direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D5×D8, D40⋊4C6, C24⋊24D10, C120⋊17C22, C60.190C23, C5⋊2(C6×D8), C8⋊4(C6×D5), C40⋊2(C2×C6), D4⋊D5⋊1C6, (D4×D5)⋊1C6, D4⋊1(C6×D5), (C5×D8)⋊2C6, (C8×D5)⋊1C6, C15⋊14(C2×D8), (C15×D8)⋊6C2, (D5×C24)⋊6C2, D20⋊1(C2×C6), (C3×D40)⋊12C2, (C3×D4)⋊16D10, (C6×D5).85D4, C10.27(C6×D4), C6.181(D4×D5), D10.23(C3×D4), C30.340(C2×D4), C20.1(C22×C6), Dic5.7(C3×D4), (D4×C15)⋊16C22, (C3×D20)⋊18C22, (C3×Dic5).54D4, C12.190(C22×D5), (D5×C12).104C22, (C3×D4×D5)⋊8C2, C4.1(D5×C2×C6), C2.15(C3×D4×D5), C5⋊2C8⋊5(C2×C6), (C3×D4⋊D5)⋊9C2, (C5×D4)⋊1(C2×C6), (C4×D5).15(C2×C6), (C3×C5⋊2C8)⋊38C22, SmallGroup(480,703)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D5×D8
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=d-1 >
Subgroups: 688 in 152 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C15, C2×C8, D8, D8, C2×D4, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C2×D8, C5⋊2C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C2×C24, C3×D8, C3×D8, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, C6×D8, C3×C5⋊2C8, C120, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, D5×C2×C6, D5×D8, D5×C24, C3×D40, C3×D4⋊D5, C15×D8, C3×D4×D5, C3×D5×D8
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, D8, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×D8, C22×D5, C3×D8, C6×D4, C6×D5, D4×D5, C6×D8, D5×C2×C6, D5×D8, C3×D4×D5, C3×D5×D8
►On 120 points
Generators in S120
(1 104 64)(2 97 57)(3 98 58)(4 99 59)(5 100 60)(6 101 61)(7 102 62)(8 103 63)(9 83 53)(10 84 54)(11 85 55)(12 86 56)(13 87 49)(14 88 50)(15 81 51)(16 82 52)(17 89 118)(18 90 119)(19 91 120)(20 92 113)(21 93 114)(22 94 115)(23 95 116)(24 96 117)(25 44 75)(26 45 76)(27 46 77)(28 47 78)(29 48 79)(30 41 80)(31 42 73)(32 43 74)(33 108 69)(34 109 70)(35 110 71)(36 111 72)(37 112 65)(38 105 66)(39 106 67)(40 107 68)
(1 105 42 20 54)(2 106 43 21 55)(3 107 44 22 56)(4 108 45 23 49)(5 109 46 24 50)(6 110 47 17 51)(7 111 48 18 52)(8 112 41 19 53)(9 103 65 80 91)(10 104 66 73 92)(11 97 67 74 93)(12 98 68 75 94)(13 99 69 76 95)(14 100 70 77 96)(15 101 71 78 89)(16 102 72 79 90)(25 115 86 58 40)(26 116 87 59 33)(27 117 88 60 34)(28 118 81 61 35)(29 119 82 62 36)(30 120 83 63 37)(31 113 84 64 38)(32 114 85 57 39)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 103)(10 104)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(33 116)(34 117)(35 118)(36 119)(37 120)(38 113)(39 114)(40 115)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)
(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 4)(2 3)(5 8)(6 7)(9 14)(10 13)(11 12)(15 16)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 38)(34 37)(35 36)(39 40)(41 46)(42 45)(43 44)(47 48)(49 54)(50 53)(51 52)(55 56)(57 58)(59 64)(60 63)(61 62)(65 70)(66 69)(67 68)(71 72)(73 76)(74 75)(77 80)(78 79)(81 82)(83 88)(84 87)(85 86)(89 90)(91 96)(92 95)(93 94)(97 98)(99 104)(100 103)(101 102)(105 108)(106 107)(109 112)(110 111)(113 116)(114 115)(117 120)(118 119)
G:=sub<Sym(120)| (1,104,64)(2,97,57)(3,98,58)(4,99,59)(5,100,60)(6,101,61)(7,102,62)(8,103,63)(9,83,53)(10,84,54)(11,85,55)(12,86,56)(13,87,49)(14,88,50)(15,81,51)(16,82,52)(17,89,118)(18,90,119)(19,91,120)(20,92,113)(21,93,114)(22,94,115)(23,95,116)(24,96,117)(25,44,75)(26,45,76)(27,46,77)(28,47,78)(29,48,79)(30,41,80)(31,42,73)(32,43,74)(33,108,69)(34,109,70)(35,110,71)(36,111,72)(37,112,65)(38,105,66)(39,106,67)(40,107,68), (1,105,42,20,54)(2,106,43,21,55)(3,107,44,22,56)(4,108,45,23,49)(5,109,46,24,50)(6,110,47,17,51)(7,111,48,18,52)(8,112,41,19,53)(9,103,65,80,91)(10,104,66,73,92)(11,97,67,74,93)(12,98,68,75,94)(13,99,69,76,95)(14,100,70,77,96)(15,101,71,78,89)(16,102,72,79,90)(25,115,86,58,40)(26,116,87,59,33)(27,117,88,60,34)(28,118,81,61,35)(29,119,82,62,36)(30,120,83,63,37)(31,113,84,64,38)(32,114,85,57,39), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,103)(10,104)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(33,116)(34,117)(35,118)(36,119)(37,120)(38,113)(39,114)(40,115)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90), (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), (1,4)(2,3)(5,8)(6,7)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,38)(34,37)(35,36)(39,40)(41,46)(42,45)(43,44)(47,48)(49,54)(50,53)(51,52)(55,56)(57,58)(59,64)(60,63)(61,62)(65,70)(66,69)(67,68)(71,72)(73,76)(74,75)(77,80)(78,79)(81,82)(83,88)(84,87)(85,86)(89,90)(91,96)(92,95)(93,94)(97,98)(99,104)(100,103)(101,102)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)>;
G:=Group( (1,104,64)(2,97,57)(3,98,58)(4,99,59)(5,100,60)(6,101,61)(7,102,62)(8,103,63)(9,83,53)(10,84,54)(11,85,55)(12,86,56)(13,87,49)(14,88,50)(15,81,51)(16,82,52)(17,89,118)(18,90,119)(19,91,120)(20,92,113)(21,93,114)(22,94,115)(23,95,116)(24,96,117)(25,44,75)(26,45,76)(27,46,77)(28,47,78)(29,48,79)(30,41,80)(31,42,73)(32,43,74)(33,108,69)(34,109,70)(35,110,71)(36,111,72)(37,112,65)(38,105,66)(39,106,67)(40,107,68), (1,105,42,20,54)(2,106,43,21,55)(3,107,44,22,56)(4,108,45,23,49)(5,109,46,24,50)(6,110,47,17,51)(7,111,48,18,52)(8,112,41,19,53)(9,103,65,80,91)(10,104,66,73,92)(11,97,67,74,93)(12,98,68,75,94)(13,99,69,76,95)(14,100,70,77,96)(15,101,71,78,89)(16,102,72,79,90)(25,115,86,58,40)(26,116,87,59,33)(27,117,88,60,34)(28,118,81,61,35)(29,119,82,62,36)(30,120,83,63,37)(31,113,84,64,38)(32,114,85,57,39), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,103)(10,104)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(33,116)(34,117)(35,118)(36,119)(37,120)(38,113)(39,114)(40,115)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90), (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), (1,4)(2,3)(5,8)(6,7)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,38)(34,37)(35,36)(39,40)(41,46)(42,45)(43,44)(47,48)(49,54)(50,53)(51,52)(55,56)(57,58)(59,64)(60,63)(61,62)(65,70)(66,69)(67,68)(71,72)(73,76)(74,75)(77,80)(78,79)(81,82)(83,88)(84,87)(85,86)(89,90)(91,96)(92,95)(93,94)(97,98)(99,104)(100,103)(101,102)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119) );
G=PermutationGroup([[(1,104,64),(2,97,57),(3,98,58),(4,99,59),(5,100,60),(6,101,61),(7,102,62),(8,103,63),(9,83,53),(10,84,54),(11,85,55),(12,86,56),(13,87,49),(14,88,50),(15,81,51),(16,82,52),(17,89,118),(18,90,119),(19,91,120),(20,92,113),(21,93,114),(22,94,115),(23,95,116),(24,96,117),(25,44,75),(26,45,76),(27,46,77),(28,47,78),(29,48,79),(30,41,80),(31,42,73),(32,43,74),(33,108,69),(34,109,70),(35,110,71),(36,111,72),(37,112,65),(38,105,66),(39,106,67),(40,107,68)], [(1,105,42,20,54),(2,106,43,21,55),(3,107,44,22,56),(4,108,45,23,49),(5,109,46,24,50),(6,110,47,17,51),(7,111,48,18,52),(8,112,41,19,53),(9,103,65,80,91),(10,104,66,73,92),(11,97,67,74,93),(12,98,68,75,94),(13,99,69,76,95),(14,100,70,77,96),(15,101,71,78,89),(16,102,72,79,90),(25,115,86,58,40),(26,116,87,59,33),(27,117,88,60,34),(28,118,81,61,35),(29,119,82,62,36),(30,120,83,63,37),(31,113,84,64,38),(32,114,85,57,39)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,103),(10,104),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,110),(18,111),(19,112),(20,105),(21,106),(22,107),(23,108),(24,109),(33,116),(34,117),(35,118),(36,119),(37,120),(38,113),(39,114),(40,115),(57,85),(58,86),(59,87),(60,88),(61,81),(62,82),(63,83),(64,84),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,89),(72,90)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12),(15,16),(17,18),(19,24),(20,23),(21,22),(25,32),(26,31),(27,30),(28,29),(33,38),(34,37),(35,36),(39,40),(41,46),(42,45),(43,44),(47,48),(49,54),(50,53),(51,52),(55,56),(57,58),(59,64),(60,63),(61,62),(65,70),(66,69),(67,68),(71,72),(73,76),(74,75),(77,80),(78,79),(81,82),(83,88),(84,87),(85,86),(89,90),(91,96),(92,95),(93,94),(97,98),(99,104),(100,103),(101,102),(105,108),(106,107),(109,112),(110,111),(113,116),(114,115),(117,120),(118,119)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 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 | 5 | 5 | 20 | 20 | 1 | 1 | 2 | 10 | 2 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | 10 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D8 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C3×D8 | C6×D5 | C6×D5 | D4×D5 | D5×D8 | C3×D4×D5 | C3×D5×D8 |
| kernel | C3×D5×D8 | D5×C24 | C3×D40 | C3×D4⋊D5 | C15×D8 | C3×D4×D5 | D5×D8 | C8×D5 | D40 | D4⋊D5 | C5×D8 | D4×D5 | C3×Dic5 | C6×D5 | C3×D8 | C3×D5 | C24 | C3×D4 | Dic5 | D10 | D8 | D5 | C8 | D4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 4 | 8 | 4 | 8 | 2 | 4 | 4 | 8 |
Matrix representation of C3×D5×D8 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 1 | 0 | 0 |
| 240 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 230 | 11 |
| 0 | 0 | 230 | 230 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 11 | 230 |
| 0 | 0 | 230 | 230 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,11,230,0,0,230,230] >;
C3×D5×D8 in GAP, Magma, Sage, TeX
C_3\times D_5\times D_8
% in TeX
G:=Group("C3xD5xD8"); // GroupNames label
G:=SmallGroup(480,703);
// by ID
G=gap.SmallGroup(480,703);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,303,1271,648,102,18822]);
// 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^-1>;
// generators/relations