direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4○D20, D20⋊28D6, Dic10⋊25D6, D60⋊33C22, C30.19C24, D30.6C23, C60.114C23, Dic30⋊30C22, Dic15.9C23, (C4×D5)⋊12D6, (C2×C12)⋊6D10, (C2×C20)⋊27D6, C5⋊D4⋊13D6, (S3×D20)⋊13C2, (C4×S3)⋊17D10, (C2×C60)⋊5C22, C15⋊Q8⋊10C22, D60⋊11C2⋊9C2, D20⋊5S3⋊13C2, D60⋊C2⋊13C2, (C6×D5).5C23, C6.19(C23×D5), (S3×C20)⋊21C22, (S3×Dic10)⋊13C2, (C2×Dic3)⋊22D10, Dic5.D6⋊7C2, (C3×D20)⋊24C22, (C4×D15)⋊15C22, (D5×C12)⋊12C22, C3⋊D20⋊12C22, C15⋊7D4⋊14C22, C5⋊D12⋊12C22, C15⋊D4⋊12C22, C10.19(S3×C23), D30.C2⋊8C22, (D5×Dic3)⋊7C22, D10.5(C22×S3), D6.25(C22×D5), D6.D10⋊10C2, (S3×C10).30C23, (C2×C30).238C23, C20.188(C22×S3), (C22×S3).83D10, C12.188(C22×D5), Dic5.8(C22×S3), (C3×Dic5).8C23, (C3×Dic10)⋊22C22, (C10×Dic3)⋊27C22, (C5×Dic3).29C23, (S3×Dic5).10C22, Dic3.34(C22×D5), (C4×S3×D5)⋊9C2, (S3×C2×C4)⋊6D5, C5⋊1(S3×C4○D4), (S3×C2×C20)⋊1C2, (C2×C4)⋊9(S3×D5), C15⋊9(C2×C4○D4), C3⋊4(C2×C4○D20), (S3×C5⋊D4)⋊7C2, C4.161(C2×S3×D5), (C3×C4○D20)⋊5C2, (C5×S3)⋊1(C4○D4), (C2×S3×D5).6C22, C2.22(C22×S3×D5), C22.14(C2×S3×D5), (C3×C5⋊D4)⋊8C22, (C2×C6).10(C22×D5), (S3×C2×C10).103C22, (C2×C10).247(C22×S3), SmallGroup(480,1091)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1612 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×2], S3 [×3], C6, C6 [×3], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×4], Dic6 [×3], C4×S3 [×4], C4×S3 [×6], D12 [×3], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3, C22×S3 [×2], C5×S3 [×2], C5×S3, C3×D5 [×2], D15 [×2], C30, C30, C2×C4○D4, Dic10, Dic10 [×3], C4×D5 [×2], C4×D5 [×6], D20, D20 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×C10, S3×C2×C4, S3×C2×C4 [×2], C4○D12 [×3], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20, C4○D20 [×7], C2×C5⋊D4 [×2], C22×C20, S3×C4○D4, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], S3×C20 [×4], C10×Dic3, Dic30, C4×D15 [×2], D60, C15⋊7D4 [×2], C2×C60, C2×S3×D5 [×2], S3×C2×C10, C2×C4○D20, D20⋊5S3, S3×Dic10, D60⋊C2, D6.D10 [×2], C4×S3×D5 [×2], S3×D20, Dic5.D6 [×2], S3×C5⋊D4 [×2], C3×C4○D20, S3×C2×C20, D60⋊11C2, S3×C4○D20
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C4○D20 [×2], C23×D5, S3×C4○D4, C2×S3×D5 [×3], C2×C4○D20, C22×S3×D5, S3×C4○D20
Generators and relations
G = < a,b,c,d,e | a3=b2=c4=e2=1, d10=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d9 >
►On 120 points
(1 38 116)(2 39 117)(3 40 118)(4 21 119)(5 22 120)(6 23 101)(7 24 102)(8 25 103)(9 26 104)(10 27 105)(11 28 106)(12 29 107)(13 30 108)(14 31 109)(15 32 110)(16 33 111)(17 34 112)(18 35 113)(19 36 114)(20 37 115)(41 63 88)(42 64 89)(43 65 90)(44 66 91)(45 67 92)(46 68 93)(47 69 94)(48 70 95)(49 71 96)(50 72 97)(51 73 98)(52 74 99)(53 75 100)(54 76 81)(55 77 82)(56 78 83)(57 79 84)(58 80 85)(59 61 86)(60 62 87)
(21 119)(22 120)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)
(1 62 11 72)(2 63 12 73)(3 64 13 74)(4 65 14 75)(5 66 15 76)(6 67 16 77)(7 68 17 78)(8 69 18 79)(9 70 19 80)(10 71 20 61)(21 90 31 100)(22 91 32 81)(23 92 33 82)(24 93 34 83)(25 94 35 84)(26 95 36 85)(27 96 37 86)(28 97 38 87)(29 98 39 88)(30 99 40 89)(41 107 51 117)(42 108 52 118)(43 109 53 119)(44 110 54 120)(45 111 55 101)(46 112 56 102)(47 113 57 103)(48 114 58 104)(49 115 59 105)(50 116 60 106)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
G:=sub<Sym(120)| (1,38,116)(2,39,117)(3,40,118)(4,21,119)(5,22,120)(6,23,101)(7,24,102)(8,25,103)(9,26,104)(10,27,105)(11,28,106)(12,29,107)(13,30,108)(14,31,109)(15,32,110)(16,33,111)(17,34,112)(18,35,113)(19,36,114)(20,37,115)(41,63,88)(42,64,89)(43,65,90)(44,66,91)(45,67,92)(46,68,93)(47,69,94)(48,70,95)(49,71,96)(50,72,97)(51,73,98)(52,74,99)(53,75,100)(54,76,81)(55,77,82)(56,78,83)(57,79,84)(58,80,85)(59,61,86)(60,62,87), (21,119)(22,120)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87), (1,62,11,72)(2,63,12,73)(3,64,13,74)(4,65,14,75)(5,66,15,76)(6,67,16,77)(7,68,17,78)(8,69,18,79)(9,70,19,80)(10,71,20,61)(21,90,31,100)(22,91,32,81)(23,92,33,82)(24,93,34,83)(25,94,35,84)(26,95,36,85)(27,96,37,86)(28,97,38,87)(29,98,39,88)(30,99,40,89)(41,107,51,117)(42,108,52,118)(43,109,53,119)(44,110,54,120)(45,111,55,101)(46,112,56,102)(47,113,57,103)(48,114,58,104)(49,115,59,105)(50,116,60,106), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)>;
G:=Group( (1,38,116)(2,39,117)(3,40,118)(4,21,119)(5,22,120)(6,23,101)(7,24,102)(8,25,103)(9,26,104)(10,27,105)(11,28,106)(12,29,107)(13,30,108)(14,31,109)(15,32,110)(16,33,111)(17,34,112)(18,35,113)(19,36,114)(20,37,115)(41,63,88)(42,64,89)(43,65,90)(44,66,91)(45,67,92)(46,68,93)(47,69,94)(48,70,95)(49,71,96)(50,72,97)(51,73,98)(52,74,99)(53,75,100)(54,76,81)(55,77,82)(56,78,83)(57,79,84)(58,80,85)(59,61,86)(60,62,87), (21,119)(22,120)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87), (1,62,11,72)(2,63,12,73)(3,64,13,74)(4,65,14,75)(5,66,15,76)(6,67,16,77)(7,68,17,78)(8,69,18,79)(9,70,19,80)(10,71,20,61)(21,90,31,100)(22,91,32,81)(23,92,33,82)(24,93,34,83)(25,94,35,84)(26,95,36,85)(27,96,37,86)(28,97,38,87)(29,98,39,88)(30,99,40,89)(41,107,51,117)(42,108,52,118)(43,109,53,119)(44,110,54,120)(45,111,55,101)(46,112,56,102)(47,113,57,103)(48,114,58,104)(49,115,59,105)(50,116,60,106), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116) );
G=PermutationGroup([(1,38,116),(2,39,117),(3,40,118),(4,21,119),(5,22,120),(6,23,101),(7,24,102),(8,25,103),(9,26,104),(10,27,105),(11,28,106),(12,29,107),(13,30,108),(14,31,109),(15,32,110),(16,33,111),(17,34,112),(18,35,113),(19,36,114),(20,37,115),(41,63,88),(42,64,89),(43,65,90),(44,66,91),(45,67,92),(46,68,93),(47,69,94),(48,70,95),(49,71,96),(50,72,97),(51,73,98),(52,74,99),(53,75,100),(54,76,81),(55,77,82),(56,78,83),(57,79,84),(58,80,85),(59,61,86),(60,62,87)], [(21,119),(22,120),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87)], [(1,62,11,72),(2,63,12,73),(3,64,13,74),(4,65,14,75),(5,66,15,76),(6,67,16,77),(7,68,17,78),(8,69,18,79),(9,70,19,80),(10,71,20,61),(21,90,31,100),(22,91,32,81),(23,92,33,82),(24,93,34,83),(25,94,35,84),(26,95,36,85),(27,96,37,86),(28,97,38,87),(29,98,39,88),(30,99,40,89),(41,107,51,117),(42,108,52,118),(43,109,53,119),(44,110,54,120),(45,111,55,101),(46,112,56,102),(47,113,57,103),(48,114,58,104),(49,115,59,105),(50,116,60,106)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,100),(94,99),(95,98),(96,97),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 59 | 15 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 49 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 54 | 29 |
| 0 | 0 | 32 | 59 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 54 | 29 |
| 0 | 0 | 32 | 7 |
G:=sub<GL(4,GF(61))| [59,12,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[1,49,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,50,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,54,32,0,0,29,59],[60,0,0,0,0,60,0,0,0,0,54,32,0,0,29,7] >;
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 10 | 10 | 30 | 30 | 2 | 1 | 1 | 2 | 3 | 3 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | S3×D5 | S3×C4○D4 | C2×S3×D5 | C2×S3×D5 | S3×C4○D20 |
| kernel | S3×C4○D20 | D20⋊5S3 | S3×Dic10 | D60⋊C2 | D6.D10 | C4×S3×D5 | S3×D20 | Dic5.D6 | S3×C5⋊D4 | C3×C4○D20 | S3×C2×C20 | D60⋊11C2 | C4○D20 | S3×C2×C4 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | C5×S3 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | S3 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
S_3\times C_4\circ D_{20} % in TeX
G:=Group("S3xC4oD20"); // GroupNames label
G:=SmallGroup(480,1091);
// by ID
G=gap.SmallGroup(480,1091);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^10=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^9>;
// generators/relations