direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×D40, D20⋊7D6, C40⋊22D6, C24⋊4D10, D120⋊4C2, C120⋊2C22, D6.11D20, D60⋊16C22, C60.92C23, Dic3.2D20, C5⋊1(S3×D8), C8⋊7(S3×D5), C3⋊1(C2×D40), C15⋊2(C2×D8), C3⋊C8⋊24D10, (C5×S3)⋊1D8, (S3×C8)⋊1D5, (S3×C40)⋊1C2, (S3×D20)⋊8C2, (C3×D40)⋊2C2, C3⋊D40⋊9C2, C10.2(S3×D4), C6.2(C2×D20), C2.7(S3×D20), C30.6(C2×D4), (S3×C10).18D4, (C4×S3).37D10, (C3×D20)⋊14C22, (C5×Dic3).21D4, C12.65(C22×D5), (S3×C20).43C22, C20.142(C22×S3), C4.91(C2×S3×D5), (C5×C3⋊C8)⋊28C22, SmallGroup(480,328)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×D40
G = < a,b,c,d | a3=b2=c40=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1308 in 152 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, D8, C2×D4, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C2×D8, C40, C40, D20, D20, C2×C20, C22×D5, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C5×Dic3, C60, S3×D5, C6×D5, S3×C10, D30, D40, D40, C2×C40, C2×D20, S3×D8, C5×C3⋊C8, C120, C3⋊D20, C3×D20, S3×C20, D60, C2×S3×D5, C2×D40, C3⋊D40, C3×D40, S3×C40, D120, S3×D20, S3×D40
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, C2×D8, D20, C22×D5, S3×D4, S3×D5, D40, C2×D20, S3×D8, C2×S3×D5, C2×D40, S3×D20, S3×D40
►On 120 points
Generators in S120
(1 120 76)(2 81 77)(3 82 78)(4 83 79)(5 84 80)(6 85 41)(7 86 42)(8 87 43)(9 88 44)(10 89 45)(11 90 46)(12 91 47)(13 92 48)(14 93 49)(15 94 50)(16 95 51)(17 96 52)(18 97 53)(19 98 54)(20 99 55)(21 100 56)(22 101 57)(23 102 58)(24 103 59)(25 104 60)(26 105 61)(27 106 62)(28 107 63)(29 108 64)(30 109 65)(31 110 66)(32 111 67)(33 112 68)(34 113 69)(35 114 70)(36 115 71)(37 116 72)(38 117 73)(39 118 74)(40 119 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 45)(42 44)(46 80)(47 79)(48 78)(49 77)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)
G:=sub<Sym(120)| (1,120,76)(2,81,77)(3,82,78)(4,83,79)(5,84,80)(6,85,41)(7,86,42)(8,87,43)(9,88,44)(10,89,45)(11,90,46)(12,91,47)(13,92,48)(14,93,49)(15,94,50)(16,95,51)(17,96,52)(18,97,53)(19,98,54)(20,99,55)(21,100,56)(22,101,57)(23,102,58)(24,103,59)(25,104,60)(26,105,61)(27,106,62)(28,107,63)(29,108,64)(30,109,65)(31,110,66)(32,111,67)(33,112,68)(34,113,69)(35,114,70)(36,115,71)(37,116,72)(38,117,73)(39,118,74)(40,119,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,45)(42,44)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)>;
G:=Group( (1,120,76)(2,81,77)(3,82,78)(4,83,79)(5,84,80)(6,85,41)(7,86,42)(8,87,43)(9,88,44)(10,89,45)(11,90,46)(12,91,47)(13,92,48)(14,93,49)(15,94,50)(16,95,51)(17,96,52)(18,97,53)(19,98,54)(20,99,55)(21,100,56)(22,101,57)(23,102,58)(24,103,59)(25,104,60)(26,105,61)(27,106,62)(28,107,63)(29,108,64)(30,109,65)(31,110,66)(32,111,67)(33,112,68)(34,113,69)(35,114,70)(36,115,71)(37,116,72)(38,117,73)(39,118,74)(40,119,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,45)(42,44)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108) );
G=PermutationGroup([[(1,120,76),(2,81,77),(3,82,78),(4,83,79),(5,84,80),(6,85,41),(7,86,42),(8,87,43),(9,88,44),(10,89,45),(11,90,46),(12,91,47),(13,92,48),(14,93,49),(15,94,50),(16,95,51),(17,96,52),(18,97,53),(19,98,54),(20,99,55),(21,100,56),(22,101,57),(23,102,58),(24,103,59),(25,104,60),(26,105,61),(27,106,62),(28,107,63),(29,108,64),(30,109,65),(31,110,66),(32,111,67),(33,112,68),(34,113,69),(35,114,70),(36,115,71),(37,116,72),(38,117,73),(39,118,74),(40,119,75)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,45),(42,44),(46,80),(47,79),(48,78),(49,77),(50,76),(51,75),(52,74),(53,73),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108)]])
69 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 30A | 30B | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 3 | 3 | 20 | 20 | 60 | 60 | 2 | 2 | 6 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 6 | 6 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
69 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | D10 | D20 | D20 | D40 | S3×D4 | S3×D5 | S3×D8 | C2×S3×D5 | S3×D20 | S3×D40 |
| kernel | S3×D40 | C3⋊D40 | C3×D40 | S3×C40 | D120 | S3×D20 | D40 | C5×Dic3 | S3×C10 | S3×C8 | C40 | D20 | C5×S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | S3 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 16 | 1 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of S3×D40 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 197 | 163 | 0 | 0 | 0 | 0 |
| 78 | 78 | 0 | 0 | 0 | 0 |
| 0 | 0 | 195 | 70 | 0 | 0 |
| 0 | 0 | 208 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 190 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 147 | 0 | 0 |
| 0 | 0 | 44 | 194 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[197,78,0,0,0,0,163,78,0,0,0,0,0,0,195,208,0,0,0,0,70,24,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,190,0,0,0,0,0,240,0,0,0,0,0,0,47,44,0,0,0,0,147,194,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
S3×D40 in GAP, Magma, Sage, TeX
S_3\times D_{40} % in TeX
G:=Group("S3xD40"); // GroupNames label
G:=SmallGroup(480,328);
// by ID
G=gap.SmallGroup(480,328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,142,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^40=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations