G = C15⋊C22≀C2 order 480 = 25·3·5
3rd semidirect product of C15 and C22≀C2 acting via C22≀C2/C23=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30)⋊7D4, C15⋊6C22≀C2, (S3×C10)⋊16D4, C5⋊5(C23⋊2D6), D6⋊7(C5⋊D4), (C2×Dic5)⋊3D6, (S3×C23)⋊1D5, (C22×D5)⋊3D6, D6⋊Dic5⋊35C2, C30.248(C2×D4), C10.163(S3×D4), C3⋊2(C24⋊2D5), C23.32(S3×D5), C22⋊4(C15⋊D4), (C6×Dic5)⋊7C22, (C22×C6).44D10, C30.38D4⋊29C2, (C2×C30).210C23, (C22×S3).78D10, (C22×C10).110D6, (C2×Dic15)⋊13C22, (C22×C30).72C22, (C6×C5⋊D4)⋊6C2, (C2×C5⋊D4)⋊6S3, (D5×C2×C6)⋊2C22, (C2×C6)⋊1(C5⋊D4), (S3×C22×C10)⋊1C2, C6.95(C2×C5⋊D4), C2.43(S3×C5⋊D4), (C2×C15⋊D4)⋊15C2, C2.25(C2×C15⋊D4), C10.95(C2×C3⋊D4), (C2×C10)⋊15(C3⋊D4), C22.239(C2×S3×D5), (S3×C2×C10).95C22, (C2×C6).222(C22×D5), (C2×C10).222(C22×S3), SmallGroup(480,644)
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, bab=a11, ac=ca, ad=da, ae=ea, faf=a4, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 1180 in 260 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C30, C22≀C2, C2×Dic5, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C3×Dic5, Dic15, C6×D5, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C23.D5, C2×C5⋊D4, C2×C5⋊D4, C23×C10, C23⋊2D6, C15⋊D4, C6×Dic5, C3×C5⋊D4, C2×Dic15, D5×C2×C6, S3×C2×C10, S3×C2×C10, C22×C30, C24⋊2D5, D6⋊Dic5, C30.38D4, C2×C15⋊D4, C6×C5⋊D4, S3×C22×C10, C15⋊C22≀C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, C5⋊D4, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, C2×C5⋊D4, C23⋊2D6, C15⋊D4, C2×S3×D5, C24⋊2D5, C2×C15⋊D4, S3×C5⋊D4, 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 28)(2 24)(3 20)(4 16)(5 27)(6 23)(7 19)(8 30)(9 26)(10 22)(11 18)(12 29)(13 25)(14 21)(15 17)(31 47)(32 58)(33 54)(34 50)(35 46)(36 57)(37 53)(38 49)(39 60)(40 56)(41 52)(42 48)(43 59)(44 55)(45 51)(61 119)(62 115)(63 111)(64 107)(65 118)(66 114)(67 110)(68 106)(69 117)(70 113)(71 109)(72 120)(73 116)(74 112)(75 108)(76 93)(77 104)(78 100)(79 96)(80 92)(81 103)(82 99)(83 95)(84 91)(85 102)(86 98)(87 94)(88 105)(89 101)(90 97)
(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)(91 108)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 106)(105 107)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 46)(28 47)(29 48)(30 49)(61 102)(62 103)(63 104)(64 105)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 106)(88 107)(89 108)(90 109)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 46)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)(91 108)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 106)(105 107)
(1 61)(2 65)(3 69)(4 73)(5 62)(6 66)(7 70)(8 74)(9 63)(10 67)(11 71)(12 75)(13 64)(14 68)(15 72)(16 82)(17 86)(18 90)(19 79)(20 83)(21 87)(22 76)(23 80)(24 84)(25 88)(26 77)(27 81)(28 85)(29 89)(30 78)(31 102)(32 91)(33 95)(34 99)(35 103)(36 92)(37 96)(38 100)(39 104)(40 93)(41 97)(42 101)(43 105)(44 94)(45 98)(46 115)(47 119)(48 108)(49 112)(50 116)(51 120)(52 109)(53 113)(54 117)(55 106)(56 110)(57 114)(58 118)(59 107)(60 111)
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), (1,28)(2,24)(3,20)(4,16)(5,27)(6,23)(7,19)(8,30)(9,26)(10,22)(11,18)(12,29)(13,25)(14,21)(15,17)(31,47)(32,58)(33,54)(34,50)(35,46)(36,57)(37,53)(38,49)(39,60)(40,56)(41,52)(42,48)(43,59)(44,55)(45,51)(61,119)(62,115)(63,111)(64,107)(65,118)(66,114)(67,110)(68,106)(69,117)(70,113)(71,109)(72,120)(73,116)(74,112)(75,108)(76,93)(77,104)(78,100)(79,96)(80,92)(81,103)(82,99)(83,95)(84,91)(85,102)(86,98)(87,94)(88,105)(89,101)(90,97), (61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,106)(105,107), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,46)(28,47)(29,48)(30,49)(61,102)(62,103)(63,104)(64,105)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,106)(88,107)(89,108)(90,109), (1,28)(2,29)(3,30)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,46)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,106)(105,107), (1,61)(2,65)(3,69)(4,73)(5,62)(6,66)(7,70)(8,74)(9,63)(10,67)(11,71)(12,75)(13,64)(14,68)(15,72)(16,82)(17,86)(18,90)(19,79)(20,83)(21,87)(22,76)(23,80)(24,84)(25,88)(26,77)(27,81)(28,85)(29,89)(30,78)(31,102)(32,91)(33,95)(34,99)(35,103)(36,92)(37,96)(38,100)(39,104)(40,93)(41,97)(42,101)(43,105)(44,94)(45,98)(46,115)(47,119)(48,108)(49,112)(50,116)(51,120)(52,109)(53,113)(54,117)(55,106)(56,110)(57,114)(58,118)(59,107)(60,111)>;
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), (1,28)(2,24)(3,20)(4,16)(5,27)(6,23)(7,19)(8,30)(9,26)(10,22)(11,18)(12,29)(13,25)(14,21)(15,17)(31,47)(32,58)(33,54)(34,50)(35,46)(36,57)(37,53)(38,49)(39,60)(40,56)(41,52)(42,48)(43,59)(44,55)(45,51)(61,119)(62,115)(63,111)(64,107)(65,118)(66,114)(67,110)(68,106)(69,117)(70,113)(71,109)(72,120)(73,116)(74,112)(75,108)(76,93)(77,104)(78,100)(79,96)(80,92)(81,103)(82,99)(83,95)(84,91)(85,102)(86,98)(87,94)(88,105)(89,101)(90,97), (61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,106)(105,107), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,46)(28,47)(29,48)(30,49)(61,102)(62,103)(63,104)(64,105)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,106)(88,107)(89,108)(90,109), (1,28)(2,29)(3,30)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,46)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,106)(105,107), (1,61)(2,65)(3,69)(4,73)(5,62)(6,66)(7,70)(8,74)(9,63)(10,67)(11,71)(12,75)(13,64)(14,68)(15,72)(16,82)(17,86)(18,90)(19,79)(20,83)(21,87)(22,76)(23,80)(24,84)(25,88)(26,77)(27,81)(28,85)(29,89)(30,78)(31,102)(32,91)(33,95)(34,99)(35,103)(36,92)(37,96)(38,100)(39,104)(40,93)(41,97)(42,101)(43,105)(44,94)(45,98)(46,115)(47,119)(48,108)(49,112)(50,116)(51,120)(52,109)(53,113)(54,117)(55,106)(56,110)(57,114)(58,118)(59,107)(60,111) );
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)], [(1,28),(2,24),(3,20),(4,16),(5,27),(6,23),(7,19),(8,30),(9,26),(10,22),(11,18),(12,29),(13,25),(14,21),(15,17),(31,47),(32,58),(33,54),(34,50),(35,46),(36,57),(37,53),(38,49),(39,60),(40,56),(41,52),(42,48),(43,59),(44,55),(45,51),(61,119),(62,115),(63,111),(64,107),(65,118),(66,114),(67,110),(68,106),(69,117),(70,113),(71,109),(72,120),(73,116),(74,112),(75,108),(76,93),(77,104),(78,100),(79,96),(80,92),(81,103),(82,99),(83,95),(84,91),(85,102),(86,98),(87,94),(88,105),(89,101),(90,97)], [(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,76),(68,77),(69,78),(70,79),(71,80),(72,81),(73,82),(74,83),(75,84),(91,108),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,106),(105,107)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,46),(28,47),(29,48),(30,49),(61,102),(62,103),(63,104),(64,105),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,110),(77,111),(78,112),(79,113),(80,114),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,106),(88,107),(89,108),(90,109)], [(1,28),(2,29),(3,30),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,46),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,76),(68,77),(69,78),(70,79),(71,80),(72,81),(73,82),(74,83),(75,84),(91,108),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,106),(105,107)], [(1,61),(2,65),(3,69),(4,73),(5,62),(6,66),(7,70),(8,74),(9,63),(10,67),(11,71),(12,75),(13,64),(14,68),(15,72),(16,82),(17,86),(18,90),(19,79),(20,83),(21,87),(22,76),(23,80),(24,84),(25,88),(26,77),(27,81),(28,85),(29,89),(30,78),(31,102),(32,91),(33,95),(34,99),(35,103),(36,92),(37,96),(38,100),(39,104),(40,93),(41,97),(42,101),(43,105),(44,94),(45,98),(46,115),(47,119),(48,108),(49,112),(50,116),(51,120),(52,109),(53,113),(54,117),(55,106),(56,110),(57,114),(58,118),(59,107),(60,111)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10N | 10O | ··· | 10AD | 12A | 12B | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 2 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 20 | 20 | 4 | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | C3⋊D4 | C5⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | C15⋊D4 | C2×S3×D5 | S3×C5⋊D4 |
| kernel | C15⋊C22≀C2 | D6⋊Dic5 | C30.38D4 | C2×C15⋊D4 | C6×C5⋊D4 | S3×C22×C10 | C2×C5⋊D4 | S3×C10 | C2×C30 | S3×C23 | C2×Dic5 | C22×D5 | C22×C10 | C22×S3 | C22×C6 | C2×C10 | D6 | C2×C6 | C10 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 16 | 8 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C15⋊C22≀C2 ►in GL4(𝔽61) generated by
| 1 | 52 | 0 | 0 |
| 41 | 59 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 48 | 9 |
| 60 | 0 | 0 | 0 |
| 20 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 38 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 34 | 40 | 0 | 0 |
| 55 | 27 | 0 | 0 |
| 0 | 0 | 22 | 47 |
| 0 | 0 | 4 | 39 |
G:=sub<GL(4,GF(61))| [1,41,0,0,52,59,0,0,0,0,34,48,0,0,0,9],[60,20,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,1,38,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[34,55,0,0,40,27,0,0,0,0,22,4,0,0,47,39] >;
C15⋊C22≀C2 in GAP, Magma, Sage, TeX
C_{15}\rtimes C_2^2\wr C_2 % in TeX
G:=Group("C15:C2^2wrC2"); // GroupNames label
G:=SmallGroup(480,644);
// by ID
G=gap.SmallGroup(480,644);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^15=b^2=c^2=d^2=e^2=f^2=1,b*a*b=a^11,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^4,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