metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40:2D6, D40:2S3, D20:8D6, C24:10D10, D6.5D20, C120:8C22, C60.94C23, Dic3.7D20, D60.26C22, Dic30:15C22, C3:C8:2D10, C8:4(S3xD5), (C3xD40):4C2, C8:S3:3D5, C5:1(D8:S3), C10.4(S3xD4), C30.8(C2xD4), C2.9(S3xD20), C6.4(C2xD20), C24:D5:3C2, (S3xD20):10C2, C3:D40:11C2, C3:2(C8:D10), C15:4(C8:C22), (C4xS3).2D10, (S3xC10).2D4, D20:5S3:8C2, (C5xDic3).2D4, C6.D20:10C2, (C3xD20):15C22, C12.67(C22xD5), (S3xC20).25C22, C20.144(C22xS3), C4.93(C2xS3xD5), (C5xC8:S3):3C2, (C5xC3:C8):16C22, SmallGroup(480,330)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40:S3
G = < a,b,c,d | a40=b2=c3=d2=1, bab=a-1, ac=ca, dad=a21, bc=cb, dbd=a20b, dcd=c-1 >
Subgroups: 1020 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2xC4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2xC6, C15, M4(2), D8, SD16, C2xD4, C4oD4, Dic5, C20, C20, D10, C2xC10, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C22xS3, C5xS3, C3xD5, D15, C30, C8:C22, C40, C40, Dic10, C4xD5, D20, D20, C5:D4, C2xC20, C22xD5, C8:S3, C24:C2, D4:S3, D4.S3, C3xD8, S3xD4, D4:2S3, C5xDic3, Dic15, C60, S3xD5, C6xD5, S3xC10, D30, C40:C2, D40, D40, C5xM4(2), C2xD20, C4oD20, D8:S3, C5xC3:C8, C120, D5xDic3, C15:D4, C3:D20, C3xD20, S3xC20, Dic30, D60, C2xS3xD5, C8:D10, C3:D40, C6.D20, C3xD40, C5xC8:S3, C24:D5, D20:5S3, S3xD20, D40:S3
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, C22xS3, C8:C22, D20, C22xD5, S3xD4, S3xD5, C2xD20, D8:S3, C2xS3xD5, C8:D10, S3xD20, D40:S3
►On 120 points
Generators in S120
(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)
(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 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(81 82)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)
(1 49 102)(2 50 103)(3 51 104)(4 52 105)(5 53 106)(6 54 107)(7 55 108)(8 56 109)(9 57 110)(10 58 111)(11 59 112)(12 60 113)(13 61 114)(14 62 115)(15 63 116)(16 64 117)(17 65 118)(18 66 119)(19 67 120)(20 68 81)(21 69 82)(22 70 83)(23 71 84)(24 72 85)(25 73 86)(26 74 87)(27 75 88)(28 76 89)(29 77 90)(30 78 91)(31 79 92)(32 80 93)(33 41 94)(34 42 95)(35 43 96)(36 44 97)(37 45 98)(38 46 99)(39 47 100)(40 48 101)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(41 94)(42 115)(43 96)(44 117)(45 98)(46 119)(47 100)(48 81)(49 102)(50 83)(51 104)(52 85)(53 106)(54 87)(55 108)(56 89)(57 110)(58 91)(59 112)(60 93)(61 114)(62 95)(63 116)(64 97)(65 118)(66 99)(67 120)(68 101)(69 82)(70 103)(71 84)(72 105)(73 86)(74 107)(75 88)(76 109)(77 90)(78 111)(79 92)(80 113)
G:=sub<Sym(120)| (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), (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)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(81,82)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102), (1,49,102)(2,50,103)(3,51,104)(4,52,105)(5,53,106)(6,54,107)(7,55,108)(8,56,109)(9,57,110)(10,58,111)(11,59,112)(12,60,113)(13,61,114)(14,62,115)(15,63,116)(16,64,117)(17,65,118)(18,66,119)(19,67,120)(20,68,81)(21,69,82)(22,70,83)(23,71,84)(24,72,85)(25,73,86)(26,74,87)(27,75,88)(28,76,89)(29,77,90)(30,78,91)(31,79,92)(32,80,93)(33,41,94)(34,42,95)(35,43,96)(36,44,97)(37,45,98)(38,46,99)(39,47,100)(40,48,101), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(41,94)(42,115)(43,96)(44,117)(45,98)(46,119)(47,100)(48,81)(49,102)(50,83)(51,104)(52,85)(53,106)(54,87)(55,108)(56,89)(57,110)(58,91)(59,112)(60,93)(61,114)(62,95)(63,116)(64,97)(65,118)(66,99)(67,120)(68,101)(69,82)(70,103)(71,84)(72,105)(73,86)(74,107)(75,88)(76,109)(77,90)(78,111)(79,92)(80,113)>;
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), (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)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(81,82)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102), (1,49,102)(2,50,103)(3,51,104)(4,52,105)(5,53,106)(6,54,107)(7,55,108)(8,56,109)(9,57,110)(10,58,111)(11,59,112)(12,60,113)(13,61,114)(14,62,115)(15,63,116)(16,64,117)(17,65,118)(18,66,119)(19,67,120)(20,68,81)(21,69,82)(22,70,83)(23,71,84)(24,72,85)(25,73,86)(26,74,87)(27,75,88)(28,76,89)(29,77,90)(30,78,91)(31,79,92)(32,80,93)(33,41,94)(34,42,95)(35,43,96)(36,44,97)(37,45,98)(38,46,99)(39,47,100)(40,48,101), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(41,94)(42,115)(43,96)(44,117)(45,98)(46,119)(47,100)(48,81)(49,102)(50,83)(51,104)(52,85)(53,106)(54,87)(55,108)(56,89)(57,110)(58,91)(59,112)(60,93)(61,114)(62,95)(63,116)(64,97)(65,118)(66,99)(67,120)(68,101)(69,82)(70,103)(71,84)(72,105)(73,86)(74,107)(75,88)(76,109)(77,90)(78,111)(79,92)(80,113) );
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)], [(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),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(81,82),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102)], [(1,49,102),(2,50,103),(3,51,104),(4,52,105),(5,53,106),(6,54,107),(7,55,108),(8,56,109),(9,57,110),(10,58,111),(11,59,112),(12,60,113),(13,61,114),(14,62,115),(15,63,116),(16,64,117),(17,65,118),(18,66,119),(19,67,120),(20,68,81),(21,69,82),(22,70,83),(23,71,84),(24,72,85),(25,73,86),(26,74,87),(27,75,88),(28,76,89),(29,77,90),(30,78,91),(31,79,92),(32,80,93),(33,41,94),(34,42,95),(35,43,96),(36,44,97),(37,45,98),(38,46,99),(39,47,100),(40,48,101)], [(2,22),(4,24),(6,26),(8,28),(10,30),(12,32),(14,34),(16,36),(18,38),(20,40),(41,94),(42,115),(43,96),(44,117),(45,98),(46,119),(47,100),(48,81),(49,102),(50,83),(51,104),(52,85),(53,106),(54,87),(55,108),(56,89),(57,110),(58,91),(59,112),(60,93),(61,114),(62,95),(63,116),(64,97),(65,118),(66,99),(67,120),(68,101),(69,82),(70,103),(71,84),(72,105),(73,86),(74,107),(75,88),(76,109),(77,90),(78,111),(79,92),(80,113)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 20 | 20 | 60 | 2 | 2 | 6 | 60 | 2 | 2 | 2 | 40 | 40 | 4 | 12 | 2 | 2 | 12 | 12 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | D20 | D20 | C8:C22 | S3xD4 | S3xD5 | D8:S3 | C2xS3xD5 | C8:D10 | S3xD20 | D40:S3 |
| kernel | D40:S3 | C3:D40 | C6.D20 | C3xD40 | C5xC8:S3 | C24:D5 | D20:5S3 | S3xD20 | D40 | C5xDic3 | S3xC10 | C8:S3 | C40 | D20 | C3:C8 | C24 | C4xS3 | Dic3 | D6 | C15 | C10 | C8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D40:S3 ►in GL4(F241) generated by
| 16 | 203 | 32 | 165 |
| 86 | 102 | 172 | 204 |
| 209 | 76 | 225 | 38 |
| 69 | 37 | 155 | 139 |
| 84 | 148 | 168 | 55 |
| 18 | 157 | 36 | 73 |
| 73 | 186 | 157 | 93 |
| 205 | 168 | 223 | 84 |
| 240 | 0 | 240 | 0 |
| 0 | 240 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 240 | 0 |
| 0 | 240 | 0 | 240 |
G:=sub<GL(4,GF(241))| [16,86,209,69,203,102,76,37,32,172,225,155,165,204,38,139],[84,18,73,205,148,157,186,168,168,36,157,223,55,73,93,84],[240,0,1,0,0,240,0,1,240,0,0,0,0,240,0,0],[1,0,240,0,0,1,0,240,0,0,240,0,0,0,0,240] >;
D40:S3 in GAP, Magma, Sage, TeX
D_{40}\rtimes S_3 % in TeX
G:=Group("D40:S3"); // GroupNames label
G:=SmallGroup(480,330);
// by ID
G=gap.SmallGroup(480,330);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,142,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^21,b*c=c*b,d*b*d=a^20*b,d*c*d=c^-1>;
// generators/relations