direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×D15, C60⋊2C23, C23⋊4D30, D30⋊7C23, D60⋊24C22, C30.58C24, Dic15⋊4C23, C6⋊4(D4×D5), (C6×D4)⋊5D5, (C2×C4)⋊6D30, C10⋊4(S3×D4), (C5×D4)⋊20D6, (D4×C30)⋊5C2, (D4×C10)⋊5S3, C30⋊13(C2×D4), (C2×C20)⋊11D6, (C3×D4)⋊20D10, (C2×D60)⋊14C2, (C2×C12)⋊11D10, (C2×C30)⋊2C23, (C2×C60)⋊7C22, C20⋊4(C22×S3), C4⋊1(C22×D15), (C22×C6)⋊8D10, C12⋊4(C22×D5), C15⋊14(C22×D4), (C23×D15)⋊4C2, (C22×C10)⋊11D6, C15⋊7D4⋊8C22, C6.58(C23×D5), C2.6(C23×D15), (D4×C15)⋊22C22, (C4×D15)⋊16C22, C10.58(S3×C23), (C22×C30)⋊4C22, C22⋊2(C22×D15), (C2×Dic15)⋊25C22, (C22×D15)⋊18C22, C5⋊5(C2×S3×D4), C3⋊5(C2×D4×D5), (C2×C4×D15)⋊3C2, (C2×C15⋊7D4)⋊9C2, (C2×C6)⋊4(C22×D5), (C2×C10)⋊7(C22×S3), SmallGroup(480,1169)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3092 in 472 conjugacy classes, 135 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×30], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], D10 [×30], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×19], C22×C6 [×2], D15 [×4], D15 [×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 [×19], C22×C10 [×2], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], Dic15 [×2], C60 [×2], D30 [×10], D30 [×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], C2×S3×D4, C4×D15 [×4], D60 [×4], C2×Dic15, C15⋊7D4 [×8], C2×C60, D4×C15 [×4], C22×D15, C22×D15 [×10], C22×D15 [×8], C22×C30 [×2], C2×D4×D5, C2×C4×D15, C2×D60, D4×D15 [×8], C2×C15⋊7D4 [×2], D4×C30, C23×D15 [×2], C2×D4×D15
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], D15, C22×D4, C22×D5 [×7], S3×D4 [×2], S3×C23, D30 [×7], D4×D5 [×2], C23×D5, C2×S3×D4, C22×D15 [×7], C2×D4×D5, D4×D15 [×2], C23×D15, C2×D4×D15
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d15=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
Generators in S120
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 61)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 91)(43 92)(44 93)(45 94)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 106)(58 107)(59 108)(60 109)
(1 92 24 110)(2 93 25 111)(3 94 26 112)(4 95 27 113)(5 96 28 114)(6 97 29 115)(7 98 30 116)(8 99 16 117)(9 100 17 118)(10 101 18 119)(11 102 19 120)(12 103 20 106)(13 104 21 107)(14 105 22 108)(15 91 23 109)(31 87 49 65)(32 88 50 66)(33 89 51 67)(34 90 52 68)(35 76 53 69)(36 77 54 70)(37 78 55 71)(38 79 56 72)(39 80 57 73)(40 81 58 74)(41 82 59 75)(42 83 60 61)(43 84 46 62)(44 85 47 63)(45 86 48 64)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 61)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 91)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(91 110)(92 109)(93 108)(94 107)(95 106)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)
G:=sub<Sym(120)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,61)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,91)(43,92)(44,93)(45,94)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,106)(58,107)(59,108)(60,109), (1,92,24,110)(2,93,25,111)(3,94,26,112)(4,95,27,113)(5,96,28,114)(6,97,29,115)(7,98,30,116)(8,99,16,117)(9,100,17,118)(10,101,18,119)(11,102,19,120)(12,103,20,106)(13,104,21,107)(14,105,22,108)(15,91,23,109)(31,87,49,65)(32,88,50,66)(33,89,51,67)(34,90,52,68)(35,76,53,69)(36,77,54,70)(37,78,55,71)(38,79,56,72)(39,80,57,73)(40,81,58,74)(41,82,59,75)(42,83,60,61)(43,84,46,62)(44,85,47,63)(45,86,48,64), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,61)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,91), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(91,110)(92,109)(93,108)(94,107)(95,106)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)>;
G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,61)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,91)(43,92)(44,93)(45,94)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,106)(58,107)(59,108)(60,109), (1,92,24,110)(2,93,25,111)(3,94,26,112)(4,95,27,113)(5,96,28,114)(6,97,29,115)(7,98,30,116)(8,99,16,117)(9,100,17,118)(10,101,18,119)(11,102,19,120)(12,103,20,106)(13,104,21,107)(14,105,22,108)(15,91,23,109)(31,87,49,65)(32,88,50,66)(33,89,51,67)(34,90,52,68)(35,76,53,69)(36,77,54,70)(37,78,55,71)(38,79,56,72)(39,80,57,73)(40,81,58,74)(41,82,59,75)(42,83,60,61)(43,84,46,62)(44,85,47,63)(45,86,48,64), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,61)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,91), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(91,110)(92,109)(93,108)(94,107)(95,106)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111) );
G=PermutationGroup([(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,61),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,91),(43,92),(44,93),(45,94),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,106),(58,107),(59,108),(60,109)], [(1,92,24,110),(2,93,25,111),(3,94,26,112),(4,95,27,113),(5,96,28,114),(6,97,29,115),(7,98,30,116),(8,99,16,117),(9,100,17,118),(10,101,18,119),(11,102,19,120),(12,103,20,106),(13,104,21,107),(14,105,22,108),(15,91,23,109),(31,87,49,65),(32,88,50,66),(33,89,51,67),(34,90,52,68),(35,76,53,69),(36,77,54,70),(37,78,55,71),(38,79,56,72),(39,80,57,73),(40,81,58,74),(41,82,59,75),(42,83,60,61),(43,84,46,62),(44,85,47,63),(45,86,48,64)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,61),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,101),(56,102),(57,103),(58,104),(59,105),(60,91)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,60),(44,59),(45,58),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(91,110),(92,109),(93,108),(94,107),(95,106),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 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 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 |
| 53 | 23 | 0 | 0 |
| 38 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 53 | 23 | 0 | 0 |
| 45 | 8 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[53,38,0,0,23,5,0,0,0,0,1,0,0,0,0,1],[53,45,0,0,23,8,0,0,0,0,60,0,0,0,0,60] >;
90 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 | 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 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 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 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | S3×D4 | D4×D5 | D4×D15 |
| kernel | C2×D4×D15 | C2×C4×D15 | C2×D60 | D4×D15 | C2×C15⋊7D4 | D4×C30 | C23×D15 | D4×C10 | D30 | C6×D4 | C2×C20 | C5×D4 | C22×C10 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C10 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 4 | 16 | 8 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4\times D_{15} % in TeX
G:=Group("C2xD4xD15"); // GroupNames label
G:=SmallGroup(480,1169);
// by ID
G=gap.SmallGroup(480,1169);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,185,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^15=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
×
⋊
ℤ
𝔽
○
≀