direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C22≀C2, C24⋊7C30, (C2×D4)⋊1C30, (C2×C30)⋊29D4, C2.4(D4×C30), (D4×C10)⋊10C6, (C6×D4)⋊10C10, (D4×C30)⋊28C2, C22⋊C4⋊2C30, (C23×C6)⋊3C10, C23⋊1(C2×C30), (C23×C30)⋊1C2, (C23×C10)⋊9C6, C10.67(C6×D4), C6.67(D4×C10), C22⋊3(D4×C15), (C2×C60)⋊38C22, C30.450(C2×D4), (C22×C30)⋊1C22, (C2×C30).455C23, C22.10(C22×C30), (C2×C6)⋊7(C5×D4), (C2×C20)⋊8(C2×C6), (C2×C4)⋊1(C2×C30), (C2×C12)⋊8(C2×C10), (C2×C10)⋊10(C3×D4), (C5×C22⋊C4)⋊10C6, (C22×C6)⋊1(C2×C10), (C22×C10)⋊2(C2×C6), (C15×C22⋊C4)⋊26C2, (C3×C22⋊C4)⋊10C10, (C2×C10).75(C22×C6), (C2×C6).75(C22×C10), SmallGroup(480,925)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C22≀C2
G = < a,b,c,d,e,f | a15=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 424 in 260 conjugacy classes, 104 normal (20 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, C23, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C30, C30, C22≀C2, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C3×C22⋊C4, C6×D4, C23×C6, C60, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, D4×C10, C23×C10, C3×C22≀C2, C2×C60, D4×C15, C22×C30, C22×C30, C22×C30, C5×C22≀C2, C15×C22⋊C4, D4×C30, C23×C30, C15×C22≀C2
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C2×C10, C3×D4, C22×C6, C30, C22≀C2, C5×D4, C22×C10, C6×D4, C2×C30, D4×C10, C3×C22≀C2, D4×C15, C22×C30, C5×C22≀C2, D4×C30, C15×C22≀C2
►On 120 points
Generators in S120
(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)
(1 102)(2 103)(3 104)(4 105)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(46 74)(47 75)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(76 120)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)
(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 46)(30 47)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(91 116)(92 117)(93 118)(94 119)(95 120)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)
(1 102)(2 103)(3 104)(4 105)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 46)(30 47)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(76 120)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 104)(17 105)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 76)(44 77)(45 78)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)
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), (1,102)(2,103)(3,104)(4,105)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(46,74)(47,75)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(76,120)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119), (16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,46)(30,47)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(91,116)(92,117)(93,118)(94,119)(95,120)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115), (1,102)(2,103)(3,104)(4,105)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,46)(30,47)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(76,120)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,104)(17,105)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,76)(44,77)(45,78)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)>;
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), (1,102)(2,103)(3,104)(4,105)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(46,74)(47,75)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(76,120)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119), (16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,46)(30,47)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(91,116)(92,117)(93,118)(94,119)(95,120)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115), (1,102)(2,103)(3,104)(4,105)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,46)(30,47)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(76,120)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,104)(17,105)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,76)(44,77)(45,78)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113) );
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)], [(1,102),(2,103),(3,104),(4,105),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(46,74),(47,75),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(76,120),(77,106),(78,107),(79,108),(80,109),(81,110),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119)], [(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,46),(30,47),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,82),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,57),(32,58),(33,59),(34,60),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(91,116),(92,117),(93,118),(94,119),(95,120),(96,106),(97,107),(98,108),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115)], [(1,102),(2,103),(3,104),(4,105),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,46),(30,47),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(76,120),(77,106),(78,107),(79,108),(80,109),(81,110),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,104),(17,105),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,97),(25,98),(26,99),(27,100),(28,101),(29,102),(30,103),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,76),(44,77),(45,78),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,113)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6R | 6S | 6T | 10A | ··· | 10L | 10M | ··· | 10AJ | 10AK | 10AL | 10AM | 10AN | 12A | ··· | 12F | 15A | ··· | 15H | 20A | ··· | 20L | 30A | ··· | 30X | 30Y | ··· | 30BT | 30BU | ··· | 30CB | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | C3×D4 | C5×D4 | D4×C15 |
| kernel | C15×C22≀C2 | C15×C22⋊C4 | D4×C30 | C23×C30 | C5×C22≀C2 | C3×C22≀C2 | C5×C22⋊C4 | D4×C10 | C23×C10 | C3×C22⋊C4 | C6×D4 | C23×C6 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C30 | C2×C10 | C2×C6 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 4 | 6 | 6 | 2 | 12 | 12 | 4 | 8 | 24 | 24 | 8 | 6 | 12 | 24 | 48 |
Matrix representation of C15×C22≀C2 ►in GL4(𝔽61) generated by
| 57 | 0 | 0 | 0 |
| 0 | 57 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 59 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [57,0,0,0,0,57,0,0,0,0,20,0,0,0,0,20],[60,0,0,0,0,60,0,0,0,0,60,60,0,0,0,1],[1,0,0,0,0,60,0,0,0,0,1,1,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,59,60] >;
C15×C22≀C2 in GAP, Magma, Sage, TeX
C_{15}\times C_2^2\wr C_2 % in TeX
G:=Group("C15xC2^2wrC2"); // GroupNames label
G:=SmallGroup(480,925);
// by ID
G=gap.SmallGroup(480,925);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,5126]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^15=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations