metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊7C4, C2.2D80, C40.79D4, C10.5D16, C10.3SD32, C20.23SD16, C22.10D40, (C2×C80)⋊3C2, (C2×C16)⋊3D5, C40⋊5C4⋊1C2, C8.19(C4×D5), C5⋊3(C2.D16), C40.90(C2×C4), (C2×D40).1C2, (C2×C4).74D20, (C2×C10).16D8, (C2×C8).299D10, (C2×C20).372D4, C4.2(C40⋊C2), C8.36(C5⋊D4), C2.3(C16⋊D5), C2.7(D20⋊5C4), C20.87(C22⋊C4), (C2×C40).372C22, C10.30(D4⋊C4), C4.16(D10⋊C4), SmallGroup(320,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊7C4
G = < a,b,c | a40=b2=c4=1, bab=cac-1=a-1, cbc-1=a23b >
Subgroups: 478 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C16, C4⋊C4, C2×C8, D8, C2×D4, Dic5, C20, D10, C2×C10, C2.D8, C2×C16, C2×D8, C40, D20, C2×Dic5, C2×C20, C22×D5, C2.D16, C80, D40, D40, C4⋊Dic5, C2×C40, C2×D20, C40⋊5C4, C2×C80, C2×D40, D40⋊7C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, D16, SD32, C4×D5, D20, C5⋊D4, C2.D16, C40⋊C2, D40, D10⋊C4, D80, C16⋊D5, D20⋊5C4, D40⋊7C4
►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 47)(42 46)(43 45)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(81 87)(82 86)(83 85)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(121 130)(122 129)(123 128)(124 127)(125 126)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)(141 150)(142 149)(143 148)(144 147)(145 146)
(1 52 146 112)(2 51 147 111)(3 50 148 110)(4 49 149 109)(5 48 150 108)(6 47 151 107)(7 46 152 106)(8 45 153 105)(9 44 154 104)(10 43 155 103)(11 42 156 102)(12 41 157 101)(13 80 158 100)(14 79 159 99)(15 78 160 98)(16 77 121 97)(17 76 122 96)(18 75 123 95)(19 74 124 94)(20 73 125 93)(21 72 126 92)(22 71 127 91)(23 70 128 90)(24 69 129 89)(25 68 130 88)(26 67 131 87)(27 66 132 86)(28 65 133 85)(29 64 134 84)(30 63 135 83)(31 62 136 82)(32 61 137 81)(33 60 138 120)(34 59 139 119)(35 58 140 118)(36 57 141 117)(37 56 142 116)(38 55 143 115)(39 54 144 114)(40 53 145 113)
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)(41,47)(42,46)(43,45)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(81,87)(82,86)(83,85)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(121,130)(122,129)(123,128)(124,127)(125,126)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146), (1,52,146,112)(2,51,147,111)(3,50,148,110)(4,49,149,109)(5,48,150,108)(6,47,151,107)(7,46,152,106)(8,45,153,105)(9,44,154,104)(10,43,155,103)(11,42,156,102)(12,41,157,101)(13,80,158,100)(14,79,159,99)(15,78,160,98)(16,77,121,97)(17,76,122,96)(18,75,123,95)(19,74,124,94)(20,73,125,93)(21,72,126,92)(22,71,127,91)(23,70,128,90)(24,69,129,89)(25,68,130,88)(26,67,131,87)(27,66,132,86)(28,65,133,85)(29,64,134,84)(30,63,135,83)(31,62,136,82)(32,61,137,81)(33,60,138,120)(34,59,139,119)(35,58,140,118)(36,57,141,117)(37,56,142,116)(38,55,143,115)(39,54,144,114)(40,53,145,113)>;
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)(41,47)(42,46)(43,45)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(81,87)(82,86)(83,85)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(121,130)(122,129)(123,128)(124,127)(125,126)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146), (1,52,146,112)(2,51,147,111)(3,50,148,110)(4,49,149,109)(5,48,150,108)(6,47,151,107)(7,46,152,106)(8,45,153,105)(9,44,154,104)(10,43,155,103)(11,42,156,102)(12,41,157,101)(13,80,158,100)(14,79,159,99)(15,78,160,98)(16,77,121,97)(17,76,122,96)(18,75,123,95)(19,74,124,94)(20,73,125,93)(21,72,126,92)(22,71,127,91)(23,70,128,90)(24,69,129,89)(25,68,130,88)(26,67,131,87)(27,66,132,86)(28,65,133,85)(29,64,134,84)(30,63,135,83)(31,62,136,82)(32,61,137,81)(33,60,138,120)(34,59,139,119)(35,58,140,118)(36,57,141,117)(37,56,142,116)(38,55,143,115)(39,54,144,114)(40,53,145,113) );
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),(41,47),(42,46),(43,45),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(81,87),(82,86),(83,85),(88,120),(89,119),(90,118),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(121,130),(122,129),(123,128),(124,127),(125,126),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151),(141,150),(142,149),(143,148),(144,147),(145,146)], [(1,52,146,112),(2,51,147,111),(3,50,148,110),(4,49,149,109),(5,48,150,108),(6,47,151,107),(7,46,152,106),(8,45,153,105),(9,44,154,104),(10,43,155,103),(11,42,156,102),(12,41,157,101),(13,80,158,100),(14,79,159,99),(15,78,160,98),(16,77,121,97),(17,76,122,96),(18,75,123,95),(19,74,124,94),(20,73,125,93),(21,72,126,92),(22,71,127,91),(23,70,128,90),(24,69,129,89),(25,68,130,88),(26,67,131,87),(27,66,132,86),(28,65,133,85),(29,64,134,84),(30,63,135,83),(31,62,136,82),(32,61,137,81),(33,60,138,120),(34,59,139,119),(35,58,140,118),(36,57,141,117),(37,56,142,116),(38,55,143,115),(39,54,144,114),(40,53,145,113)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | SD16 | D8 | D10 | D16 | SD32 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | D40 | D80 | C16⋊D5 |
| kernel | D40⋊7C4 | C40⋊5C4 | C2×C80 | C2×D40 | D40 | C40 | C2×C20 | C2×C16 | C20 | C2×C10 | C2×C8 | C10 | C10 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of D40⋊7C4 ►in GL4(𝔽241) generated by
| 11 | 230 | 0 | 0 |
| 11 | 11 | 0 | 0 |
| 0 | 0 | 20 | 194 |
| 0 | 0 | 47 | 227 |
| 11 | 230 | 0 | 0 |
| 230 | 230 | 0 | 0 |
| 0 | 0 | 199 | 29 |
| 0 | 0 | 14 | 42 |
| 214 | 85 | 0 | 0 |
| 85 | 27 | 0 | 0 |
| 0 | 0 | 3 | 170 |
| 0 | 0 | 85 | 238 |
G:=sub<GL(4,GF(241))| [11,11,0,0,230,11,0,0,0,0,20,47,0,0,194,227],[11,230,0,0,230,230,0,0,0,0,199,14,0,0,29,42],[214,85,0,0,85,27,0,0,0,0,3,85,0,0,170,238] >;
D40⋊7C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_7C_4 % in TeX
G:=Group("D40:7C4"); // GroupNames label
G:=SmallGroup(320,67);
// by ID
G=gap.SmallGroup(320,67);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,204,422,268,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^23*b>;
// generators/relations