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×C20)⋊27D6, (C2×C12)⋊6D10, C5⋊D4⋊13D6, (C4×S3)⋊17D10, (S3×D20)⋊13C2, (C2×C60)⋊5C22, C15⋊Q8⋊10C22, D20⋊5S3⋊13C2, D60⋊11C2⋊9C2, 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, C15⋊D4⋊12C22, C5⋊D12⋊12C22, C15⋊7D4⋊14C22, C3⋊D20⋊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, C20.188(C22×S3), (C2×C30).238C23, (C22×S3).83D10, C12.188(C22×D5), Dic5.8(C22×S3), (C3×Dic5).8C23, (C10×Dic3)⋊27C22, (C3×Dic10)⋊22C22, (S3×Dic5).10C22, (C5×Dic3).29C23, Dic3.34(C22×D5), (S3×C2×C4)⋊6D5, (C4×S3×D5)⋊9C2, C5⋊1(S3×C4○D4), (S3×C2×C20)⋊1C2, (C2×C4)⋊9(S3×D5), C3⋊4(C2×C4○D20), C15⋊9(C2×C4○D4), (S3×C5⋊D4)⋊7C2, C4.161(C2×S3×D5), (C3×C4○D20)⋊5C2, (C5×S3)⋊1(C4○D4), (C2×S3×D5).6C22, C22.14(C2×S3×D5), C2.22(C22×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
Generators and relations for S3×C4○D20
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 >
Subgroups: 1612 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C5×S3, C5×S3, C3×D5, D15, C30, C30, C2×C4○D4, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, S3×C2×C4, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, S3×C10, D30, C2×C30, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, S3×C4○D4, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, S3×C20, C10×Dic3, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C2×S3×D5, S3×C2×C10, C2×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, S3×C4○D20
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, S3×C23, S3×D5, C4○D20, C23×D5, S3×C4○D4, C2×S3×D5, C2×C4○D20, C22×S3×D5, S3×C4○D20
►On 120 points
(1 113 70)(2 114 71)(3 115 72)(4 116 73)(5 117 74)(6 118 75)(7 119 76)(8 120 77)(9 101 78)(10 102 79)(11 103 80)(12 104 61)(13 105 62)(14 106 63)(15 107 64)(16 108 65)(17 109 66)(18 110 67)(19 111 68)(20 112 69)(21 54 84)(22 55 85)(23 56 86)(24 57 87)(25 58 88)(26 59 89)(27 60 90)(28 41 91)(29 42 92)(30 43 93)(31 44 94)(32 45 95)(33 46 96)(34 47 97)(35 48 98)(36 49 99)(37 50 100)(38 51 81)(39 52 82)(40 53 83)
(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 113)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 101)(79 102)(80 103)
(1 34 11 24)(2 35 12 25)(3 36 13 26)(4 37 14 27)(5 38 15 28)(6 39 16 29)(7 40 17 30)(8 21 18 31)(9 22 19 32)(10 23 20 33)(41 117 51 107)(42 118 52 108)(43 119 53 109)(44 120 54 110)(45 101 55 111)(46 102 56 112)(47 103 57 113)(48 104 58 114)(49 105 59 115)(50 106 60 116)(61 88 71 98)(62 89 72 99)(63 90 73 100)(64 91 74 81)(65 92 75 82)(66 93 76 83)(67 94 77 84)(68 95 78 85)(69 96 79 86)(70 97 80 87)
(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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
G:=sub<Sym(120)| (1,113,70)(2,114,71)(3,115,72)(4,116,73)(5,117,74)(6,118,75)(7,119,76)(8,120,77)(9,101,78)(10,102,79)(11,103,80)(12,104,61)(13,105,62)(14,106,63)(15,107,64)(16,108,65)(17,109,66)(18,110,67)(19,111,68)(20,112,69)(21,54,84)(22,55,85)(23,56,86)(24,57,87)(25,58,88)(26,59,89)(27,60,90)(28,41,91)(29,42,92)(30,43,93)(31,44,94)(32,45,95)(33,46,96)(34,47,97)(35,48,98)(36,49,99)(37,50,100)(38,51,81)(39,52,82)(40,53,83), (41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,101)(79,102)(80,103), (1,34,11,24)(2,35,12,25)(3,36,13,26)(4,37,14,27)(5,38,15,28)(6,39,16,29)(7,40,17,30)(8,21,18,31)(9,22,19,32)(10,23,20,33)(41,117,51,107)(42,118,52,108)(43,119,53,109)(44,120,54,110)(45,101,55,111)(46,102,56,112)(47,103,57,113)(48,104,58,114)(49,105,59,115)(50,106,60,116)(61,88,71,98)(62,89,72,99)(63,90,73,100)(64,91,74,81)(65,92,75,82)(66,93,76,83)(67,94,77,84)(68,95,78,85)(69,96,79,86)(70,97,80,87), (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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)>;
G:=Group( (1,113,70)(2,114,71)(3,115,72)(4,116,73)(5,117,74)(6,118,75)(7,119,76)(8,120,77)(9,101,78)(10,102,79)(11,103,80)(12,104,61)(13,105,62)(14,106,63)(15,107,64)(16,108,65)(17,109,66)(18,110,67)(19,111,68)(20,112,69)(21,54,84)(22,55,85)(23,56,86)(24,57,87)(25,58,88)(26,59,89)(27,60,90)(28,41,91)(29,42,92)(30,43,93)(31,44,94)(32,45,95)(33,46,96)(34,47,97)(35,48,98)(36,49,99)(37,50,100)(38,51,81)(39,52,82)(40,53,83), (41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,101)(79,102)(80,103), (1,34,11,24)(2,35,12,25)(3,36,13,26)(4,37,14,27)(5,38,15,28)(6,39,16,29)(7,40,17,30)(8,21,18,31)(9,22,19,32)(10,23,20,33)(41,117,51,107)(42,118,52,108)(43,119,53,109)(44,120,54,110)(45,101,55,111)(46,102,56,112)(47,103,57,113)(48,104,58,114)(49,105,59,115)(50,106,60,116)(61,88,71,98)(62,89,72,99)(63,90,73,100)(64,91,74,81)(65,92,75,82)(66,93,76,83)(67,94,77,84)(68,95,78,85)(69,96,79,86)(70,97,80,87), (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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113) );
G=PermutationGroup([[(1,113,70),(2,114,71),(3,115,72),(4,116,73),(5,117,74),(6,118,75),(7,119,76),(8,120,77),(9,101,78),(10,102,79),(11,103,80),(12,104,61),(13,105,62),(14,106,63),(15,107,64),(16,108,65),(17,109,66),(18,110,67),(19,111,68),(20,112,69),(21,54,84),(22,55,85),(23,56,86),(24,57,87),(25,58,88),(26,59,89),(27,60,90),(28,41,91),(29,42,92),(30,43,93),(31,44,94),(32,45,95),(33,46,96),(34,47,97),(35,48,98),(36,49,99),(37,50,100),(38,51,81),(39,52,82),(40,53,83)], [(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,111),(69,112),(70,113),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,101),(79,102),(80,103)], [(1,34,11,24),(2,35,12,25),(3,36,13,26),(4,37,14,27),(5,38,15,28),(6,39,16,29),(7,40,17,30),(8,21,18,31),(9,22,19,32),(10,23,20,33),(41,117,51,107),(42,118,52,108),(43,119,53,109),(44,120,54,110),(45,101,55,111),(46,102,56,112),(47,103,57,113),(48,104,58,114),(49,105,59,115),(50,106,60,116),(61,88,71,98),(62,89,72,99),(63,90,73,100),(64,91,74,81),(65,92,75,82),(66,93,76,83),(67,94,77,84),(68,95,78,85),(69,96,79,86),(70,97,80,87)], [(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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,80),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,100),(94,99),(95,98),(96,97),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)]])
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 |
Matrix representation of S3×C4○D20 ►in 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] >;
S3×C4○D20 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