metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40.5C4, C20.56D8, C40.39D4, C8.23D20, Dic20.5C4, C8.14(C4×D5), C40.53(C2×C4), (C2×C20).97D4, C8.C4⋊2D5, C4.29(D4⋊D5), (C2×C8).249D10, C5⋊3(D8.C4), (C2×C10).6SD16, D40⋊7C2.3C2, (C2×C40).37C22, C22.1(Q8⋊D5), C4.6(D10⋊C4), C20.53(C22⋊C4), C10.24(D4⋊C4), C2.11(D20⋊6C4), (C2×C5⋊2C16)⋊1C2, (C5×C8.C4)⋊1C2, (C2×C4).118(C5⋊D4), SmallGroup(320,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40.5C4
G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a31, cbc-1=a15b >
2C2
40C2
20C22
20C4
2C10
8D5
4C8
10Q8
10D4
20D4
20C2×C4
4Dic5
4D10
5Q16
5D8
10C16
10SD16
10C4○D4
2D20
4C40
Smallest permutation representation of D40.5C4
►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 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(77 80)(78 79)(81 117)(82 116)(83 115)(84 114)(85 113)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(118 120)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)(127 153)(128 152)(129 151)(130 150)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)
(1 128 59 107 21 148 79 87)(2 159 60 98 22 139 80 118)(3 150 61 89 23 130 41 109)(4 141 62 120 24 121 42 100)(5 132 63 111 25 152 43 91)(6 123 64 102 26 143 44 82)(7 154 65 93 27 134 45 113)(8 145 66 84 28 125 46 104)(9 136 67 115 29 156 47 95)(10 127 68 106 30 147 48 86)(11 158 69 97 31 138 49 117)(12 149 70 88 32 129 50 108)(13 140 71 119 33 160 51 99)(14 131 72 110 34 151 52 90)(15 122 73 101 35 142 53 81)(16 153 74 92 36 133 54 112)(17 144 75 83 37 124 55 103)(18 135 76 114 38 155 56 94)(19 126 77 105 39 146 57 85)(20 157 78 96 40 137 58 116)
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)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(77,80)(78,79)(81,117)(82,116)(83,115)(84,114)(85,113)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(118,120)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141), (1,128,59,107,21,148,79,87)(2,159,60,98,22,139,80,118)(3,150,61,89,23,130,41,109)(4,141,62,120,24,121,42,100)(5,132,63,111,25,152,43,91)(6,123,64,102,26,143,44,82)(7,154,65,93,27,134,45,113)(8,145,66,84,28,125,46,104)(9,136,67,115,29,156,47,95)(10,127,68,106,30,147,48,86)(11,158,69,97,31,138,49,117)(12,149,70,88,32,129,50,108)(13,140,71,119,33,160,51,99)(14,131,72,110,34,151,52,90)(15,122,73,101,35,142,53,81)(16,153,74,92,36,133,54,112)(17,144,75,83,37,124,55,103)(18,135,76,114,38,155,56,94)(19,126,77,105,39,146,57,85)(20,157,78,96,40,137,58,116)>;
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)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(77,80)(78,79)(81,117)(82,116)(83,115)(84,114)(85,113)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(118,120)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141), (1,128,59,107,21,148,79,87)(2,159,60,98,22,139,80,118)(3,150,61,89,23,130,41,109)(4,141,62,120,24,121,42,100)(5,132,63,111,25,152,43,91)(6,123,64,102,26,143,44,82)(7,154,65,93,27,134,45,113)(8,145,66,84,28,125,46,104)(9,136,67,115,29,156,47,95)(10,127,68,106,30,147,48,86)(11,158,69,97,31,138,49,117)(12,149,70,88,32,129,50,108)(13,140,71,119,33,160,51,99)(14,131,72,110,34,151,52,90)(15,122,73,101,35,142,53,81)(16,153,74,92,36,133,54,112)(17,144,75,83,37,124,55,103)(18,135,76,114,38,155,56,94)(19,126,77,105,39,146,57,85)(20,157,78,96,40,137,58,116) );
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),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(77,80),(78,79),(81,117),(82,116),(83,115),(84,114),(85,113),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(118,120),(121,159),(122,158),(123,157),(124,156),(125,155),(126,154),(127,153),(128,152),(129,151),(130,150),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141)], [(1,128,59,107,21,148,79,87),(2,159,60,98,22,139,80,118),(3,150,61,89,23,130,41,109),(4,141,62,120,24,121,42,100),(5,132,63,111,25,152,43,91),(6,123,64,102,26,143,44,82),(7,154,65,93,27,134,45,113),(8,145,66,84,28,125,46,104),(9,136,67,115,29,156,47,95),(10,127,68,106,30,147,48,86),(11,158,69,97,31,138,49,117),(12,149,70,88,32,129,50,108),(13,140,71,119,33,160,51,99),(14,131,72,110,34,151,52,90),(15,122,73,101,35,142,53,81),(16,153,74,92,36,133,54,112),(17,144,75,83,37,124,55,103),(18,135,76,114,38,155,56,94),(19,126,77,105,39,146,57,85),(20,157,78,96,40,137,58,116)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 40 | 1 | 1 | 2 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | D20 | C5⋊D4 | D8.C4 | D4⋊D5 | Q8⋊D5 | D40.5C4 |
| kernel | D40.5C4 | C2×C5⋊2C16 | C5×C8.C4 | D40⋊7C2 | D40 | Dic20 | C40 | C2×C20 | C8.C4 | 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 | 2 | 2 | 8 |
Matrix representation of D40.5C4 ►in GL4(𝔽241) generated by
| 0 | 219 | 0 | 0 |
| 11 | 219 | 0 | 0 |
| 0 | 0 | 190 | 190 |
| 0 | 0 | 51 | 240 |
| 0 | 219 | 0 | 0 |
| 230 | 0 | 0 | 0 |
| 0 | 0 | 190 | 190 |
| 0 | 0 | 240 | 51 |
| 173 | 50 | 0 | 0 |
| 198 | 68 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
G:=sub<GL(4,GF(241))| [0,11,0,0,219,219,0,0,0,0,190,51,0,0,190,240],[0,230,0,0,219,0,0,0,0,0,190,240,0,0,190,51],[173,198,0,0,50,68,0,0,0,0,177,0,0,0,0,177] >;
D40.5C4 in GAP, Magma, Sage, TeX
D_{40}._5C_4 % in TeX
G:=Group("D40.5C4"); // GroupNames label
G:=SmallGroup(320,55);
// by ID
G=gap.SmallGroup(320,55);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,184,675,346,192,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^15*b>;
// generators/relations
Export