direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×S3×D20, C60⋊3C23, D30⋊2C23, D60⋊32C22, C30.16C24, C30⋊2(C2×D4), C6⋊1(C2×D20), C10⋊1(S3×D4), (C6×D20)⋊8C2, (C2×C20)⋊26D6, (C2×C12)⋊4D10, C15⋊2(C22×D4), C3⋊1(C22×D20), (C2×D60)⋊24C2, (S3×C10)⋊12D4, (C4×S3)⋊16D10, C20⋊5(C22×S3), (C6×D5)⋊1C23, (C22×D5)⋊9D6, C12⋊2(C22×D5), (C2×C60)⋊12C22, D10⋊1(C22×S3), C6.16(C23×D5), (S3×C20)⋊18C22, (C2×Dic3)⋊21D10, (C3×D20)⋊23C22, C3⋊D20⋊10C22, C10.16(S3×C23), (C5×Dic3)⋊5C23, Dic3⋊4(C22×D5), D6.32(C22×D5), (S3×C10).29C23, (C2×C30).235C23, (C22×S3).91D10, (C22×D15)⋊9C22, (C10×Dic3)⋊26C22, C5⋊1(C2×S3×D4), C4⋊3(C2×S3×D5), (S3×C2×C4)⋊5D5, (S3×C2×C20)⋊6C2, (C2×C4)⋊8(S3×D5), (C5×S3)⋊1(C2×D4), (D5×C2×C6)⋊4C22, (C22×S3×D5)⋊6C2, (C2×S3×D5)⋊10C22, (C2×C3⋊D20)⋊19C2, C2.19(C22×S3×D5), C22.104(C2×S3×D5), (S3×C2×C10).102C22, (C2×C6).245(C22×D5), (C2×C10).245(C22×S3), SmallGroup(480,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×S3×D20
G = < a,b,c,d,e | a2=b3=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 3004 in 472 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C20, C20, D10, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, C22×D4, D20, D20, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C60, S3×D5, C6×D5, C6×D5, S3×C10, D30, D30, C2×C30, C2×D20, C2×D20, C22×C20, C23×D5, C2×S3×D4, C3⋊D20, C3×D20, S3×C20, C10×Dic3, D60, C2×C60, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22×D20, S3×D20, C2×C3⋊D20, C6×D20, S3×C2×C20, C2×D60, C22×S3×D5, C2×S3×D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C22×S3, C22×D4, D20, C22×D5, S3×D4, S3×C23, S3×D5, C2×D20, C23×D5, C2×S3×D4, C2×S3×D5, C22×D20, S3×D20, C22×S3×D5, C2×S3×D20
►On 120 points
Generators in S120
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 96)(22 97)(23 98)(24 99)(25 100)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 61)(60 62)
(1 77 36)(2 78 37)(3 79 38)(4 80 39)(5 61 40)(6 62 21)(7 63 22)(8 64 23)(9 65 24)(10 66 25)(11 67 26)(12 68 27)(13 69 28)(14 70 29)(15 71 30)(16 72 31)(17 73 32)(18 74 33)(19 75 34)(20 76 35)(41 97 119)(42 98 120)(43 99 101)(44 100 102)(45 81 103)(46 82 104)(47 83 105)(48 84 106)(49 85 107)(50 86 108)(51 87 109)(52 88 110)(53 89 111)(54 90 112)(55 91 113)(56 92 114)(57 93 115)(58 94 116)(59 95 117)(60 96 118)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(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)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,96)(22,97)(23,98)(24,99)(25,100)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,61)(60,62), (1,77,36)(2,78,37)(3,79,38)(4,80,39)(5,61,40)(6,62,21)(7,63,22)(8,64,23)(9,65,24)(10,66,25)(11,67,26)(12,68,27)(13,69,28)(14,70,29)(15,71,30)(16,72,31)(17,73,32)(18,74,33)(19,75,34)(20,76,35)(41,97,119)(42,98,120)(43,99,101)(44,100,102)(45,81,103)(46,82,104)(47,83,105)(48,84,106)(49,85,107)(50,86,108)(51,87,109)(52,88,110)(53,89,111)(54,90,112)(55,91,113)(56,92,114)(57,93,115)(58,94,116)(59,95,117)(60,96,118), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)>;
G:=Group( (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,96)(22,97)(23,98)(24,99)(25,100)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,61)(60,62), (1,77,36)(2,78,37)(3,79,38)(4,80,39)(5,61,40)(6,62,21)(7,63,22)(8,64,23)(9,65,24)(10,66,25)(11,67,26)(12,68,27)(13,69,28)(14,70,29)(15,71,30)(16,72,31)(17,73,32)(18,74,33)(19,75,34)(20,76,35)(41,97,119)(42,98,120)(43,99,101)(44,100,102)(45,81,103)(46,82,104)(47,83,105)(48,84,106)(49,85,107)(50,86,108)(51,87,109)(52,88,110)(53,89,111)(54,90,112)(55,91,113)(56,92,114)(57,93,115)(58,94,116)(59,95,117)(60,96,118), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113) );
G=PermutationGroup([[(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,96),(22,97),(23,98),(24,99),(25,100),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,61),(60,62)], [(1,77,36),(2,78,37),(3,79,38),(4,80,39),(5,61,40),(6,62,21),(7,63,22),(8,64,23),(9,65,24),(10,66,25),(11,67,26),(12,68,27),(13,69,28),(14,70,29),(15,71,30),(16,72,31),(17,73,32),(18,74,33),(19,75,34),(20,76,35),(41,97,119),(42,98,120),(43,99,101),(44,100,102),(45,81,103),(46,82,104),(47,83,105),(48,84,106),(49,85,107),(50,86,108),(51,87,109),(52,88,110),(53,89,111),(54,90,112),(55,91,113),(56,92,114),(57,93,115),(58,94,116),(59,95,117),(60,96,118)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,80),(74,79),(75,78),(76,77),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(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 | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D20 | S3×D4 | S3×D5 | C2×S3×D5 | C2×S3×D5 | S3×D20 |
| kernel | C2×S3×D20 | S3×D20 | C2×C3⋊D20 | C6×D20 | S3×C2×C20 | C2×D60 | C22×S3×D5 | C2×D20 | S3×C10 | S3×C2×C4 | D20 | C2×C20 | C22×D5 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 1 | 2 | 8 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C2×S3×D20 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 34 | 32 | 0 | 0 | 0 | 0 |
| 2 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 29 | 0 | 0 |
| 0 | 0 | 59 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 25 | 32 | 0 | 0 | 0 | 0 |
| 11 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 29 | 0 | 0 |
| 0 | 0 | 50 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[34,2,0,0,0,0,32,36,0,0,0,0,0,0,27,59,0,0,0,0,29,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[25,11,0,0,0,0,32,36,0,0,0,0,0,0,36,50,0,0,0,0,29,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C2×S3×D20 in GAP, Magma, Sage, TeX
C_2\times S_3\times D_{20} % in TeX
G:=Group("C2xS3xD20"); // GroupNames label
G:=SmallGroup(480,1088);
// by ID
G=gap.SmallGroup(480,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations