direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D5×SD16, C24⋊27D10, C120⋊30C22, C60.193C23, C8⋊5(C6×D5), C40⋊5(C2×C6), (C8×D5)⋊4C6, Q8⋊D5⋊1C6, Q8⋊2(C6×D5), (Q8×D5)⋊4C6, C5⋊2(C6×SD16), C40⋊C2⋊5C6, D4.D5⋊3C6, D4.2(C6×D5), (D4×D5).1C6, (D5×C24)⋊13C2, (C3×Q8)⋊14D10, (C5×SD16)⋊3C6, D20.2(C2×C6), (C6×D5).86D4, C10.30(C6×D4), C6.184(D4×D5), C15⋊20(C2×SD16), Dic10⋊2(C2×C6), (C15×SD16)⋊9C2, (C3×D4).26D10, D10.24(C3×D4), C30.343(C2×D4), C20.4(C22×C6), Dic5.8(C3×D4), (C3×Dic5).55D4, (Q8×C15)⋊14C22, (C3×D20).31C22, (D4×C15).26C22, C12.193(C22×D5), (C3×Dic10)⋊17C22, (D5×C12).106C22, (C3×Q8×D5)⋊8C2, C4.4(D5×C2×C6), (C3×D4×D5).4C2, C2.18(C3×D4×D5), C5⋊2C8⋊6(C2×C6), (C3×Q8⋊D5)⋊9C2, (C5×Q8)⋊2(C2×C6), (C5×D4).2(C2×C6), (C3×C40⋊C2)⋊13C2, (C3×D4.D5)⋊11C2, (C4×D5).17(C2×C6), (C3×C5⋊2C8)⋊39C22, SmallGroup(480,706)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D5×SD16
G = < a,b,c,d,e | a3=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 528 in 136 conjugacy classes, 58 normal (54 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C12, C12, C2×C6, C15, C2×C8, SD16, SD16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C3×D5, C3×D5, C30, C30, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×C24, C3×SD16, C3×SD16, C6×D4, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C6×SD16, C3×C5⋊2C8, C120, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, Q8×C15, D5×C2×C6, D5×SD16, D5×C24, C3×C40⋊C2, C3×D4.D5, C3×Q8⋊D5, C15×SD16, C3×D4×D5, C3×Q8×D5, C3×D5×SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, SD16, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×SD16, C22×D5, C3×SD16, C6×D4, C6×D5, D4×D5, C6×SD16, D5×C2×C6, D5×SD16, C3×D4×D5, C3×D5×SD16
►On 120 points
Generators in S120
(1 104 65)(2 97 66)(3 98 67)(4 99 68)(5 100 69)(6 101 70)(7 102 71)(8 103 72)(9 85 106)(10 86 107)(11 87 108)(12 88 109)(13 81 110)(14 82 111)(15 83 112)(16 84 105)(17 42 58)(18 43 59)(19 44 60)(20 45 61)(21 46 62)(22 47 63)(23 48 64)(24 41 57)(25 54 75)(26 55 76)(27 56 77)(28 49 78)(29 50 79)(30 51 80)(31 52 73)(32 53 74)(33 95 120)(34 96 113)(35 89 114)(36 90 115)(37 91 116)(38 92 117)(39 93 118)(40 94 119)
(1 54 46 38 108)(2 55 47 39 109)(3 56 48 40 110)(4 49 41 33 111)(5 50 42 34 112)(6 51 43 35 105)(7 52 44 36 106)(8 53 45 37 107)(9 102 73 60 90)(10 103 74 61 91)(11 104 75 62 92)(12 97 76 63 93)(13 98 77 64 94)(14 99 78 57 95)(15 100 79 58 96)(16 101 80 59 89)(17 113 83 69 29)(18 114 84 70 30)(19 115 85 71 31)(20 116 86 72 32)(21 117 87 65 25)(22 118 88 66 26)(23 119 81 67 27)(24 120 82 68 28)
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 102)(10 103)(11 104)(12 97)(13 98)(14 99)(15 100)(16 101)(25 117)(26 118)(27 119)(28 120)(29 113)(30 114)(31 115)(32 116)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(65 87)(66 88)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 89)
(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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(57 63)(59 61)(60 64)(66 68)(67 71)(70 72)(73 77)(74 80)(76 78)(81 85)(82 88)(84 86)(89 91)(90 94)(93 95)(97 99)(98 102)(101 103)(105 107)(106 110)(109 111)(114 116)(115 119)(118 120)
G:=sub<Sym(120)| (1,104,65)(2,97,66)(3,98,67)(4,99,68)(5,100,69)(6,101,70)(7,102,71)(8,103,72)(9,85,106)(10,86,107)(11,87,108)(12,88,109)(13,81,110)(14,82,111)(15,83,112)(16,84,105)(17,42,58)(18,43,59)(19,44,60)(20,45,61)(21,46,62)(22,47,63)(23,48,64)(24,41,57)(25,54,75)(26,55,76)(27,56,77)(28,49,78)(29,50,79)(30,51,80)(31,52,73)(32,53,74)(33,95,120)(34,96,113)(35,89,114)(36,90,115)(37,91,116)(38,92,117)(39,93,118)(40,94,119), (1,54,46,38,108)(2,55,47,39,109)(3,56,48,40,110)(4,49,41,33,111)(5,50,42,34,112)(6,51,43,35,105)(7,52,44,36,106)(8,53,45,37,107)(9,102,73,60,90)(10,103,74,61,91)(11,104,75,62,92)(12,97,76,63,93)(13,98,77,64,94)(14,99,78,57,95)(15,100,79,58,96)(16,101,80,59,89)(17,113,83,69,29)(18,114,84,70,30)(19,115,85,71,31)(20,116,86,72,32)(21,117,87,65,25)(22,118,88,66,26)(23,119,81,67,27)(24,120,82,68,28), (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,102)(10,103)(11,104)(12,97)(13,98)(14,99)(15,100)(16,101)(25,117)(26,118)(27,119)(28,120)(29,113)(30,114)(31,115)(32,116)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,89), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)(89,91)(90,94)(93,95)(97,99)(98,102)(101,103)(105,107)(106,110)(109,111)(114,116)(115,119)(118,120)>;
G:=Group( (1,104,65)(2,97,66)(3,98,67)(4,99,68)(5,100,69)(6,101,70)(7,102,71)(8,103,72)(9,85,106)(10,86,107)(11,87,108)(12,88,109)(13,81,110)(14,82,111)(15,83,112)(16,84,105)(17,42,58)(18,43,59)(19,44,60)(20,45,61)(21,46,62)(22,47,63)(23,48,64)(24,41,57)(25,54,75)(26,55,76)(27,56,77)(28,49,78)(29,50,79)(30,51,80)(31,52,73)(32,53,74)(33,95,120)(34,96,113)(35,89,114)(36,90,115)(37,91,116)(38,92,117)(39,93,118)(40,94,119), (1,54,46,38,108)(2,55,47,39,109)(3,56,48,40,110)(4,49,41,33,111)(5,50,42,34,112)(6,51,43,35,105)(7,52,44,36,106)(8,53,45,37,107)(9,102,73,60,90)(10,103,74,61,91)(11,104,75,62,92)(12,97,76,63,93)(13,98,77,64,94)(14,99,78,57,95)(15,100,79,58,96)(16,101,80,59,89)(17,113,83,69,29)(18,114,84,70,30)(19,115,85,71,31)(20,116,86,72,32)(21,117,87,65,25)(22,118,88,66,26)(23,119,81,67,27)(24,120,82,68,28), (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,102)(10,103)(11,104)(12,97)(13,98)(14,99)(15,100)(16,101)(25,117)(26,118)(27,119)(28,120)(29,113)(30,114)(31,115)(32,116)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,89), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)(89,91)(90,94)(93,95)(97,99)(98,102)(101,103)(105,107)(106,110)(109,111)(114,116)(115,119)(118,120) );
G=PermutationGroup([[(1,104,65),(2,97,66),(3,98,67),(4,99,68),(5,100,69),(6,101,70),(7,102,71),(8,103,72),(9,85,106),(10,86,107),(11,87,108),(12,88,109),(13,81,110),(14,82,111),(15,83,112),(16,84,105),(17,42,58),(18,43,59),(19,44,60),(20,45,61),(21,46,62),(22,47,63),(23,48,64),(24,41,57),(25,54,75),(26,55,76),(27,56,77),(28,49,78),(29,50,79),(30,51,80),(31,52,73),(32,53,74),(33,95,120),(34,96,113),(35,89,114),(36,90,115),(37,91,116),(38,92,117),(39,93,118),(40,94,119)], [(1,54,46,38,108),(2,55,47,39,109),(3,56,48,40,110),(4,49,41,33,111),(5,50,42,34,112),(6,51,43,35,105),(7,52,44,36,106),(8,53,45,37,107),(9,102,73,60,90),(10,103,74,61,91),(11,104,75,62,92),(12,97,76,63,93),(13,98,77,64,94),(14,99,78,57,95),(15,100,79,58,96),(16,101,80,59,89),(17,113,83,69,29),(18,114,84,70,30),(19,115,85,71,31),(20,116,86,72,32),(21,117,87,65,25),(22,118,88,66,26),(23,119,81,67,27),(24,120,82,68,28)], [(1,108),(2,109),(3,110),(4,111),(5,112),(6,105),(7,106),(8,107),(9,102),(10,103),(11,104),(12,97),(13,98),(14,99),(15,100),(16,101),(25,117),(26,118),(27,119),(28,120),(29,113),(30,114),(31,115),(32,116),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(65,87),(66,88),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,89)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(26,28),(27,31),(30,32),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(49,55),(51,53),(52,56),(57,63),(59,61),(60,64),(66,68),(67,71),(70,72),(73,77),(74,80),(76,78),(81,85),(82,88),(84,86),(89,91),(90,94),(93,95),(97,99),(98,102),(101,103),(105,107),(106,110),(109,111),(114,116),(115,119),(118,120)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 5 | 5 | 20 | 1 | 1 | 2 | 4 | 10 | 20 | 2 | 2 | 1 | 1 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 2 | 2 | 10 | 10 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C3×SD16 | C6×D5 | C6×D5 | C6×D5 | D4×D5 | D5×SD16 | C3×D4×D5 | C3×D5×SD16 |
| kernel | C3×D5×SD16 | D5×C24 | C3×C40⋊C2 | C3×D4.D5 | C3×Q8⋊D5 | C15×SD16 | C3×D4×D5 | C3×Q8×D5 | D5×SD16 | C8×D5 | C40⋊C2 | D4.D5 | Q8⋊D5 | C5×SD16 | D4×D5 | Q8×D5 | C3×Dic5 | C6×D5 | C3×SD16 | C3×D5 | C24 | C3×D4 | C3×Q8 | Dic5 | D10 | SD16 | D5 | C8 | D4 | Q8 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C3×D5×SD16 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 51 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 51 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 19 | 19 |
| 0 | 0 | 222 | 19 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[51,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,51,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,19,222,0,0,19,19],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,240] >;
C3×D5×SD16 in GAP, Magma, Sage, TeX
C_3\times D_5\times {\rm SD}_{16} % in TeX
G:=Group("C3xD5xSD16"); // GroupNames label
G:=SmallGroup(480,706);
// by ID
G=gap.SmallGroup(480,706);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,303,268,1271,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,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=d^3>;
// generators/relations