G = C15⋊9(C23⋊C4) order 480 = 25·3·5
3rd semidirect product of C15 and C23⋊C4 acting via C23⋊C4/C23=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊9(C23⋊C4), (C2×C6).37D20, (C2×C30).38D4, C23.D5⋊2S3, (C2×Dic15)⋊2C4, C6.D4⋊2D5, (C2×C10).37D12, (C22×D15)⋊2C4, C23.10(S3×D5), C10.26(D6⋊C4), (C22×C6).14D10, (C22×C10).29D6, C5⋊3(C23.6D6), C30.68(C22⋊C4), C22.3(C5⋊D12), C3⋊1(C23.1D10), C22.3(C3⋊D20), C2.11(D30⋊4C4), C6.11(D10⋊C4), (C22×C30).28C22, C22.5(D30.C2), (C2×C6).3(C4×D5), (C2×C10).26(C4×S3), (C2×C30).92(C2×C4), (C2×C15⋊7D4).7C2, (C2×C6).6(C5⋊D4), (C3×C23.D5)⋊2C2, (C5×C6.D4)⋊2C2, (C2×C10).5(C3⋊D4), SmallGroup(480,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊9(C23⋊C4)
G = < a,b,c,d,e | a15=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=a11, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 684 in 104 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C23⋊C4, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, D30, C2×C30, C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.6D6, C6×Dic5, C10×Dic3, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, C23.1D10, C3×C23.D5, C5×C6.D4, C2×C15⋊7D4, C15⋊9(C23⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C23⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C23.6D6, D30.C2, C3⋊D20, C5⋊D12, C23.1D10, D30⋊4C4, C15⋊9(C23⋊C4)
►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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 75)(12 74)(13 73)(14 72)(15 71)(16 76)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 108)(32 107)(33 106)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(41 113)(42 112)(43 111)(44 110)(45 109)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 105)(57 104)(58 103)(59 102)(60 101)
(1 58)(2 59)(3 60)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 91)(74 92)(75 93)(76 116)(77 117)(78 118)(79 119)(80 120)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(31 58)(32 59)(33 60)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(61 89)(62 90)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 106)(102 107)(103 108)(104 109)(105 110)
(1 103 31 70)(2 99 32 66)(3 95 33 62)(4 91 34 73)(5 102 35 69)(6 98 36 65)(7 94 37 61)(8 105 38 72)(9 101 39 68)(10 97 40 64)(11 93 41 75)(12 104 42 71)(13 100 43 67)(14 96 44 63)(15 92 45 74)(16 110 50 85)(17 106 51 81)(18 117 52 77)(19 113 53 88)(20 109 54 84)(21 120 55 80)(22 116 56 76)(23 112 57 87)(24 108 58 83)(25 119 59 79)(26 115 60 90)(27 111 46 86)(28 107 47 82)(29 118 48 78)(30 114 49 89)
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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,75)(12,74)(13,73)(14,72)(15,71)(16,76)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,108)(32,107)(33,106)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(41,113)(42,112)(43,111)(44,110)(45,109)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,105)(57,104)(58,103)(59,102)(60,101), (1,58)(2,59)(3,60)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93)(76,116)(77,117)(78,118)(79,119)(80,120)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(31,58)(32,59)(33,60)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(61,89)(62,90)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,106)(102,107)(103,108)(104,109)(105,110), (1,103,31,70)(2,99,32,66)(3,95,33,62)(4,91,34,73)(5,102,35,69)(6,98,36,65)(7,94,37,61)(8,105,38,72)(9,101,39,68)(10,97,40,64)(11,93,41,75)(12,104,42,71)(13,100,43,67)(14,96,44,63)(15,92,45,74)(16,110,50,85)(17,106,51,81)(18,117,52,77)(19,113,53,88)(20,109,54,84)(21,120,55,80)(22,116,56,76)(23,112,57,87)(24,108,58,83)(25,119,59,79)(26,115,60,90)(27,111,46,86)(28,107,47,82)(29,118,48,78)(30,114,49,89)>;
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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,75)(12,74)(13,73)(14,72)(15,71)(16,76)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,108)(32,107)(33,106)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(41,113)(42,112)(43,111)(44,110)(45,109)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,105)(57,104)(58,103)(59,102)(60,101), (1,58)(2,59)(3,60)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,91)(74,92)(75,93)(76,116)(77,117)(78,118)(79,119)(80,120)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(31,58)(32,59)(33,60)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(61,89)(62,90)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,106)(102,107)(103,108)(104,109)(105,110), (1,103,31,70)(2,99,32,66)(3,95,33,62)(4,91,34,73)(5,102,35,69)(6,98,36,65)(7,94,37,61)(8,105,38,72)(9,101,39,68)(10,97,40,64)(11,93,41,75)(12,104,42,71)(13,100,43,67)(14,96,44,63)(15,92,45,74)(16,110,50,85)(17,106,51,81)(18,117,52,77)(19,113,53,88)(20,109,54,84)(21,120,55,80)(22,116,56,76)(23,112,57,87)(24,108,58,83)(25,119,59,79)(26,115,60,90)(27,111,46,86)(28,107,47,82)(29,118,48,78)(30,114,49,89) );
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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,75),(12,74),(13,73),(14,72),(15,71),(16,76),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(29,78),(30,77),(31,108),(32,107),(33,106),(34,120),(35,119),(36,118),(37,117),(38,116),(39,115),(40,114),(41,113),(42,112),(43,111),(44,110),(45,109),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,105),(57,104),(58,103),(59,102),(60,101)], [(1,58),(2,59),(3,60),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,31),(25,32),(26,33),(27,34),(28,35),(29,36),(30,37),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,91),(74,92),(75,93),(76,116),(77,117),(78,118),(79,119),(80,120),(81,106),(82,107),(83,108),(84,109),(85,110),(86,111),(87,112),(88,113),(89,114),(90,115)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(31,58),(32,59),(33,60),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(61,89),(62,90),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,106),(102,107),(103,108),(104,109),(105,110)], [(1,103,31,70),(2,99,32,66),(3,95,33,62),(4,91,34,73),(5,102,35,69),(6,98,36,65),(7,94,37,61),(8,105,38,72),(9,101,39,68),(10,97,40,64),(11,93,41,75),(12,104,42,71),(13,100,43,67),(14,96,44,63),(15,92,45,74),(16,110,50,85),(17,106,51,81),(18,117,52,77),(19,113,53,88),(20,109,54,84),(21,120,55,80),(22,116,56,76),(23,112,57,87),(24,108,58,83),(25,119,59,79),(26,115,60,90),(27,111,46,86),(28,107,47,82),(29,118,48,78),(30,114,49,89)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 2 | 60 | 2 | 12 | 12 | 20 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D5 | D6 | D10 | C4×S3 | D12 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | S3×D5 | C23.6D6 | D30.C2 | C3⋊D20 | C5⋊D12 | C23.1D10 | C15⋊9(C23⋊C4) |
| kernel | C15⋊9(C23⋊C4) | C3×C23.D5 | C5×C6.D4 | C2×C15⋊7D4 | C2×Dic15 | C22×D15 | C23.D5 | C2×C30 | C6.D4 | C22×C10 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | C2×C6 | C2×C6 | C2×C6 | C15 | C23 | C5 | C22 | C22 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C15⋊9(C23⋊C4) ►in GL6(𝔽61)
| 47 | 6 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 43 | 0 | 0 |
| 0 | 0 | 18 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 43 |
| 0 | 0 | 0 | 0 | 18 | 60 |
| 14 | 55 | 0 | 0 | 0 | 0 |
| 2 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 43 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 44 | 0 | 0 |
| 0 | 0 | 17 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 17 |
| 0 | 0 | 0 | 0 | 44 | 31 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 32 | 56 | 0 | 0 | 0 | 0 |
| 22 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 30 | 17 | 0 | 0 |
| 0 | 0 | 44 | 31 | 0 | 0 |
G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,6,13,0,0,0,0,0,0,18,18,0,0,0,0,43,60,0,0,0,0,0,0,18,18,0,0,0,0,43,60],[14,2,0,0,0,0,55,47,0,0,0,0,0,0,0,0,1,0,0,0,0,0,43,60,0,0,1,0,0,0,0,0,43,60,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,17,0,0,0,0,44,30,0,0,0,0,0,0,30,44,0,0,0,0,17,31],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[32,22,0,0,0,0,56,29,0,0,0,0,0,0,0,0,30,44,0,0,0,0,17,31,0,0,1,0,0,0,0,0,0,1,0,0] >;
C15⋊9(C23⋊C4) in GAP, Magma, Sage, TeX
C_{15}\rtimes_9(C_2^3\rtimes C_4) % in TeX
G:=Group("C15:9(C2^3:C4)"); // GroupNames label
G:=SmallGroup(480,73);
// by ID
G=gap.SmallGroup(480,73);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,219,675,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^2=c^2=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^11,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations