direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×D12, C60⋊4C23, D30⋊1C23, D60⋊38C22, C30.15C24, C6⋊1(D4×D5), (C2×C20)⋊4D6, C30⋊1(C2×D4), (C6×D5)⋊12D4, (C4×D5)⋊16D6, C10⋊1(C2×D12), C5⋊1(C22×D12), C15⋊1(C22×D4), (C2×D60)⋊29C2, (C10×D12)⋊8C2, (C2×C12)⋊26D10, C20⋊2(C22×S3), D6⋊1(C22×D5), C12⋊5(C22×D5), (S3×C10)⋊1C23, (C2×C60)⋊13C22, (C2×Dic5)⋊21D6, (C22×S3)⋊8D10, C6.15(C23×D5), (D5×C12)⋊18C22, (C5×D12)⋊23C22, C5⋊D12⋊10C22, C10.15(S3×C23), (C3×Dic5)⋊5C23, Dic5⋊4(C22×S3), (C6×D5).42C23, (C2×C30).234C23, (C6×Dic5)⋊26C22, D10.54(C22×S3), (C22×D5).113D6, (C22×D15)⋊8C22, C3⋊1(C2×D4×D5), C4⋊2(C2×S3×D5), (C2×C4×D5)⋊5S3, (D5×C2×C12)⋊6C2, (C3×D5)⋊1(C2×D4), (C2×C4)⋊11(S3×D5), (C2×S3×D5)⋊9C22, (C22×S3×D5)⋊5C2, (S3×C2×C10)⋊4C22, (C2×C5⋊D12)⋊19C2, C2.18(C22×S3×D5), C22.103(C2×S3×D5), (D5×C2×C6).119C22, (C2×C6).244(C22×D5), (C2×C10).244(C22×S3), SmallGroup(480,1087)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2972 in 472 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], C12 [×2], C12 [×2], D6 [×4], D6 [×28], C2×C6, C2×C6 [×6], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], C20 [×2], D10 [×6], D10 [×24], C2×C10, C2×C10 [×8], D12 [×4], D12 [×12], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×S3 [×18], C22×C6, C5×S3 [×4], C3×D5 [×4], D15 [×4], C30, C30 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×18], C22×C10 [×2], C2×D12, C2×D12 [×11], C22×C12, S3×C23 [×2], C3×Dic5 [×2], C60 [×2], S3×D5 [×16], C6×D5 [×6], S3×C10 [×4], S3×C10 [×4], D30 [×4], D30 [×4], C2×C30, C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C22×D12, C5⋊D12 [×8], D5×C12 [×4], C6×Dic5, C5×D12 [×4], D60 [×4], C2×C60, C2×S3×D5 [×8], C2×S3×D5 [×8], D5×C2×C6, S3×C2×C10 [×2], C22×D15 [×2], C2×D4×D5, D5×D12 [×8], C2×C5⋊D12 [×2], D5×C2×C12, C10×D12, C2×D60, C22×S3×D5 [×2], C2×D5×D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], D12 [×4], C22×S3 [×7], C22×D4, C22×D5 [×7], C2×D12 [×6], S3×C23, S3×D5, D4×D5 [×2], C23×D5, C22×D12, C2×S3×D5 [×3], C2×D4×D5, D5×D12 [×2], C22×S3×D5, C2×D5×D12
Generators and relations
G = < a,b,c,d,e | a2=b5=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►On 120 points
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 72)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 109)(96 110)
(1 113 102 50 66)(2 114 103 51 67)(3 115 104 52 68)(4 116 105 53 69)(5 117 106 54 70)(6 118 107 55 71)(7 119 108 56 72)(8 120 97 57 61)(9 109 98 58 62)(10 110 99 59 63)(11 111 100 60 64)(12 112 101 49 65)(13 92 83 36 48)(14 93 84 25 37)(15 94 73 26 38)(16 95 74 27 39)(17 96 75 28 40)(18 85 76 29 41)(19 86 77 30 42)(20 87 78 31 43)(21 88 79 32 44)(22 89 80 33 45)(23 90 81 34 46)(24 91 82 35 47)
(1 72)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 85)(36 86)(49 118)(50 119)(51 120)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 17)(14 16)(18 24)(19 23)(20 22)(25 27)(28 36)(29 35)(30 34)(31 33)(37 39)(40 48)(41 47)(42 46)(43 45)(49 53)(50 52)(54 60)(55 59)(56 58)(62 72)(63 71)(64 70)(65 69)(66 68)(74 84)(75 83)(76 82)(77 81)(78 80)(85 91)(86 90)(87 89)(92 96)(93 95)(98 108)(99 107)(100 106)(101 105)(102 104)(109 119)(110 118)(111 117)(112 116)(113 115)
G:=sub<Sym(120)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,109)(96,110), (1,113,102,50,66)(2,114,103,51,67)(3,115,104,52,68)(4,116,105,53,69)(5,117,106,54,70)(6,118,107,55,71)(7,119,108,56,72)(8,120,97,57,61)(9,109,98,58,62)(10,110,99,59,63)(11,111,100,60,64)(12,112,101,49,65)(13,92,83,36,48)(14,93,84,25,37)(15,94,73,26,38)(16,95,74,27,39)(17,96,75,28,40)(18,85,76,29,41)(19,86,77,30,42)(20,87,78,31,43)(21,88,79,32,44)(22,89,80,33,45)(23,90,81,34,46)(24,91,82,35,47), (1,72)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,85)(36,86)(49,118)(50,119)(51,120)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,39)(40,48)(41,47)(42,46)(43,45)(49,53)(50,52)(54,60)(55,59)(56,58)(62,72)(63,71)(64,70)(65,69)(66,68)(74,84)(75,83)(76,82)(77,81)(78,80)(85,91)(86,90)(87,89)(92,96)(93,95)(98,108)(99,107)(100,106)(101,105)(102,104)(109,119)(110,118)(111,117)(112,116)(113,115)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,109)(96,110), (1,113,102,50,66)(2,114,103,51,67)(3,115,104,52,68)(4,116,105,53,69)(5,117,106,54,70)(6,118,107,55,71)(7,119,108,56,72)(8,120,97,57,61)(9,109,98,58,62)(10,110,99,59,63)(11,111,100,60,64)(12,112,101,49,65)(13,92,83,36,48)(14,93,84,25,37)(15,94,73,26,38)(16,95,74,27,39)(17,96,75,28,40)(18,85,76,29,41)(19,86,77,30,42)(20,87,78,31,43)(21,88,79,32,44)(22,89,80,33,45)(23,90,81,34,46)(24,91,82,35,47), (1,72)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,85)(36,86)(49,118)(50,119)(51,120)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,39)(40,48)(41,47)(42,46)(43,45)(49,53)(50,52)(54,60)(55,59)(56,58)(62,72)(63,71)(64,70)(65,69)(66,68)(74,84)(75,83)(76,82)(77,81)(78,80)(85,91)(86,90)(87,89)(92,96)(93,95)(98,108)(99,107)(100,106)(101,105)(102,104)(109,119)(110,118)(111,117)(112,116)(113,115) );
G=PermutationGroup([(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,72),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,109),(96,110)], [(1,113,102,50,66),(2,114,103,51,67),(3,115,104,52,68),(4,116,105,53,69),(5,117,106,54,70),(6,118,107,55,71),(7,119,108,56,72),(8,120,97,57,61),(9,109,98,58,62),(10,110,99,59,63),(11,111,100,60,64),(12,112,101,49,65),(13,92,83,36,48),(14,93,84,25,37),(15,94,73,26,38),(16,95,74,27,39),(17,96,75,28,40),(18,85,76,29,41),(19,86,77,30,42),(20,87,78,31,43),(21,88,79,32,44),(22,89,80,33,45),(23,90,81,34,46),(24,91,82,35,47)], [(1,72),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,37),(21,38),(22,39),(23,40),(24,41),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,85),(36,86),(49,118),(50,119),(51,120),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,17),(14,16),(18,24),(19,23),(20,22),(25,27),(28,36),(29,35),(30,34),(31,33),(37,39),(40,48),(41,47),(42,46),(43,45),(49,53),(50,52),(54,60),(55,59),(56,58),(62,72),(63,71),(64,70),(65,69),(66,68),(74,84),(75,83),(76,82),(77,81),(78,80),(85,91),(86,90),(87,89),(92,96),(93,95),(98,108),(99,107),(100,106),(101,105),(102,104),(109,119),(110,118),(111,117),(112,116),(113,115)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 17 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 15 | 23 | 0 | 0 |
| 38 | 38 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,17],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0],[15,38,0,0,23,38,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1] >;
72 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 | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
72 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 | D6 | D10 | D10 | D10 | D12 | S3×D5 | D4×D5 | C2×S3×D5 | C2×S3×D5 | D5×D12 |
| kernel | C2×D5×D12 | D5×D12 | C2×C5⋊D12 | D5×C2×C12 | C10×D12 | C2×D60 | C22×S3×D5 | C2×C4×D5 | C6×D5 | C2×D12 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | D12 | C2×C12 | C22×S3 | D10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 8 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_5\times D_{12} % in TeX
G:=Group("C2xD5xD12"); // GroupNames label
G:=SmallGroup(480,1087);
// by ID
G=gap.SmallGroup(480,1087);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^12=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