direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C5⋊C8, C5⋊C24, C15⋊2C8, C10.C12, C6.2F5, C30.2C4, Dic5.2C6, C2.(C3×F5), (C3×Dic5).4C2, SmallGroup(120,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×C5⋊C8 |
Generators and relations for C3×C5⋊C8
G = < a,b,c | a3=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C3×C5⋊C8
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | i | -i | -i | i | -i | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | -i | i | i | -i | i | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | -i | i | -i | i | 1 | 1 | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ83 | -1 | -1 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | i | -i | i | -i | 1 | 1 | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ85 | -1 | -1 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | -i | i | -i | i | 1 | 1 | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ87 | -1 | -1 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | i | -i | i | -i | 1 | 1 | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ8 | -1 | -1 | linear of order 8 |
| ρ13 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | i | -i | i | -i | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ14 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | i | -i | i | -i | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | -i | i | -i | i | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ16 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | -i | i | -i | i | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ17 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | ζ6 | ζ65 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | ζ82ζ32 | ζ86ζ3 | ζ82ζ3 | ζ86ζ32 | ζ32 | ζ3 | ζ83ζ3 | ζ8ζ32 | ζ87ζ32 | ζ83ζ32 | ζ85ζ3 | ζ8ζ3 | ζ87ζ3 | ζ85ζ32 | ζ6 | ζ65 | linear of order 24 |
| ρ18 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | ζ65 | ζ6 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | ζ82ζ3 | ζ86ζ32 | ζ82ζ32 | ζ86ζ3 | ζ3 | ζ32 | ζ87ζ32 | ζ85ζ3 | ζ83ζ3 | ζ87ζ3 | ζ8ζ32 | ζ85ζ32 | ζ83ζ32 | ζ8ζ3 | ζ65 | ζ6 | linear of order 24 |
| ρ19 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | ζ65 | ζ6 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | ζ86ζ3 | ζ82ζ32 | ζ86ζ32 | ζ82ζ3 | ζ3 | ζ32 | ζ8ζ32 | ζ83ζ3 | ζ85ζ3 | ζ8ζ3 | ζ87ζ32 | ζ83ζ32 | ζ85ζ32 | ζ87ζ3 | ζ65 | ζ6 | linear of order 24 |
| ρ20 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | ζ65 | ζ6 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | ζ82ζ3 | ζ86ζ32 | ζ82ζ32 | ζ86ζ3 | ζ3 | ζ32 | ζ83ζ32 | ζ8ζ3 | ζ87ζ3 | ζ83ζ3 | ζ85ζ32 | ζ8ζ32 | ζ87ζ32 | ζ85ζ3 | ζ65 | ζ6 | linear of order 24 |
| ρ21 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | ζ65 | ζ6 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | ζ86ζ3 | ζ82ζ32 | ζ86ζ32 | ζ82ζ3 | ζ3 | ζ32 | ζ85ζ32 | ζ87ζ3 | ζ8ζ3 | ζ85ζ3 | ζ83ζ32 | ζ87ζ32 | ζ8ζ32 | ζ83ζ3 | ζ65 | ζ6 | linear of order 24 |
| ρ22 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | ζ6 | ζ65 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | ζ82ζ32 | ζ86ζ3 | ζ82ζ3 | ζ86ζ32 | ζ32 | ζ3 | ζ87ζ3 | ζ85ζ32 | ζ83ζ32 | ζ87ζ32 | ζ8ζ3 | ζ85ζ3 | ζ83ζ3 | ζ8ζ32 | ζ6 | ζ65 | linear of order 24 |
| ρ23 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | ζ6 | ζ65 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | ζ86ζ32 | ζ82ζ3 | ζ86ζ3 | ζ82ζ32 | ζ32 | ζ3 | ζ85ζ3 | ζ87ζ32 | ζ8ζ32 | ζ85ζ32 | ζ83ζ3 | ζ87ζ3 | ζ8ζ3 | ζ83ζ32 | ζ6 | ζ65 | linear of order 24 |
| ρ24 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | ζ6 | ζ65 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | ζ86ζ32 | ζ82ζ3 | ζ86ζ3 | ζ82ζ32 | ζ32 | ζ3 | ζ8ζ3 | ζ83ζ32 | ζ85ζ32 | ζ8ζ32 | ζ87ζ3 | ζ83ζ3 | ζ85ζ3 | ζ87ζ32 | ζ6 | ζ65 | linear of order 24 |
| ρ25 | 4 | 4 | 4 | 4 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ26 | 4 | -4 | 4 | 4 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ27 | 4 | 4 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex lifted from C3×F5 |
| ρ28 | 4 | -4 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 2-2√-3 | 2+2√-3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | complex faithful |
| ρ29 | 4 | -4 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 2+2√-3 | 2-2√-3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | complex faithful |
| ρ30 | 4 | 4 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex lifted from C3×F5 |
►Regular action on 120 points
Generators in S120
(1 100 65)(2 101 66)(3 102 67)(4 103 68)(5 104 69)(6 97 70)(7 98 71)(8 99 72)(9 86 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 107 58)(18 108 59)(19 109 60)(20 110 61)(21 111 62)(22 112 63)(23 105 64)(24 106 57)(25 96 113)(26 89 114)(27 90 115)(28 91 116)(29 92 117)(30 93 118)(31 94 119)(32 95 120)(33 43 74)(34 44 75)(35 45 76)(36 46 77)(37 47 78)(38 48 79)(39 41 80)(40 42 73)
(1 105 29 51 44)(2 52 106 45 30)(3 46 53 31 107)(4 32 47 108 54)(5 109 25 55 48)(6 56 110 41 26)(7 42 49 27 111)(8 28 43 112 50)(9 75 100 64 92)(10 57 76 93 101)(11 94 58 102 77)(12 103 95 78 59)(13 79 104 60 96)(14 61 80 89 97)(15 90 62 98 73)(16 99 91 74 63)(17 67 36 88 119)(18 81 68 120 37)(19 113 82 38 69)(20 39 114 70 83)(21 71 40 84 115)(22 85 72 116 33)(23 117 86 34 65)(24 35 118 66 87)
(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)
G:=sub<Sym(120)| (1,100,65)(2,101,66)(3,102,67)(4,103,68)(5,104,69)(6,97,70)(7,98,71)(8,99,72)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,107,58)(18,108,59)(19,109,60)(20,110,61)(21,111,62)(22,112,63)(23,105,64)(24,106,57)(25,96,113)(26,89,114)(27,90,115)(28,91,116)(29,92,117)(30,93,118)(31,94,119)(32,95,120)(33,43,74)(34,44,75)(35,45,76)(36,46,77)(37,47,78)(38,48,79)(39,41,80)(40,42,73), (1,105,29,51,44)(2,52,106,45,30)(3,46,53,31,107)(4,32,47,108,54)(5,109,25,55,48)(6,56,110,41,26)(7,42,49,27,111)(8,28,43,112,50)(9,75,100,64,92)(10,57,76,93,101)(11,94,58,102,77)(12,103,95,78,59)(13,79,104,60,96)(14,61,80,89,97)(15,90,62,98,73)(16,99,91,74,63)(17,67,36,88,119)(18,81,68,120,37)(19,113,82,38,69)(20,39,114,70,83)(21,71,40,84,115)(22,85,72,116,33)(23,117,86,34,65)(24,35,118,66,87), (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)>;
G:=Group( (1,100,65)(2,101,66)(3,102,67)(4,103,68)(5,104,69)(6,97,70)(7,98,71)(8,99,72)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,107,58)(18,108,59)(19,109,60)(20,110,61)(21,111,62)(22,112,63)(23,105,64)(24,106,57)(25,96,113)(26,89,114)(27,90,115)(28,91,116)(29,92,117)(30,93,118)(31,94,119)(32,95,120)(33,43,74)(34,44,75)(35,45,76)(36,46,77)(37,47,78)(38,48,79)(39,41,80)(40,42,73), (1,105,29,51,44)(2,52,106,45,30)(3,46,53,31,107)(4,32,47,108,54)(5,109,25,55,48)(6,56,110,41,26)(7,42,49,27,111)(8,28,43,112,50)(9,75,100,64,92)(10,57,76,93,101)(11,94,58,102,77)(12,103,95,78,59)(13,79,104,60,96)(14,61,80,89,97)(15,90,62,98,73)(16,99,91,74,63)(17,67,36,88,119)(18,81,68,120,37)(19,113,82,38,69)(20,39,114,70,83)(21,71,40,84,115)(22,85,72,116,33)(23,117,86,34,65)(24,35,118,66,87), (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) );
G=PermutationGroup([[(1,100,65),(2,101,66),(3,102,67),(4,103,68),(5,104,69),(6,97,70),(7,98,71),(8,99,72),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,107,58),(18,108,59),(19,109,60),(20,110,61),(21,111,62),(22,112,63),(23,105,64),(24,106,57),(25,96,113),(26,89,114),(27,90,115),(28,91,116),(29,92,117),(30,93,118),(31,94,119),(32,95,120),(33,43,74),(34,44,75),(35,45,76),(36,46,77),(37,47,78),(38,48,79),(39,41,80),(40,42,73)], [(1,105,29,51,44),(2,52,106,45,30),(3,46,53,31,107),(4,32,47,108,54),(5,109,25,55,48),(6,56,110,41,26),(7,42,49,27,111),(8,28,43,112,50),(9,75,100,64,92),(10,57,76,93,101),(11,94,58,102,77),(12,103,95,78,59),(13,79,104,60,96),(14,61,80,89,97),(15,90,62,98,73),(16,99,91,74,63),(17,67,36,88,119),(18,81,68,120,37),(19,113,82,38,69),(20,39,114,70,83),(21,71,40,84,115),(22,85,72,116,33),(23,117,86,34,65),(24,35,118,66,87)], [(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)]])
C3×C5⋊C8 is a maximal subgroup of
D15⋊C8 D6.F5 Dic3.F5 SL2(𝔽3).F5
Matrix representation of C3×C5⋊C8 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 3 | 3 | 3 |
| 6 | 3 | 6 | 6 |
| 1 | 3 | 3 | 2 |
| 6 | 0 | 5 | 5 |
| 6 | 4 | 0 | 4 |
| 5 | 6 | 4 | 3 |
| 0 | 0 | 1 | 3 |
| 6 | 5 | 3 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,6,1,6,3,3,3,0,3,6,3,5,3,6,2,5],[6,5,0,6,4,6,0,5,0,4,1,3,4,3,3,1] >;
C3×C5⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes C_8
% in TeX
G:=Group("C3xC5:C8"); // GroupNames label
G:=SmallGroup(120,6);
// by ID
G=gap.SmallGroup(120,6);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-5,30,42,1204,414]);
// Polycyclic
G:=Group<a,b,c|a^3=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C3×C5⋊C8 in TeX
Character table of C3×C5⋊C8 in TeX