metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40.3C4, C20.37D8, C40.85D4, C4.19D40, Dic20.3C4, (C2×C80)⋊4C2, (C2×C16)⋊4D5, C8.20(C4×D5), C40.91(C2×C4), (C2×C4).75D20, C40.6C4⋊1C2, (C2×C8).312D10, (C2×C20).394D4, C5⋊4(D8.C4), C8.42(C5⋊D4), D40⋊7C2.1C2, (C2×C10).18SD16, C2.8(D20⋊5C4), C20.88(C22⋊C4), (C2×C40).384C22, C22.1(C40⋊C2), C4.17(D10⋊C4), C10.31(D4⋊C4), SmallGroup(320,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40.3C4
G = < a,b,c | a40=b2=1, c4=a10, bab=a-1, ac=ca, cbc-1=a15b >
2C2
40C2
20C4
20C22
2C10
8D5
10Q8
10D4
20D4
20C2×C4
20C8
4D10
4Dic5
2C16
5D8
5Q16
10M4(2)
10SD16
10C4○D4
2D20
2C80
Smallest permutation representation of D40.3C4
►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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(154 160)(155 159)(156 158)
(1 57 88 160 11 67 98 130 21 77 108 140 31 47 118 150)(2 58 89 121 12 68 99 131 22 78 109 141 32 48 119 151)(3 59 90 122 13 69 100 132 23 79 110 142 33 49 120 152)(4 60 91 123 14 70 101 133 24 80 111 143 34 50 81 153)(5 61 92 124 15 71 102 134 25 41 112 144 35 51 82 154)(6 62 93 125 16 72 103 135 26 42 113 145 36 52 83 155)(7 63 94 126 17 73 104 136 27 43 114 146 37 53 84 156)(8 64 95 127 18 74 105 137 28 44 115 147 38 54 85 157)(9 65 96 128 19 75 106 138 29 45 116 148 39 55 86 158)(10 66 97 129 20 76 107 139 30 46 117 149 40 56 87 159)
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(154,160)(155,159)(156,158), (1,57,88,160,11,67,98,130,21,77,108,140,31,47,118,150)(2,58,89,121,12,68,99,131,22,78,109,141,32,48,119,151)(3,59,90,122,13,69,100,132,23,79,110,142,33,49,120,152)(4,60,91,123,14,70,101,133,24,80,111,143,34,50,81,153)(5,61,92,124,15,71,102,134,25,41,112,144,35,51,82,154)(6,62,93,125,16,72,103,135,26,42,113,145,36,52,83,155)(7,63,94,126,17,73,104,136,27,43,114,146,37,53,84,156)(8,64,95,127,18,74,105,137,28,44,115,147,38,54,85,157)(9,65,96,128,19,75,106,138,29,45,116,148,39,55,86,158)(10,66,97,129,20,76,107,139,30,46,117,149,40,56,87,159)>;
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(154,160)(155,159)(156,158), (1,57,88,160,11,67,98,130,21,77,108,140,31,47,118,150)(2,58,89,121,12,68,99,131,22,78,109,141,32,48,119,151)(3,59,90,122,13,69,100,132,23,79,110,142,33,49,120,152)(4,60,91,123,14,70,101,133,24,80,111,143,34,50,81,153)(5,61,92,124,15,71,102,134,25,41,112,144,35,51,82,154)(6,62,93,125,16,72,103,135,26,42,113,145,36,52,83,155)(7,63,94,126,17,73,104,136,27,43,114,146,37,53,84,156)(8,64,95,127,18,74,105,137,28,44,115,147,38,54,85,157)(9,65,96,128,19,75,106,138,29,45,116,148,39,55,86,158)(10,66,97,129,20,76,107,139,30,46,117,149,40,56,87,159) );
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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(81,104),(82,103),(83,102),(84,101),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(121,153),(122,152),(123,151),(124,150),(125,149),(126,148),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138),(154,160),(155,159),(156,158)], [(1,57,88,160,11,67,98,130,21,77,108,140,31,47,118,150),(2,58,89,121,12,68,99,131,22,78,109,141,32,48,119,151),(3,59,90,122,13,69,100,132,23,79,110,142,33,49,120,152),(4,60,91,123,14,70,101,133,24,80,111,143,34,50,81,153),(5,61,92,124,15,71,102,134,25,41,112,144,35,51,82,154),(6,62,93,125,16,72,103,135,26,42,113,145,36,52,83,155),(7,63,94,126,17,73,104,136,27,43,114,146,37,53,84,156),(8,64,95,127,18,74,105,137,28,44,115,147,38,54,85,157),(9,65,96,128,19,75,106,138,29,45,116,148,39,55,86,158),(10,66,97,129,20,76,107,139,30,46,117,149,40,56,87,159)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 40 | 1 | 1 | 2 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 40 | 40 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | C5⋊D4 | D20 | D8.C4 | D40 | C40⋊C2 | D40.3C4 |
| kernel | D40.3C4 | C40.6C4 | C2×C80 | D40⋊7C2 | D40 | Dic20 | C40 | C2×C20 | C2×C16 | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 32 |
Matrix representation of D40.3C4 ►in GL2(𝔽241) generated by
| 20 | 194 |
| 47 | 227 |
| 14 | 42 |
| 47 | 227 |
| 101 | 201 |
| 40 | 190 |
G:=sub<GL(2,GF(241))| [20,47,194,227],[14,47,42,227],[101,40,201,190] >;
D40.3C4 in GAP, Magma, Sage, TeX
D_{40}._3C_4 % in TeX
G:=Group("D40.3C4"); // GroupNames label
G:=SmallGroup(320,68);
// by ID
G=gap.SmallGroup(320,68);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,268,1123,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations
Export