direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C5⋊F5, C15⋊3F5, C52⋊5C12, (C5×C15)⋊7C4, C5⋊1(C3×F5), C5⋊D5.2C6, (C3×C5⋊D5).3C2, SmallGroup(300,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3×C5⋊F5 |
Generators and relations for C3×C5⋊F5
G = < a,b,c,d | a3=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c3 >
Character table of C3×C5⋊F5
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 6A | 6B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 15I | 15J | 15K | 15L | |
| size | 1 | 25 | 1 | 1 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 25 | 25 | 25 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ15 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ16 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ17 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | orthogonal lifted from F5 |
| ρ18 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | -1 | orthogonal lifted from F5 |
| ρ19 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | -2+2√-3 | ζ65 | ζ65 | ζ6 | -2-2√-3 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×F5 |
| ρ20 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | -2+2√-3 | ζ65 | ζ6 | ζ6 | -2-2√-3 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×F5 |
| ρ21 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | -2+2√-3 | ζ65 | -2-2√-3 | ζ6 | complex lifted from C3×F5 |
| ρ22 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ65 | -2+2√-3 | ζ6 | -2-2√-3 | complex lifted from C3×F5 |
| ρ23 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ6 | -2-2√-3 | ζ65 | -2+2√-3 | complex lifted from C3×F5 |
| ρ24 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | -2-2√-3 | ζ6 | ζ6 | ζ65 | -2+2√-3 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×F5 |
| ρ25 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ65 | -2+2√-3 | ζ6 | ζ6 | ζ6 | -2-2√-3 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×F5 |
| ρ26 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2-2√-3 | ζ6 | ζ6 | ζ6 | -2+2√-3 | ζ65 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×F5 |
| ρ27 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | -2-2√-3 | ζ6 | -2+2√-3 | ζ65 | complex lifted from C3×F5 |
| ρ28 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ6 | -2-2√-3 | ζ65 | ζ65 | ζ65 | -2+2√-3 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×F5 |
| ρ29 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2+2√-3 | ζ65 | ζ65 | ζ65 | -2-2√-3 | ζ6 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×F5 |
| ρ30 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | -2-2√-3 | ζ6 | ζ65 | ζ65 | -2+2√-3 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×F5 |
►On 75 points
Generators in S75
(1 12 56)(2 13 57)(3 14 58)(4 15 59)(5 11 60)(6 55 34)(7 51 35)(8 52 31)(9 53 32)(10 54 33)(16 65 40)(17 61 36)(18 62 37)(19 63 38)(20 64 39)(21 70 45)(22 66 41)(23 67 42)(24 68 43)(25 69 44)(26 75 50)(27 71 46)(28 72 47)(29 73 48)(30 74 49)
(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)
(1 42 32 39 26)(2 43 33 40 27)(3 44 34 36 28)(4 45 35 37 29)(5 41 31 38 30)(6 17 72 14 25)(7 18 73 15 21)(8 19 74 11 22)(9 20 75 12 23)(10 16 71 13 24)(46 57 68 54 65)(47 58 69 55 61)(48 59 70 51 62)(49 60 66 52 63)(50 56 67 53 64)
(2 3 5 4)(6 74 18 24)(7 71 17 22)(8 73 16 25)(9 75 20 23)(10 72 19 21)(11 15 13 14)(26 39 42 32)(27 36 41 35)(28 38 45 33)(29 40 44 31)(30 37 43 34)(46 61 66 51)(47 63 70 54)(48 65 69 52)(49 62 68 55)(50 64 67 53)(57 58 60 59)
G:=sub<Sym(75)| (1,12,56)(2,13,57)(3,14,58)(4,15,59)(5,11,60)(6,55,34)(7,51,35)(8,52,31)(9,53,32)(10,54,33)(16,65,40)(17,61,36)(18,62,37)(19,63,38)(20,64,39)(21,70,45)(22,66,41)(23,67,42)(24,68,43)(25,69,44)(26,75,50)(27,71,46)(28,72,47)(29,73,48)(30,74,49), (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), (1,42,32,39,26)(2,43,33,40,27)(3,44,34,36,28)(4,45,35,37,29)(5,41,31,38,30)(6,17,72,14,25)(7,18,73,15,21)(8,19,74,11,22)(9,20,75,12,23)(10,16,71,13,24)(46,57,68,54,65)(47,58,69,55,61)(48,59,70,51,62)(49,60,66,52,63)(50,56,67,53,64), (2,3,5,4)(6,74,18,24)(7,71,17,22)(8,73,16,25)(9,75,20,23)(10,72,19,21)(11,15,13,14)(26,39,42,32)(27,36,41,35)(28,38,45,33)(29,40,44,31)(30,37,43,34)(46,61,66,51)(47,63,70,54)(48,65,69,52)(49,62,68,55)(50,64,67,53)(57,58,60,59)>;
G:=Group( (1,12,56)(2,13,57)(3,14,58)(4,15,59)(5,11,60)(6,55,34)(7,51,35)(8,52,31)(9,53,32)(10,54,33)(16,65,40)(17,61,36)(18,62,37)(19,63,38)(20,64,39)(21,70,45)(22,66,41)(23,67,42)(24,68,43)(25,69,44)(26,75,50)(27,71,46)(28,72,47)(29,73,48)(30,74,49), (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), (1,42,32,39,26)(2,43,33,40,27)(3,44,34,36,28)(4,45,35,37,29)(5,41,31,38,30)(6,17,72,14,25)(7,18,73,15,21)(8,19,74,11,22)(9,20,75,12,23)(10,16,71,13,24)(46,57,68,54,65)(47,58,69,55,61)(48,59,70,51,62)(49,60,66,52,63)(50,56,67,53,64), (2,3,5,4)(6,74,18,24)(7,71,17,22)(8,73,16,25)(9,75,20,23)(10,72,19,21)(11,15,13,14)(26,39,42,32)(27,36,41,35)(28,38,45,33)(29,40,44,31)(30,37,43,34)(46,61,66,51)(47,63,70,54)(48,65,69,52)(49,62,68,55)(50,64,67,53)(57,58,60,59) );
G=PermutationGroup([[(1,12,56),(2,13,57),(3,14,58),(4,15,59),(5,11,60),(6,55,34),(7,51,35),(8,52,31),(9,53,32),(10,54,33),(16,65,40),(17,61,36),(18,62,37),(19,63,38),(20,64,39),(21,70,45),(22,66,41),(23,67,42),(24,68,43),(25,69,44),(26,75,50),(27,71,46),(28,72,47),(29,73,48),(30,74,49)], [(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)], [(1,42,32,39,26),(2,43,33,40,27),(3,44,34,36,28),(4,45,35,37,29),(5,41,31,38,30),(6,17,72,14,25),(7,18,73,15,21),(8,19,74,11,22),(9,20,75,12,23),(10,16,71,13,24),(46,57,68,54,65),(47,58,69,55,61),(48,59,70,51,62),(49,60,66,52,63),(50,56,67,53,64)], [(2,3,5,4),(6,74,18,24),(7,71,17,22),(8,73,16,25),(9,75,20,23),(10,72,19,21),(11,15,13,14),(26,39,42,32),(27,36,41,35),(28,38,45,33),(29,40,44,31),(30,37,43,34),(46,61,66,51),(47,63,70,54),(48,65,69,52),(49,62,68,55),(50,64,67,53),(57,58,60,59)]])
Matrix representation of C3×C5⋊F5 ►in GL8(𝔽61)
| 47 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 47 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 60 |
G:=sub<GL(8,GF(61))| [47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,60,60,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,60,60,60] >;
C3×C5⋊F5 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes F_5
% in TeX
G:=Group("C3xC5:F5"); // GroupNames label
G:=SmallGroup(300,30);
// by ID
G=gap.SmallGroup(300,30);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-5,30,483,173,3004,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C3×C5⋊F5 in TeX
Character table of C3×C5⋊F5 in TeX