,78,120,137,26)>;
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)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151), (1,87,151,46,79,111,138,37,11,97,141,56,69,101,128,27)(2,98,152,57,80,102,139,28,12,88,142,47,70,112,129,38)(3,89,153,48,61,113,140,39,13,99,143,58,71,103,130,29)(4,100,154,59,62,104,121,30,14,90,144,49,72,114,131,40)(5,91,155,50,63,115,122,21,15,81,145,60,73,105,132,31)(6,82,156,41,64,106,123,32,16,92,146,51,74,116,133,22)(7,93,157,52,65,117,124,23,17,83,147,42,75,107,134,33)(8,84,158,43,66,108,125,34,18,94,148,53,76,118,135,24)(9,95,159,54,67,119,126,25,19,85,149,44,77,109,136,35)(10,86,160,45,68,110,127,36,20,96,150,55,78,120,137,26) );
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),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,105),(102,104),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(135,140),(136,139),(137,138),(141,160),(142,159),(143,158),(144,157),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151)], [(1,87,151,46,79,111,138,37,11,97,141,56,69,101,128,27),(2,98,152,57,80,102,139,28,12,88,142,47,70,112,129,38),(3,89,153,48,61,113,140,39,13,99,143,58,71,103,130,29),(4,100,154,59,62,104,121,30,14,90,144,49,72,114,131,40),(5,91,155,50,63,115,122,21,15,81,145,60,73,105,132,31),(6,82,156,41,64,106,123,32,16,92,146,51,74,116,133,22),(7,93,157,52,65,117,124,23,17,83,147,42,75,107,134,33),(8,84,158,43,66,108,125,34,18,94,148,53,76,118,135,24),(9,95,159,54,67,119,126,25,19,85,149,44,77,109,136,35),(10,86,160,45,68,110,127,36,20,96,150,55,78,120,137,26)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 16E | ··· | 16L | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| 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 | D20.4C8 |
| kernel | D20.4C8 | C2×C5⋊2C16 | C5×M5(2) | D20.3C4 | C4.Dic5 | C4○D20 | Dic10 | D20 | C40 | M5(2) | 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 | 8 |
Matrix representation of D20.4C8 ►in GL4(𝔽241) generated by
| 0 | 240 | 0 | 0 |
| 1 | 52 | 0 | 0 |
| 0 | 0 | 1 | 116 |
| 0 | 0 | 54 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 116 |
| 0 | 0 | 0 | 240 |
| 87 | 198 | 0 | 0 |
| 43 | 154 | 0 | 0 |
| 0 | 0 | 121 | 29 |
| 0 | 0 | 134 | 120 |
G:=sub<GL(4,GF(241))| [0,1,0,0,240,52,0,0,0,0,1,54,0,0,116,240],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,116,240],[87,43,0,0,198,154,0,0,0,0,121,134,0,0,29,120] >;
D20.4C8 in GAP, Magma, Sage, TeX
D_{20}._4C_8 % in TeX
G:=Group("D20.4C8"); // GroupNames label
G:=SmallGroup(320,73);
// by ID
G=gap.SmallGroup(320,73);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,100,1123,570,136,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations
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