?
G = C15⋊2+ (1+4) order 480 = 25·3·5
The semidirect product of C15 and 2+ (1+4) acting via 2+ (1+4)/C23=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.53C24, C15⋊82+ (1+4), D30.23C23, Dic15.25C23, C5⋊D4⋊12D6, C23⋊3(S3×D5), C15⋊Q8⋊5C22, C5⋊4(D4⋊6D6), C3⋊D4⋊15D10, (C2×Dic5)⋊9D6, (C22×D5)⋊8D6, (C22×C6)⋊7D10, D10⋊D6⋊6C2, C3⋊4(D4⋊6D10), (C2×Dic3)⋊9D10, (C22×S3)⋊7D10, (C22×C10)⋊10D6, C3⋊D20⋊7C22, C15⋊D4⋊7C22, C5⋊D12⋊7C22, C6.53(C23×D5), Dic5.D6⋊6C2, Dic3.D10⋊6C2, C30.C23⋊6C2, C15⋊7D4⋊21C22, C10.53(S3×C23), D30.C2⋊4C22, (D5×Dic3)⋊4C22, (S3×Dic5)⋊4C22, (C6×D5).20C23, D6.21(C22×D5), (S3×C10).23C23, (C2×C30).253C23, (C22×C30)⋊10C22, (C6×Dic5)⋊17C22, D10.23(C22×S3), (C10×Dic3)⋊17C22, (C2×Dic15)⋊21C22, (C22×D15)⋊16C22, Dic3.23(C22×D5), Dic5.24(C22×S3), (C5×Dic3).24C23, (C3×Dic5).22C23, (D5×C3⋊D4)⋊6C2, (S3×C5⋊D4)⋊6C2, (C2×S3×D5)⋊8C22, (C6×C5⋊D4)⋊15C2, (C2×C3⋊D4)⋊13D5, (C2×C5⋊D4)⋊13S3, (D5×C2×C6)⋊12C22, (C10×C3⋊D4)⋊15C2, (C2×C15⋊7D4)⋊26C2, C22.12(C2×S3×D5), C2.53(C22×S3×D5), (S3×C2×C10)⋊12C22, (C3×C5⋊D4)⋊16C22, (C5×C3⋊D4)⋊16C22, (C2×C6).14(C22×D5), (C2×C10).16(C22×S3), SmallGroup(480,1125)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1788 in 332 conjugacy classes, 108 normal (40 characteristic)
C1, C2, C2 [×9], C3, C4 [×6], C22 [×3], C22 [×12], C5, S3 [×4], C6, C6 [×5], C2×C4 [×9], D4 [×18], Q8 [×2], C23, C23 [×5], D5 [×4], C10, C10 [×5], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6 [×3], C2×C6 [×4], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10 [×3], C2×C10 [×4], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3, C2×Dic3 [×3], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×3], C22×C6, C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C30 [×3], 2+ (1+4), Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×3], C5⋊D4 [×4], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×3], C22×C10, C22×C10, C4○D12 [×2], S3×D4 [×4], D4⋊2S3 [×4], C2×C3⋊D4, C2×C3⋊D4 [×3], C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], S3×D5 [×2], C6×D5 [×2], C6×D5, S3×C10 [×2], S3×C10, D30 [×2], D30, C2×C30 [×3], C2×C30, C4○D20 [×2], D4×D5 [×4], D4⋊2D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×3], D4×C10, D4⋊6D6, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3, C5×C3⋊D4 [×4], C2×Dic15, C15⋊7D4 [×4], C2×S3×D5 [×2], D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, D4⋊6D10, Dic5.D6 [×2], C30.C23 [×2], Dic3.D10 [×2], D5×C3⋊D4 [×2], S3×C5⋊D4 [×2], D10⋊D6 [×2], C6×C5⋊D4, C10×C3⋊D4, C2×C15⋊7D4, C15⋊2+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4⋊6D6, C2×S3×D5 [×3], D4⋊6D10, C22×S3×D5, C15⋊2+ (1+4)
Generators and relations
G = < a,b,c,d,e | a15=b4=c2=e2=1, d2=b2, bab-1=cac=a11, dad-1=a4, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►On 120 points
(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 84 17 65)(2 80 18 61)(3 76 19 72)(4 87 20 68)(5 83 21 64)(6 79 22 75)(7 90 23 71)(8 86 24 67)(9 82 25 63)(10 78 26 74)(11 89 27 70)(12 85 28 66)(13 81 29 62)(14 77 30 73)(15 88 16 69)(31 106 58 101)(32 117 59 97)(33 113 60 93)(34 109 46 104)(35 120 47 100)(36 116 48 96)(37 112 49 92)(38 108 50 103)(39 119 51 99)(40 115 52 95)(41 111 53 91)(42 107 54 102)(43 118 55 98)(44 114 56 94)(45 110 57 105)
(1 65)(2 61)(3 72)(4 68)(5 64)(6 75)(7 71)(8 67)(9 63)(10 74)(11 70)(12 66)(13 62)(14 73)(15 69)(16 88)(17 84)(18 80)(19 76)(20 87)(21 83)(22 79)(23 90)(24 86)(25 82)(26 78)(27 89)(28 85)(29 81)(30 77)(31 101)(32 97)(33 93)(34 104)(35 100)(36 96)(37 92)(38 103)(39 99)(40 95)(41 91)(42 102)(43 98)(44 94)(45 105)(46 109)(47 120)(48 116)(49 112)(50 108)(51 119)(52 115)(53 111)(54 107)(55 118)(56 114)(57 110)(58 106)(59 117)(60 113)
(1 58 17 31)(2 47 18 35)(3 51 19 39)(4 55 20 43)(5 59 21 32)(6 48 22 36)(7 52 23 40)(8 56 24 44)(9 60 25 33)(10 49 26 37)(11 53 27 41)(12 57 28 45)(13 46 29 34)(14 50 30 38)(15 54 16 42)(61 120 80 100)(62 109 81 104)(63 113 82 93)(64 117 83 97)(65 106 84 101)(66 110 85 105)(67 114 86 94)(68 118 87 98)(69 107 88 102)(70 111 89 91)(71 115 90 95)(72 119 76 99)(73 108 77 103)(74 112 78 92)(75 116 79 96)
(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 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 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(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)
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), (1,84,17,65)(2,80,18,61)(3,76,19,72)(4,87,20,68)(5,83,21,64)(6,79,22,75)(7,90,23,71)(8,86,24,67)(9,82,25,63)(10,78,26,74)(11,89,27,70)(12,85,28,66)(13,81,29,62)(14,77,30,73)(15,88,16,69)(31,106,58,101)(32,117,59,97)(33,113,60,93)(34,109,46,104)(35,120,47,100)(36,116,48,96)(37,112,49,92)(38,108,50,103)(39,119,51,99)(40,115,52,95)(41,111,53,91)(42,107,54,102)(43,118,55,98)(44,114,56,94)(45,110,57,105), (1,65)(2,61)(3,72)(4,68)(5,64)(6,75)(7,71)(8,67)(9,63)(10,74)(11,70)(12,66)(13,62)(14,73)(15,69)(16,88)(17,84)(18,80)(19,76)(20,87)(21,83)(22,79)(23,90)(24,86)(25,82)(26,78)(27,89)(28,85)(29,81)(30,77)(31,101)(32,97)(33,93)(34,104)(35,100)(36,96)(37,92)(38,103)(39,99)(40,95)(41,91)(42,102)(43,98)(44,94)(45,105)(46,109)(47,120)(48,116)(49,112)(50,108)(51,119)(52,115)(53,111)(54,107)(55,118)(56,114)(57,110)(58,106)(59,117)(60,113), (1,58,17,31)(2,47,18,35)(3,51,19,39)(4,55,20,43)(5,59,21,32)(6,48,22,36)(7,52,23,40)(8,56,24,44)(9,60,25,33)(10,49,26,37)(11,53,27,41)(12,57,28,45)(13,46,29,34)(14,50,30,38)(15,54,16,42)(61,120,80,100)(62,109,81,104)(63,113,82,93)(64,117,83,97)(65,106,84,101)(66,110,85,105)(67,114,86,94)(68,118,87,98)(69,107,88,102)(70,111,89,91)(71,115,90,95)(72,119,76,99)(73,108,77,103)(74,112,78,92)(75,116,79,96), (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)(16,57)(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)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(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)>;
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), (1,84,17,65)(2,80,18,61)(3,76,19,72)(4,87,20,68)(5,83,21,64)(6,79,22,75)(7,90,23,71)(8,86,24,67)(9,82,25,63)(10,78,26,74)(11,89,27,70)(12,85,28,66)(13,81,29,62)(14,77,30,73)(15,88,16,69)(31,106,58,101)(32,117,59,97)(33,113,60,93)(34,109,46,104)(35,120,47,100)(36,116,48,96)(37,112,49,92)(38,108,50,103)(39,119,51,99)(40,115,52,95)(41,111,53,91)(42,107,54,102)(43,118,55,98)(44,114,56,94)(45,110,57,105), (1,65)(2,61)(3,72)(4,68)(5,64)(6,75)(7,71)(8,67)(9,63)(10,74)(11,70)(12,66)(13,62)(14,73)(15,69)(16,88)(17,84)(18,80)(19,76)(20,87)(21,83)(22,79)(23,90)(24,86)(25,82)(26,78)(27,89)(28,85)(29,81)(30,77)(31,101)(32,97)(33,93)(34,104)(35,100)(36,96)(37,92)(38,103)(39,99)(40,95)(41,91)(42,102)(43,98)(44,94)(45,105)(46,109)(47,120)(48,116)(49,112)(50,108)(51,119)(52,115)(53,111)(54,107)(55,118)(56,114)(57,110)(58,106)(59,117)(60,113), (1,58,17,31)(2,47,18,35)(3,51,19,39)(4,55,20,43)(5,59,21,32)(6,48,22,36)(7,52,23,40)(8,56,24,44)(9,60,25,33)(10,49,26,37)(11,53,27,41)(12,57,28,45)(13,46,29,34)(14,50,30,38)(15,54,16,42)(61,120,80,100)(62,109,81,104)(63,113,82,93)(64,117,83,97)(65,106,84,101)(66,110,85,105)(67,114,86,94)(68,118,87,98)(69,107,88,102)(70,111,89,91)(71,115,90,95)(72,119,76,99)(73,108,77,103)(74,112,78,92)(75,116,79,96), (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)(16,57)(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)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(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) );
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)], [(1,84,17,65),(2,80,18,61),(3,76,19,72),(4,87,20,68),(5,83,21,64),(6,79,22,75),(7,90,23,71),(8,86,24,67),(9,82,25,63),(10,78,26,74),(11,89,27,70),(12,85,28,66),(13,81,29,62),(14,77,30,73),(15,88,16,69),(31,106,58,101),(32,117,59,97),(33,113,60,93),(34,109,46,104),(35,120,47,100),(36,116,48,96),(37,112,49,92),(38,108,50,103),(39,119,51,99),(40,115,52,95),(41,111,53,91),(42,107,54,102),(43,118,55,98),(44,114,56,94),(45,110,57,105)], [(1,65),(2,61),(3,72),(4,68),(5,64),(6,75),(7,71),(8,67),(9,63),(10,74),(11,70),(12,66),(13,62),(14,73),(15,69),(16,88),(17,84),(18,80),(19,76),(20,87),(21,83),(22,79),(23,90),(24,86),(25,82),(26,78),(27,89),(28,85),(29,81),(30,77),(31,101),(32,97),(33,93),(34,104),(35,100),(36,96),(37,92),(38,103),(39,99),(40,95),(41,91),(42,102),(43,98),(44,94),(45,105),(46,109),(47,120),(48,116),(49,112),(50,108),(51,119),(52,115),(53,111),(54,107),(55,118),(56,114),(57,110),(58,106),(59,117),(60,113)], [(1,58,17,31),(2,47,18,35),(3,51,19,39),(4,55,20,43),(5,59,21,32),(6,48,22,36),(7,52,23,40),(8,56,24,44),(9,60,25,33),(10,49,26,37),(11,53,27,41),(12,57,28,45),(13,46,29,34),(14,50,30,38),(15,54,16,42),(61,120,80,100),(62,109,81,104),(63,113,82,93),(64,117,83,97),(65,106,84,101),(66,110,85,105),(67,114,86,94),(68,118,87,98),(69,107,88,102),(70,111,89,91),(71,115,90,95),(72,119,76,99),(73,108,77,103),(74,112,78,92),(75,116,79,96)], [(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),(16,57),(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),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(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)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 8 | 6 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 53 | 56 | 48 | 38 |
| 44 | 35 | 23 | 23 |
| 0 | 0 | 60 | 1 |
| 11 | 46 | 59 | 17 |
| 4 | 50 | 15 | 0 |
| 3 | 50 | 15 | 0 |
| 0 | 0 | 60 | 1 |
| 11 | 46 | 59 | 17 |
| 18 | 42 | 15 | 0 |
| 19 | 42 | 15 | 0 |
| 8 | 14 | 0 | 0 |
| 52 | 53 | 0 | 0 |
| 4 | 0 | 14 | 45 |
| 36 | 34 | 39 | 47 |
| 30 | 45 | 0 | 0 |
| 60 | 31 | 0 | 0 |
| 4 | 0 | 14 | 45 |
| 11 | 57 | 16 | 47 |
G:=sub<GL(4,GF(61))| [8,8,53,44,6,0,56,35,0,0,48,23,0,0,38,23],[0,11,4,3,0,46,50,50,60,59,15,15,1,17,0,0],[0,11,18,19,0,46,42,42,60,59,15,15,1,17,0,0],[8,52,4,36,14,53,0,34,0,0,14,39,0,0,45,47],[30,60,4,11,45,31,0,57,0,0,14,16,0,0,45,47] >;
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | 2+ (1+4) | S3×D5 | D4⋊6D6 | C2×S3×D5 | D4⋊6D10 | C15⋊2+ (1+4) |
| kernel | C15⋊2+ (1+4) | Dic5.D6 | C30.C23 | Dic3.D10 | D5×C3⋊D4 | S3×C5⋊D4 | D10⋊D6 | C6×C5⋊D4 | C10×C3⋊D4 | C2×C15⋊7D4 | C2×C5⋊D4 | C2×C3⋊D4 | C2×Dic5 | C5⋊D4 | C22×D5 | C22×C10 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C15 | C23 | C5 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 8 | 2 | 2 | 1 | 2 | 2 | 6 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_{15}\rtimes 2_+^{(1+4)} % in TeX
G:=Group("C15:ES+(2,2)"); // GroupNames label
G:=SmallGroup(480,1125);
// by ID
G=gap.SmallGroup(480,1125);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^4=c^2=e^2=1,d^2=b^2,b*a*b^-1=c*a*c=a^11,d*a*d^-1=a^4,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=b^2*d>;
// generators/relations