direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D8⋊D5, C24⋊17D10, C120⋊29C22, C60.191C23, C8⋊2(C6×D5), C40⋊4(C2×C6), D4⋊D5⋊2C6, D8⋊2(C3×D5), (C3×D8)⋊6D5, D4⋊2(C6×D5), (C5×D8)⋊4C6, (D4×D5)⋊2C6, C8⋊D5⋊3C6, C40⋊C2⋊3C6, D4.D5⋊1C6, (C3×D4)⋊17D10, D4⋊2D5⋊1C6, (C15×D8)⋊11C2, (C6×D5).71D4, D20.1(C2×C6), C10.28(C6×D4), C6.182(D4×D5), C15⋊29(C8⋊C22), Dic10⋊1(C2×C6), D10.13(C3×D4), C30.341(C2×D4), C20.2(C22×C6), (D4×C15)⋊17C22, Dic5.16(C3×D4), (C3×Dic5).77D4, (D5×C12).76C22, (C3×D20).30C22, C12.191(C22×D5), (C3×Dic10)⋊16C22, (C3×D4×D5)⋊9C2, C4.2(D5×C2×C6), C5⋊2(C3×C8⋊C22), C2.16(C3×D4×D5), C5⋊2C8⋊1(C2×C6), (C5×D4)⋊2(C2×C6), (C3×D4⋊D5)⋊10C2, (C3×D4.D5)⋊9C2, (C4×D5).1(C2×C6), (C3×D4⋊2D5)⋊8C2, (C3×C8⋊D5)⋊11C2, (C3×C40⋊C2)⋊11C2, (C3×C5⋊2C8)⋊23C22, SmallGroup(480,704)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8⋊D5
G = < a,b,c,d,e | a3=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
Subgroups: 544 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C15, M4(2), D8, D8, SD16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C3×M4(2), C3×D8, C3×D8, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, C3×C8⋊C22, C3×C5⋊2C8, C120, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, D4×C15, D5×C2×C6, D8⋊D5, C3×C8⋊D5, C3×C40⋊C2, C3×D4⋊D5, C3×D4.D5, C15×D8, C3×D4×D5, C3×D4⋊2D5, C3×D8⋊D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8⋊C22, D5×C2×C6, D8⋊D5, C3×D4×D5, C3×D8⋊D5
►On 120 points
Generators in S120
(1 104 79)(2 97 80)(3 98 73)(4 99 74)(5 100 75)(6 101 76)(7 102 77)(8 103 78)(9 31 108)(10 32 109)(11 25 110)(12 26 111)(13 27 112)(14 28 105)(15 29 106)(16 30 107)(17 94 115)(18 95 116)(19 96 117)(20 89 118)(21 90 119)(22 91 120)(23 92 113)(24 93 114)(33 56 69)(34 49 70)(35 50 71)(36 51 72)(37 52 65)(38 53 66)(39 54 67)(40 55 68)(41 57 85)(42 58 86)(43 59 87)(44 60 88)(45 61 81)(46 62 82)(47 63 83)(48 64 84)
(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 3)(4 8)(5 7)(9 13)(10 12)(14 16)(17 23)(18 22)(19 21)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(57 61)(58 60)(62 64)(65 69)(66 68)(70 72)(73 79)(74 78)(75 77)(81 85)(82 84)(86 88)(90 96)(91 95)(92 94)(98 104)(99 103)(100 102)(105 107)(108 112)(109 111)(113 115)(116 120)(117 119)
(1 109 23 42 53)(2 110 24 43 54)(3 111 17 44 55)(4 112 18 45 56)(5 105 19 46 49)(6 106 20 47 50)(7 107 21 48 51)(8 108 22 41 52)(9 91 57 65 103)(10 92 58 66 104)(11 93 59 67 97)(12 94 60 68 98)(13 95 61 69 99)(14 96 62 70 100)(15 89 63 71 101)(16 90 64 72 102)(25 114 87 39 80)(26 115 88 40 73)(27 116 81 33 74)(28 117 82 34 75)(29 118 83 35 76)(30 119 84 36 77)(31 120 85 37 78)(32 113 86 38 79)
(1 53)(2 50)(3 55)(4 52)(5 49)(6 54)(7 51)(8 56)(9 61)(10 58)(11 63)(12 60)(13 57)(14 62)(15 59)(16 64)(18 22)(20 24)(25 83)(26 88)(27 85)(28 82)(29 87)(30 84)(31 81)(32 86)(33 78)(34 75)(35 80)(36 77)(37 74)(38 79)(39 76)(40 73)(41 112)(42 109)(43 106)(44 111)(45 108)(46 105)(47 110)(48 107)(65 99)(66 104)(67 101)(68 98)(69 103)(70 100)(71 97)(72 102)(89 93)(91 95)(114 118)(116 120)
G:=sub<Sym(120)| (1,104,79)(2,97,80)(3,98,73)(4,99,74)(5,100,75)(6,101,76)(7,102,77)(8,103,78)(9,31,108)(10,32,109)(11,25,110)(12,26,111)(13,27,112)(14,28,105)(15,29,106)(16,30,107)(17,94,115)(18,95,116)(19,96,117)(20,89,118)(21,90,119)(22,91,120)(23,92,113)(24,93,114)(33,56,69)(34,49,70)(35,50,71)(36,51,72)(37,52,65)(38,53,66)(39,54,67)(40,55,68)(41,57,85)(42,58,86)(43,59,87)(44,60,88)(45,61,81)(46,62,82)(47,63,83)(48,64,84), (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), (1,3)(4,8)(5,7)(9,13)(10,12)(14,16)(17,23)(18,22)(19,21)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(73,79)(74,78)(75,77)(81,85)(82,84)(86,88)(90,96)(91,95)(92,94)(98,104)(99,103)(100,102)(105,107)(108,112)(109,111)(113,115)(116,120)(117,119), (1,109,23,42,53)(2,110,24,43,54)(3,111,17,44,55)(4,112,18,45,56)(5,105,19,46,49)(6,106,20,47,50)(7,107,21,48,51)(8,108,22,41,52)(9,91,57,65,103)(10,92,58,66,104)(11,93,59,67,97)(12,94,60,68,98)(13,95,61,69,99)(14,96,62,70,100)(15,89,63,71,101)(16,90,64,72,102)(25,114,87,39,80)(26,115,88,40,73)(27,116,81,33,74)(28,117,82,34,75)(29,118,83,35,76)(30,119,84,36,77)(31,120,85,37,78)(32,113,86,38,79), (1,53)(2,50)(3,55)(4,52)(5,49)(6,54)(7,51)(8,56)(9,61)(10,58)(11,63)(12,60)(13,57)(14,62)(15,59)(16,64)(18,22)(20,24)(25,83)(26,88)(27,85)(28,82)(29,87)(30,84)(31,81)(32,86)(33,78)(34,75)(35,80)(36,77)(37,74)(38,79)(39,76)(40,73)(41,112)(42,109)(43,106)(44,111)(45,108)(46,105)(47,110)(48,107)(65,99)(66,104)(67,101)(68,98)(69,103)(70,100)(71,97)(72,102)(89,93)(91,95)(114,118)(116,120)>;
G:=Group( (1,104,79)(2,97,80)(3,98,73)(4,99,74)(5,100,75)(6,101,76)(7,102,77)(8,103,78)(9,31,108)(10,32,109)(11,25,110)(12,26,111)(13,27,112)(14,28,105)(15,29,106)(16,30,107)(17,94,115)(18,95,116)(19,96,117)(20,89,118)(21,90,119)(22,91,120)(23,92,113)(24,93,114)(33,56,69)(34,49,70)(35,50,71)(36,51,72)(37,52,65)(38,53,66)(39,54,67)(40,55,68)(41,57,85)(42,58,86)(43,59,87)(44,60,88)(45,61,81)(46,62,82)(47,63,83)(48,64,84), (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), (1,3)(4,8)(5,7)(9,13)(10,12)(14,16)(17,23)(18,22)(19,21)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(73,79)(74,78)(75,77)(81,85)(82,84)(86,88)(90,96)(91,95)(92,94)(98,104)(99,103)(100,102)(105,107)(108,112)(109,111)(113,115)(116,120)(117,119), (1,109,23,42,53)(2,110,24,43,54)(3,111,17,44,55)(4,112,18,45,56)(5,105,19,46,49)(6,106,20,47,50)(7,107,21,48,51)(8,108,22,41,52)(9,91,57,65,103)(10,92,58,66,104)(11,93,59,67,97)(12,94,60,68,98)(13,95,61,69,99)(14,96,62,70,100)(15,89,63,71,101)(16,90,64,72,102)(25,114,87,39,80)(26,115,88,40,73)(27,116,81,33,74)(28,117,82,34,75)(29,118,83,35,76)(30,119,84,36,77)(31,120,85,37,78)(32,113,86,38,79), (1,53)(2,50)(3,55)(4,52)(5,49)(6,54)(7,51)(8,56)(9,61)(10,58)(11,63)(12,60)(13,57)(14,62)(15,59)(16,64)(18,22)(20,24)(25,83)(26,88)(27,85)(28,82)(29,87)(30,84)(31,81)(32,86)(33,78)(34,75)(35,80)(36,77)(37,74)(38,79)(39,76)(40,73)(41,112)(42,109)(43,106)(44,111)(45,108)(46,105)(47,110)(48,107)(65,99)(66,104)(67,101)(68,98)(69,103)(70,100)(71,97)(72,102)(89,93)(91,95)(114,118)(116,120) );
G=PermutationGroup([[(1,104,79),(2,97,80),(3,98,73),(4,99,74),(5,100,75),(6,101,76),(7,102,77),(8,103,78),(9,31,108),(10,32,109),(11,25,110),(12,26,111),(13,27,112),(14,28,105),(15,29,106),(16,30,107),(17,94,115),(18,95,116),(19,96,117),(20,89,118),(21,90,119),(22,91,120),(23,92,113),(24,93,114),(33,56,69),(34,49,70),(35,50,71),(36,51,72),(37,52,65),(38,53,66),(39,54,67),(40,55,68),(41,57,85),(42,58,86),(43,59,87),(44,60,88),(45,61,81),(46,62,82),(47,63,83),(48,64,84)], [(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)], [(1,3),(4,8),(5,7),(9,13),(10,12),(14,16),(17,23),(18,22),(19,21),(26,32),(27,31),(28,30),(33,37),(34,36),(38,40),(41,45),(42,44),(46,48),(49,51),(52,56),(53,55),(57,61),(58,60),(62,64),(65,69),(66,68),(70,72),(73,79),(74,78),(75,77),(81,85),(82,84),(86,88),(90,96),(91,95),(92,94),(98,104),(99,103),(100,102),(105,107),(108,112),(109,111),(113,115),(116,120),(117,119)], [(1,109,23,42,53),(2,110,24,43,54),(3,111,17,44,55),(4,112,18,45,56),(5,105,19,46,49),(6,106,20,47,50),(7,107,21,48,51),(8,108,22,41,52),(9,91,57,65,103),(10,92,58,66,104),(11,93,59,67,97),(12,94,60,68,98),(13,95,61,69,99),(14,96,62,70,100),(15,89,63,71,101),(16,90,64,72,102),(25,114,87,39,80),(26,115,88,40,73),(27,116,81,33,74),(28,117,82,34,75),(29,118,83,35,76),(30,119,84,36,77),(31,120,85,37,78),(32,113,86,38,79)], [(1,53),(2,50),(3,55),(4,52),(5,49),(6,54),(7,51),(8,56),(9,61),(10,58),(11,63),(12,60),(13,57),(14,62),(15,59),(16,64),(18,22),(20,24),(25,83),(26,88),(27,85),(28,82),(29,87),(30,84),(31,81),(32,86),(33,78),(34,75),(35,80),(36,77),(37,74),(38,79),(39,76),(40,73),(41,112),(42,109),(43,106),(44,111),(45,108),(46,105),(47,110),(48,107),(65,99),(66,104),(67,101),(68,98),(69,103),(70,100),(71,97),(72,102),(89,93),(91,95),(114,118),(116,120)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 4 | 10 | 20 | 1 | 1 | 2 | 10 | 20 | 2 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 4 | 20 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
75 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 | 4 | 4 | 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 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C6×D5 | C6×D5 | C8⋊C22 | D4×D5 | C3×C8⋊C22 | D8⋊D5 | C3×D4×D5 | C3×D8⋊D5 |
| kernel | C3×D8⋊D5 | C3×C8⋊D5 | C3×C40⋊C2 | C3×D4⋊D5 | C3×D4.D5 | C15×D8 | C3×D4×D5 | C3×D4⋊2D5 | D8⋊D5 | C8⋊D5 | C40⋊C2 | D4⋊D5 | D4.D5 | C5×D8 | D4×D5 | D4⋊2D5 | C3×Dic5 | C6×D5 | C3×D8 | C24 | C3×D4 | Dic5 | D10 | D8 | C8 | D4 | C15 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×D8⋊D5 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 173 | 151 |
| 0 | 0 | 173 | 22 |
| 55 | 225 | 15 | 144 |
| 170 | 71 | 211 | 226 |
| 1 | 0 | 7 | 7 |
| 0 | 1 | 234 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 189 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 |
| 0 | 0 | 191 | 190 |
| 189 | 240 | 0 | 0 |
| 52 | 52 | 0 | 0 |
| 0 | 0 | 0 | 189 |
| 0 | 0 | 190 | 0 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,0,55,170,0,0,225,71,173,173,15,211,151,22,144,226],[1,0,0,0,0,1,0,0,7,234,240,0,7,0,0,240],[189,1,0,0,240,0,0,0,0,0,240,191,0,0,240,190],[189,52,0,0,240,52,0,0,0,0,0,190,0,0,189,0] >;
C3×D8⋊D5 in GAP, Magma, Sage, TeX
C_3\times D_8\rtimes D_5
% in TeX
G:=Group("C3xD8:D5"); // GroupNames label
G:=SmallGroup(480,704);
// by ID
G=gap.SmallGroup(480,704);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,303,1271,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^5=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,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations