direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×D4×D5, C60⋊7C23, C30.73C24, C10⋊2(C6×D4), C20⋊(C22×C6), (D4×C10)⋊5C6, D20⋊7(C2×C6), C30⋊14(C2×D4), C23⋊4(C6×D5), (C2×D20)⋊11C6, (C6×D20)⋊27C2, (D4×C30)⋊12C2, (C2×C12)⋊29D10, (C2×C30)⋊7C23, (C23×D5)⋊7C6, C12⋊7(C22×D5), C15⋊15(C22×D4), (C2×C60)⋊21C22, (C6×D5)⋊11C23, D10⋊2(C22×C6), (C22×C6)⋊10D10, C10.5(C23×C6), C6.73(C23×D5), (C3×D20)⋊37C22, (D5×C12)⋊22C22, (D4×C15)⋊29C22, (C3×Dic5)⋊9C23, Dic5⋊1(C22×C6), (C22×C30)⋊13C22, (C6×Dic5)⋊35C22, C5⋊2(D4×C2×C6), C4⋊1(D5×C2×C6), (C2×C4×D5)⋊3C6, (C2×C4)⋊6(C6×D5), C22⋊2(D5×C2×C6), (C2×C20)⋊2(C2×C6), (C4×D5)⋊3(C2×C6), (C5×D4)⋊5(C2×C6), (D5×C2×C12)⋊13C2, C5⋊D4⋊1(C2×C6), (C2×C5⋊D4)⋊9C6, (C6×C5⋊D4)⋊24C2, C2.6(D5×C22×C6), (D5×C2×C6)⋊22C22, (D5×C22×C6)⋊10C2, (C2×C6)⋊5(C22×D5), (C22×C10)⋊5(C2×C6), (C2×C10)⋊2(C22×C6), (C2×Dic5)⋊8(C2×C6), (C22×D5)⋊7(C2×C6), (C3×C5⋊D4)⋊17C22, SmallGroup(480,1139)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1616 in 472 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, C6, C6 [×2], C6 [×12], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×34], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], D10 [×10], D10 [×20], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C12, C2×C12 [×5], C3×D4 [×4], C3×D4 [×12], C22×C6 [×2], C22×C6 [×19], C3×D5 [×4], C3×D5 [×4], C30, C30 [×2], C30 [×4], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×10], C22×D5 [×8], C22×C10 [×2], C22×C12, C6×D4, C6×D4 [×11], C23×C6 [×2], C3×Dic5 [×2], C60 [×2], C6×D5 [×10], C6×D5 [×20], C2×C30, C2×C30 [×4], C2×C30 [×4], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], D4×C2×C6, D5×C12 [×4], C3×D20 [×4], C6×Dic5, C3×C5⋊D4 [×8], C2×C60, D4×C15 [×4], D5×C2×C6, D5×C2×C6 [×10], D5×C2×C6 [×8], C22×C30 [×2], C2×D4×D5, D5×C2×C12, C6×D20, C3×D4×D5 [×8], C6×C5⋊D4 [×2], D4×C30, D5×C22×C6 [×2], C6×D4×D5
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], D5, C2×C6 [×35], C2×D4 [×6], C24, D10 [×7], C3×D4 [×4], C22×C6 [×15], C3×D5, C22×D4, C22×D5 [×7], C6×D4 [×6], C23×C6, C6×D5 [×7], D4×D5 [×2], C23×D5, D4×C2×C6, D5×C2×C6 [×7], C2×D4×D5, C3×D4×D5 [×2], D5×C22×C6, C6×D4×D5
Generators and relations
G = < a,b,c,d,e | a6=b4=c2=d5=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 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 81 97 108)(2 82 98 103)(3 83 99 104)(4 84 100 105)(5 79 101 106)(6 80 102 107)(7 109 35 17)(8 110 36 18)(9 111 31 13)(10 112 32 14)(11 113 33 15)(12 114 34 16)(19 55 52 89)(20 56 53 90)(21 57 54 85)(22 58 49 86)(23 59 50 87)(24 60 51 88)(25 69 115 96)(26 70 116 91)(27 71 117 92)(28 72 118 93)(29 67 119 94)(30 68 120 95)(37 75 43 61)(38 76 44 62)(39 77 45 63)(40 78 46 64)(41 73 47 65)(42 74 48 66)
(1 108)(2 103)(3 104)(4 105)(5 106)(6 107)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 96)(26 91)(27 92)(28 93)(29 94)(30 95)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 75)(44 76)(45 77)(46 78)(47 73)(48 74)(49 86)(50 87)(51 88)(52 89)(53 90)(54 85)(67 119)(68 120)(69 115)(70 116)(71 117)(72 118)(79 101)(80 102)(81 97)(82 98)(83 99)(84 100)
(1 25 41 87 109)(2 26 42 88 110)(3 27 37 89 111)(4 28 38 90 112)(5 29 39 85 113)(6 30 40 86 114)(7 108 96 65 50)(8 103 91 66 51)(9 104 92 61 52)(10 105 93 62 53)(11 106 94 63 54)(12 107 95 64 49)(13 99 117 43 55)(14 100 118 44 56)(15 101 119 45 57)(16 102 120 46 58)(17 97 115 47 59)(18 98 116 48 60)(19 31 83 71 75)(20 32 84 72 76)(21 33 79 67 77)(22 34 80 68 78)(23 35 81 69 73)(24 36 82 70 74)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 84)(8 79)(9 80)(10 81)(11 82)(12 83)(19 95)(20 96)(21 91)(22 92)(23 93)(24 94)(25 56)(26 57)(27 58)(28 59)(29 60)(30 55)(31 107)(32 108)(33 103)(34 104)(35 105)(36 106)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)(49 71)(50 72)(51 67)(52 68)(53 69)(54 70)(61 78)(62 73)(63 74)(64 75)(65 76)(66 77)(85 116)(86 117)(87 118)(88 119)(89 120)(90 115)(97 112)(98 113)(99 114)(100 109)(101 110)(102 111)
G:=sub<Sym(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), (1,81,97,108)(2,82,98,103)(3,83,99,104)(4,84,100,105)(5,79,101,106)(6,80,102,107)(7,109,35,17)(8,110,36,18)(9,111,31,13)(10,112,32,14)(11,113,33,15)(12,114,34,16)(19,55,52,89)(20,56,53,90)(21,57,54,85)(22,58,49,86)(23,59,50,87)(24,60,51,88)(25,69,115,96)(26,70,116,91)(27,71,117,92)(28,72,118,93)(29,67,119,94)(30,68,120,95)(37,75,43,61)(38,76,44,62)(39,77,45,63)(40,78,46,64)(41,73,47,65)(42,74,48,66), (1,108)(2,103)(3,104)(4,105)(5,106)(6,107)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,96)(26,91)(27,92)(28,93)(29,94)(30,95)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,75)(44,76)(45,77)(46,78)(47,73)(48,74)(49,86)(50,87)(51,88)(52,89)(53,90)(54,85)(67,119)(68,120)(69,115)(70,116)(71,117)(72,118)(79,101)(80,102)(81,97)(82,98)(83,99)(84,100), (1,25,41,87,109)(2,26,42,88,110)(3,27,37,89,111)(4,28,38,90,112)(5,29,39,85,113)(6,30,40,86,114)(7,108,96,65,50)(8,103,91,66,51)(9,104,92,61,52)(10,105,93,62,53)(11,106,94,63,54)(12,107,95,64,49)(13,99,117,43,55)(14,100,118,44,56)(15,101,119,45,57)(16,102,120,46,58)(17,97,115,47,59)(18,98,116,48,60)(19,31,83,71,75)(20,32,84,72,76)(21,33,79,67,77)(22,34,80,68,78)(23,35,81,69,73)(24,36,82,70,74), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,84)(8,79)(9,80)(10,81)(11,82)(12,83)(19,95)(20,96)(21,91)(22,92)(23,93)(24,94)(25,56)(26,57)(27,58)(28,59)(29,60)(30,55)(31,107)(32,108)(33,103)(34,104)(35,105)(36,106)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45)(49,71)(50,72)(51,67)(52,68)(53,69)(54,70)(61,78)(62,73)(63,74)(64,75)(65,76)(66,77)(85,116)(86,117)(87,118)(88,119)(89,120)(90,115)(97,112)(98,113)(99,114)(100,109)(101,110)(102,111)>;
G:=Group( (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), (1,81,97,108)(2,82,98,103)(3,83,99,104)(4,84,100,105)(5,79,101,106)(6,80,102,107)(7,109,35,17)(8,110,36,18)(9,111,31,13)(10,112,32,14)(11,113,33,15)(12,114,34,16)(19,55,52,89)(20,56,53,90)(21,57,54,85)(22,58,49,86)(23,59,50,87)(24,60,51,88)(25,69,115,96)(26,70,116,91)(27,71,117,92)(28,72,118,93)(29,67,119,94)(30,68,120,95)(37,75,43,61)(38,76,44,62)(39,77,45,63)(40,78,46,64)(41,73,47,65)(42,74,48,66), (1,108)(2,103)(3,104)(4,105)(5,106)(6,107)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,96)(26,91)(27,92)(28,93)(29,94)(30,95)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,75)(44,76)(45,77)(46,78)(47,73)(48,74)(49,86)(50,87)(51,88)(52,89)(53,90)(54,85)(67,119)(68,120)(69,115)(70,116)(71,117)(72,118)(79,101)(80,102)(81,97)(82,98)(83,99)(84,100), (1,25,41,87,109)(2,26,42,88,110)(3,27,37,89,111)(4,28,38,90,112)(5,29,39,85,113)(6,30,40,86,114)(7,108,96,65,50)(8,103,91,66,51)(9,104,92,61,52)(10,105,93,62,53)(11,106,94,63,54)(12,107,95,64,49)(13,99,117,43,55)(14,100,118,44,56)(15,101,119,45,57)(16,102,120,46,58)(17,97,115,47,59)(18,98,116,48,60)(19,31,83,71,75)(20,32,84,72,76)(21,33,79,67,77)(22,34,80,68,78)(23,35,81,69,73)(24,36,82,70,74), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,84)(8,79)(9,80)(10,81)(11,82)(12,83)(19,95)(20,96)(21,91)(22,92)(23,93)(24,94)(25,56)(26,57)(27,58)(28,59)(29,60)(30,55)(31,107)(32,108)(33,103)(34,104)(35,105)(36,106)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45)(49,71)(50,72)(51,67)(52,68)(53,69)(54,70)(61,78)(62,73)(63,74)(64,75)(65,76)(66,77)(85,116)(86,117)(87,118)(88,119)(89,120)(90,115)(97,112)(98,113)(99,114)(100,109)(101,110)(102,111) );
G=PermutationGroup([(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)], [(1,81,97,108),(2,82,98,103),(3,83,99,104),(4,84,100,105),(5,79,101,106),(6,80,102,107),(7,109,35,17),(8,110,36,18),(9,111,31,13),(10,112,32,14),(11,113,33,15),(12,114,34,16),(19,55,52,89),(20,56,53,90),(21,57,54,85),(22,58,49,86),(23,59,50,87),(24,60,51,88),(25,69,115,96),(26,70,116,91),(27,71,117,92),(28,72,118,93),(29,67,119,94),(30,68,120,95),(37,75,43,61),(38,76,44,62),(39,77,45,63),(40,78,46,64),(41,73,47,65),(42,74,48,66)], [(1,108),(2,103),(3,104),(4,105),(5,106),(6,107),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,96),(26,91),(27,92),(28,93),(29,94),(30,95),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,75),(44,76),(45,77),(46,78),(47,73),(48,74),(49,86),(50,87),(51,88),(52,89),(53,90),(54,85),(67,119),(68,120),(69,115),(70,116),(71,117),(72,118),(79,101),(80,102),(81,97),(82,98),(83,99),(84,100)], [(1,25,41,87,109),(2,26,42,88,110),(3,27,37,89,111),(4,28,38,90,112),(5,29,39,85,113),(6,30,40,86,114),(7,108,96,65,50),(8,103,91,66,51),(9,104,92,61,52),(10,105,93,62,53),(11,106,94,63,54),(12,107,95,64,49),(13,99,117,43,55),(14,100,118,44,56),(15,101,119,45,57),(16,102,120,46,58),(17,97,115,47,59),(18,98,116,48,60),(19,31,83,71,75),(20,32,84,72,76),(21,33,79,67,77),(22,34,80,68,78),(23,35,81,69,73),(24,36,82,70,74)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(7,84),(8,79),(9,80),(10,81),(11,82),(12,83),(19,95),(20,96),(21,91),(22,92),(23,93),(24,94),(25,56),(26,57),(27,58),(28,59),(29,60),(30,55),(31,107),(32,108),(33,103),(34,104),(35,105),(36,106),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45),(49,71),(50,72),(51,67),(52,68),(53,69),(54,70),(61,78),(62,73),(63,74),(64,75),(65,76),(66,77),(85,116),(86,117),(87,118),(88,119),(89,120),(90,115),(97,112),(98,113),(99,114),(100,109),(101,110),(102,111)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 14 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 53 |
| 0 | 0 | 46 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,14,0,0,0,0,14],[60,0,0,0,0,60,0,0,0,0,60,46,0,0,53,1],[60,0,0,0,0,60,0,0,0,0,1,15,0,0,0,60],[0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,0,1] >;
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6V | 6W | ··· | 6AD | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D5 | D10 | D10 | D10 | C3×D4 | C3×D5 | C6×D5 | C6×D5 | C6×D5 | D4×D5 | C3×D4×D5 |
| kernel | C6×D4×D5 | D5×C2×C12 | C6×D20 | C3×D4×D5 | C6×C5⋊D4 | D4×C30 | D5×C22×C6 | C2×D4×D5 | C2×C4×D5 | C2×D20 | D4×D5 | C2×C5⋊D4 | D4×C10 | C23×D5 | C6×D5 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | D10 | C2×D4 | C2×C4 | D4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 2 | 2 | 16 | 4 | 2 | 4 | 4 | 2 | 2 | 8 | 4 | 8 | 4 | 4 | 16 | 8 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_6\times D_4\times D_5
% in TeX
G:=Group("C6xD4xD5"); // GroupNames label
G:=SmallGroup(480,1139);
// by ID
G=gap.SmallGroup(480,1139);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,409,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^5=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