direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C40⋊C2, C40⋊23D6, C24⋊13D10, Dic10⋊7D6, D20.18D6, D6.10D20, C120⋊12C22, C60.91C23, Dic3.1D20, D60.25C22, Dic30⋊14C22, C8⋊8(S3×D5), C3⋊C8⋊23D10, (S3×C8)⋊4D5, C5⋊1(S3×SD16), (S3×C40)⋊4C2, C2.6(S3×D20), C30.5(C2×D4), C6.1(C2×D20), C10.1(S3×D4), C15⋊2(C2×SD16), (C5×S3)⋊1SD16, (S3×D20).2C2, C24⋊D5⋊11C2, (S3×Dic10)⋊8C2, (S3×C10).17D4, (C4×S3).36D10, C6.D20⋊9C2, C15⋊SD16⋊9C2, (C5×Dic3).20D4, C12.64(C22×D5), (S3×C20).42C22, C20.141(C22×S3), (C3×D20).20C22, (C3×Dic10)⋊12C22, C3⋊1(C2×C40⋊C2), C4.90(C2×S3×D5), (C3×C40⋊C2)⋊3C2, (C5×C3⋊C8)⋊27C22, SmallGroup(480,327)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C40⋊C2
G = < a,b,c,d | a3=b2=c40=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c19 >
Subgroups: 972 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C40, C40, Dic10, Dic10, D20, D20, C2×Dic5, C2×C20, C22×D5, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C40⋊C2, C40⋊C2, C2×C40, C2×Dic10, C2×D20, S3×SD16, C5×C3⋊C8, C120, S3×Dic5, C3⋊D20, C15⋊Q8, C3×Dic10, C3×D20, S3×C20, Dic30, D60, C2×S3×D5, C2×C40⋊C2, C6.D20, C15⋊SD16, C3×C40⋊C2, S3×C40, C24⋊D5, S3×Dic10, S3×D20, S3×C40⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, C2×SD16, D20, C22×D5, S3×D4, S3×D5, C40⋊C2, C2×D20, S3×SD16, C2×S3×D5, C2×C40⋊C2, S3×D20, S3×C40⋊C2
►On 120 points
(1 68 99)(2 69 100)(3 70 101)(4 71 102)(5 72 103)(6 73 104)(7 74 105)(8 75 106)(9 76 107)(10 77 108)(11 78 109)(12 79 110)(13 80 111)(14 41 112)(15 42 113)(16 43 114)(17 44 115)(18 45 116)(19 46 117)(20 47 118)(21 48 119)(22 49 120)(23 50 81)(24 51 82)(25 52 83)(26 53 84)(27 54 85)(28 55 86)(29 56 87)(30 57 88)(31 58 89)(32 59 90)(33 60 91)(34 61 92)(35 62 93)(36 63 94)(37 64 95)(38 65 96)(39 66 97)(40 67 98)
(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 75)(42 54)(43 73)(44 52)(45 71)(46 50)(47 69)(49 67)(51 65)(53 63)(55 61)(56 80)(57 59)(58 78)(60 76)(62 74)(64 72)(66 70)(77 79)(81 117)(82 96)(83 115)(84 94)(85 113)(86 92)(87 111)(88 90)(89 109)(91 107)(93 105)(95 103)(97 101)(98 120)(100 118)(102 116)(104 114)(106 112)(108 110)
G:=sub<Sym(120)| (1,68,99)(2,69,100)(3,70,101)(4,71,102)(5,72,103)(6,73,104)(7,74,105)(8,75,106)(9,76,107)(10,77,108)(11,78,109)(12,79,110)(13,80,111)(14,41,112)(15,42,113)(16,43,114)(17,44,115)(18,45,116)(19,46,117)(20,47,118)(21,48,119)(22,49,120)(23,50,81)(24,51,82)(25,52,83)(26,53,84)(27,54,85)(28,55,86)(29,56,87)(30,57,88)(31,58,89)(32,59,90)(33,60,91)(34,61,92)(35,62,93)(36,63,94)(37,64,95)(38,65,96)(39,66,97)(40,67,98), (41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111), (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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,75)(42,54)(43,73)(44,52)(45,71)(46,50)(47,69)(49,67)(51,65)(53,63)(55,61)(56,80)(57,59)(58,78)(60,76)(62,74)(64,72)(66,70)(77,79)(81,117)(82,96)(83,115)(84,94)(85,113)(86,92)(87,111)(88,90)(89,109)(91,107)(93,105)(95,103)(97,101)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110)>;
G:=Group( (1,68,99)(2,69,100)(3,70,101)(4,71,102)(5,72,103)(6,73,104)(7,74,105)(8,75,106)(9,76,107)(10,77,108)(11,78,109)(12,79,110)(13,80,111)(14,41,112)(15,42,113)(16,43,114)(17,44,115)(18,45,116)(19,46,117)(20,47,118)(21,48,119)(22,49,120)(23,50,81)(24,51,82)(25,52,83)(26,53,84)(27,54,85)(28,55,86)(29,56,87)(30,57,88)(31,58,89)(32,59,90)(33,60,91)(34,61,92)(35,62,93)(36,63,94)(37,64,95)(38,65,96)(39,66,97)(40,67,98), (41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111), (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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,75)(42,54)(43,73)(44,52)(45,71)(46,50)(47,69)(49,67)(51,65)(53,63)(55,61)(56,80)(57,59)(58,78)(60,76)(62,74)(64,72)(66,70)(77,79)(81,117)(82,96)(83,115)(84,94)(85,113)(86,92)(87,111)(88,90)(89,109)(91,107)(93,105)(95,103)(97,101)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110) );
G=PermutationGroup([[(1,68,99),(2,69,100),(3,70,101),(4,71,102),(5,72,103),(6,73,104),(7,74,105),(8,75,106),(9,76,107),(10,77,108),(11,78,109),(12,79,110),(13,80,111),(14,41,112),(15,42,113),(16,43,114),(17,44,115),(18,45,116),(19,46,117),(20,47,118),(21,48,119),(22,49,120),(23,50,81),(24,51,82),(25,52,83),(26,53,84),(27,54,85),(28,55,86),(29,56,87),(30,57,88),(31,58,89),(32,59,90),(33,60,91),(34,61,92),(35,62,93),(36,63,94),(37,64,95),(38,65,96),(39,66,97),(40,67,98)], [(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111)], [(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)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,75),(42,54),(43,73),(44,52),(45,71),(46,50),(47,69),(49,67),(51,65),(53,63),(55,61),(56,80),(57,59),(58,78),(60,76),(62,74),(64,72),(66,70),(77,79),(81,117),(82,96),(83,115),(84,94),(85,113),(86,92),(87,111),(88,90),(89,109),(91,107),(93,105),(95,103),(97,101),(98,120),(100,118),(102,116),(104,114),(106,112),(108,110)]])
69 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 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 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 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 | 60 | 2 | 2 | 6 | 20 | 60 | 2 | 2 | 2 | 40 | 2 | 2 | 6 | 6 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 40 | 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 | 1 | 1 | 2 | 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 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | SD16 | D10 | D10 | D10 | D20 | D20 | C40⋊C2 | S3×D4 | S3×D5 | S3×SD16 | C2×S3×D5 | S3×D20 | S3×C40⋊C2 |
| kernel | S3×C40⋊C2 | C6.D20 | C15⋊SD16 | C3×C40⋊C2 | S3×C40 | C24⋊D5 | S3×Dic10 | S3×D20 | C40⋊C2 | C5×Dic3 | S3×C10 | S3×C8 | C40 | Dic10 | D20 | C5×S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | S3 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 16 | 1 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of S3×C40⋊C2 ►in GL4(𝔽241) generated by
| 240 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 200 | 31 |
| 0 | 0 | 210 | 94 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 190 | 1 |
G:=sub<GL(4,GF(241))| [240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,200,210,0,0,31,94],[1,0,0,0,0,1,0,0,0,0,240,190,0,0,0,1] >;
S3×C40⋊C2 in GAP, Magma, Sage, TeX
S_3\times C_{40}\rtimes C_2 % in TeX
G:=Group("S3xC40:C2"); // GroupNames label
G:=SmallGroup(480,327);
// by ID
G=gap.SmallGroup(480,327);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,58,675,80,1356,18822]);
// 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^19>;
// generators/relations