G = C15⋊8(C23⋊C4) order 480 = 25·3·5
2nd semidirect product of C15 and C23⋊C4 acting via C23⋊C4/C23=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊8(C23⋊C4), (C2×Dic3)⋊Dic5, (C2×C30).37D4, (C2×C10).2D12, (C22×S3)⋊Dic5, C23.D5⋊1S3, C23.9(S3×D5), (C10×Dic3)⋊1C4, C3⋊1(C23⋊Dic5), C10.47(D6⋊C4), (C22×C10).28D6, (C22×C6).13D10, C5⋊5(C23.6D6), C30.38D4⋊15C2, C22.5(S3×Dic5), C30.67(C22⋊C4), C2.11(D6⋊Dic5), C22.3(C15⋊D4), C22.8(C5⋊D12), C6.11(C23.D5), (C22×C30).27C22, (S3×C2×C10)⋊1C4, (C2×C3⋊D4).1D5, (C2×C30).91(C2×C4), (C2×C10).73(C4×S3), (C10×C3⋊D4).1C2, (C2×C6).5(C5⋊D4), (C3×C23.D5)⋊1C2, (C2×C6).3(C2×Dic5), (C2×C10).4(C3⋊D4), SmallGroup(480,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊8(C23⋊C4)
G = < a,b,c,d,e | a15=b2=c2=d2=e4=1, bab=a11, ac=ca, ad=da, eae-1=a4, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 492 in 104 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C23⋊C4, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10, C2×C30, C2×C30, C23.D5, C23.D5, D4×C10, C23.6D6, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, S3×C2×C10, C22×C30, C23⋊Dic5, C3×C23.D5, C30.38D4, C10×C3⋊D4, C15⋊8(C23⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, Dic5, D10, C4×S3, D12, C3⋊D4, C23⋊C4, C2×Dic5, C5⋊D4, D6⋊C4, S3×D5, C23.D5, C23.6D6, S3×Dic5, C15⋊D4, C5⋊D12, C23⋊Dic5, D6⋊Dic5, C15⋊8(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 68)(2 64)(3 75)(4 71)(5 67)(6 63)(7 74)(8 70)(9 66)(10 62)(11 73)(12 69)(13 65)(14 61)(15 72)(16 76)(17 87)(18 83)(19 79)(20 90)(21 86)(22 82)(23 78)(24 89)(25 85)(26 81)(27 77)(28 88)(29 84)(30 80)(31 102)(32 98)(33 94)(34 105)(35 101)(36 97)(37 93)(38 104)(39 100)(40 96)(41 92)(42 103)(43 99)(44 95)(45 91)(46 112)(47 108)(48 119)(49 115)(50 111)(51 107)(52 118)(53 114)(54 110)(55 106)(56 117)(57 113)(58 109)(59 120)(60 116)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 31)(13 32)(14 33)(15 34)(16 57)(17 58)(18 59)(19 60)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(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)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)
(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 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 76)(71 77)(72 78)(73 79)(74 80)(75 81)(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 68 35 111)(2 72 36 115)(3 61 37 119)(4 65 38 108)(5 69 39 112)(6 73 40 116)(7 62 41 120)(8 66 42 109)(9 70 43 113)(10 74 44 117)(11 63 45 106)(12 67 31 110)(13 71 32 114)(14 75 33 118)(15 64 34 107)(16 87 57 99)(17 76 58 103)(18 80 59 92)(19 84 60 96)(20 88 46 100)(21 77 47 104)(22 81 48 93)(23 85 49 97)(24 89 50 101)(25 78 51 105)(26 82 52 94)(27 86 53 98)(28 90 54 102)(29 79 55 91)(30 83 56 95)
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), (1,68)(2,64)(3,75)(4,71)(5,67)(6,63)(7,74)(8,70)(9,66)(10,62)(11,73)(12,69)(13,65)(14,61)(15,72)(16,76)(17,87)(18,83)(19,79)(20,90)(21,86)(22,82)(23,78)(24,89)(25,85)(26,81)(27,77)(28,88)(29,84)(30,80)(31,102)(32,98)(33,94)(34,105)(35,101)(36,97)(37,93)(38,104)(39,100)(40,96)(41,92)(42,103)(43,99)(44,95)(45,91)(46,112)(47,108)(48,119)(49,115)(50,111)(51,107)(52,118)(53,114)(54,110)(55,106)(56,117)(57,113)(58,109)(59,120)(60,116), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,31)(13,32)(14,33)(15,34)(16,57)(17,58)(18,59)(19,60)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(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)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112), (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)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,76)(71,77)(72,78)(73,79)(74,80)(75,81)(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), (1,68,35,111)(2,72,36,115)(3,61,37,119)(4,65,38,108)(5,69,39,112)(6,73,40,116)(7,62,41,120)(8,66,42,109)(9,70,43,113)(10,74,44,117)(11,63,45,106)(12,67,31,110)(13,71,32,114)(14,75,33,118)(15,64,34,107)(16,87,57,99)(17,76,58,103)(18,80,59,92)(19,84,60,96)(20,88,46,100)(21,77,47,104)(22,81,48,93)(23,85,49,97)(24,89,50,101)(25,78,51,105)(26,82,52,94)(27,86,53,98)(28,90,54,102)(29,79,55,91)(30,83,56,95)>;
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), (1,68)(2,64)(3,75)(4,71)(5,67)(6,63)(7,74)(8,70)(9,66)(10,62)(11,73)(12,69)(13,65)(14,61)(15,72)(16,76)(17,87)(18,83)(19,79)(20,90)(21,86)(22,82)(23,78)(24,89)(25,85)(26,81)(27,77)(28,88)(29,84)(30,80)(31,102)(32,98)(33,94)(34,105)(35,101)(36,97)(37,93)(38,104)(39,100)(40,96)(41,92)(42,103)(43,99)(44,95)(45,91)(46,112)(47,108)(48,119)(49,115)(50,111)(51,107)(52,118)(53,114)(54,110)(55,106)(56,117)(57,113)(58,109)(59,120)(60,116), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,31)(13,32)(14,33)(15,34)(16,57)(17,58)(18,59)(19,60)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(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)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112), (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)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,76)(71,77)(72,78)(73,79)(74,80)(75,81)(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), (1,68,35,111)(2,72,36,115)(3,61,37,119)(4,65,38,108)(5,69,39,112)(6,73,40,116)(7,62,41,120)(8,66,42,109)(9,70,43,113)(10,74,44,117)(11,63,45,106)(12,67,31,110)(13,71,32,114)(14,75,33,118)(15,64,34,107)(16,87,57,99)(17,76,58,103)(18,80,59,92)(19,84,60,96)(20,88,46,100)(21,77,47,104)(22,81,48,93)(23,85,49,97)(24,89,50,101)(25,78,51,105)(26,82,52,94)(27,86,53,98)(28,90,54,102)(29,79,55,91)(30,83,56,95) );
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)], [(1,68),(2,64),(3,75),(4,71),(5,67),(6,63),(7,74),(8,70),(9,66),(10,62),(11,73),(12,69),(13,65),(14,61),(15,72),(16,76),(17,87),(18,83),(19,79),(20,90),(21,86),(22,82),(23,78),(24,89),(25,85),(26,81),(27,77),(28,88),(29,84),(30,80),(31,102),(32,98),(33,94),(34,105),(35,101),(36,97),(37,93),(38,104),(39,100),(40,96),(41,92),(42,103),(43,99),(44,95),(45,91),(46,112),(47,108),(48,119),(49,115),(50,111),(51,107),(52,118),(53,114),(54,110),(55,106),(56,117),(57,113),(58,109),(59,120),(60,116)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,31),(13,32),(14,33),(15,34),(16,57),(17,58),(18,59),(19,60),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(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),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,106),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112)], [(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),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,76),(71,77),(72,78),(73,79),(74,80),(75,81),(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)], [(1,68,35,111),(2,72,36,115),(3,61,37,119),(4,65,38,108),(5,69,39,112),(6,73,40,116),(7,62,41,120),(8,66,42,109),(9,70,43,113),(10,74,44,117),(11,63,45,106),(12,67,31,110),(13,71,32,114),(14,75,33,118),(15,64,34,107),(16,87,57,99),(17,76,58,103),(18,80,59,92),(19,84,60,96),(20,88,46,100),(21,77,47,104),(22,81,48,93),(23,85,49,97),(24,89,50,101),(25,78,51,105),(26,82,52,94),(27,86,53,98),(28,90,54,102),(29,79,55,91),(30,83,56,95)]])
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 | 10K | 10L | 10M | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 2 | 12 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 20 | 20 | 4 | 4 | 12 | 12 | 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 | Dic5 | Dic5 | D10 | C4×S3 | D12 | C3⋊D4 | C5⋊D4 | C23⋊C4 | S3×D5 | C23.6D6 | S3×Dic5 | C15⋊D4 | C5⋊D12 | C23⋊Dic5 | C15⋊8(C23⋊C4) |
| kernel | C15⋊8(C23⋊C4) | C3×C23.D5 | C30.38D4 | C10×C3⋊D4 | C10×Dic3 | S3×C2×C10 | C23.D5 | C2×C30 | C2×C3⋊D4 | C22×C10 | C2×Dic3 | C22×S3 | C22×C6 | C2×C10 | C2×C10 | C2×C10 | 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 | 2 | 2 | 8 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C15⋊8(C23⋊C4) ►in GL6(𝔽61)
| 0 | 17 | 0 | 0 | 0 | 0 |
| 43 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 30 | 16 | 0 | 0 | 0 | 0 |
| 1 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 | 0 | 0 |
| 0 | 0 | 43 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 43 | 52 |
| 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 |
| 27 | 59 | 0 | 0 | 0 | 0 |
| 59 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 43 |
| 0 | 0 | 0 | 0 | 52 | 9 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
G:=sub<GL(6,GF(61))| [0,43,0,0,0,0,17,43,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[30,1,0,0,0,0,16,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,9,43,0,0,0,0,18,52,0,0,0,0,0,0,9,43,0,0,0,0,18,52],[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],[27,59,0,0,0,0,59,34,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,52,52,0,0,0,0,43,9,0,0] >;
C15⋊8(C23⋊C4) in GAP, Magma, Sage, TeX
C_{15}\rtimes_8(C_2^3\rtimes C_4) % in TeX
G:=Group("C15:8(C2^3:C4)"); // GroupNames label
G:=SmallGroup(480,72);
// by ID
G=gap.SmallGroup(480,72);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^2=c^2=d^2=e^4=1,b*a*b=a^11,a*c=c*a,a*d=d*a,e*a*e^-1=a^4,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