direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3xF5, C15:C42, D10.5D6, Dic15:1C4, C3:F5:C4, (C3xF5):C4, C3:2(C4xF5), C5:1(C4xDic3), C2.1(S3xF5), C6.1(C2xF5), C10.1(C4xS3), C30.1(C2xC4), D5.1(C4xS3), D5.(C2xDic3), (C6xF5).1C2, (C2xF5).2S3, (C5xDic3):1C4, (D5xDic3).2C2, (C6xD5).5C22, (C3xD5).(C2xC4), (C2xC3:F5).1C2, SmallGroup(240,95)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — Dic3xF5 |
Generators and relations for Dic3xF5
G = < a,b,c,d | a6=c5=d4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 232 in 60 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2xC4, D5, C10, Dic3, Dic3, C12, C2xC6, C15, C42, Dic5, C20, F5, F5, D10, C2xDic3, C2xC12, C3xD5, C30, C4xD5, C2xF5, C2xF5, C4xDic3, C5xDic3, Dic15, C3xF5, C3:F5, C6xD5, C4xF5, D5xDic3, C6xF5, C2xC3:F5, Dic3xF5
Quotients: C1, C2, C4, C22, S3, C2xC4, Dic3, D6, C42, F5, C4xS3, C2xDic3, C2xF5, C4xDic3, C4xF5, S3xF5, Dic3xF5
Character table of Dic3xF5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 6A | 6B | 6C | 10 | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 30 | |
| size | 1 | 1 | 5 | 5 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | 15 | 15 | 15 | 15 | 15 | 4 | 2 | 10 | 10 | 4 | 10 | 10 | 10 | 10 | 8 | 12 | 12 | 8 | |
| 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 | 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 |
| ρ4 | 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 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | i | -1 | i | -1 | 1 | 1 | -i | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | i | -i | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | i | -1 | -i | -i | i | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | i | -i | -1 | 1 | -1 | 1 | i | -i | -i | i | -i | i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | -i | -1 | i | i | -i | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | i | -i | 1 | -1 | 1 | -1 | -i | -i | i | -i | i | i | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | -1 | linear of order 4 |
| ρ10 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | -i | 1 | i | 1 | -1 | -1 | -i | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | i | -i | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | i | -i | -i | i | 1 | -i | -i | i | 1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | i | -i | -1 | -i | -1 | 1 | 1 | i | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | -i | i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -i | i | 1 | -1 | 1 | -1 | i | i | -i | i | -i | -i | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | -i | i | i | -i | 1 | i | i | -i | 1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -i | i | -1 | 1 | -1 | 1 | -i | i | i | -i | i | -i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | i | 1 | -i | 1 | -1 | -1 | i | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -i | i | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | -1 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ19 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 2 | -i | -i | i | i | -1 | 0 | 0 | -1 | complex lifted from C4xS3 |
| ρ22 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 2 | i | i | -i | -i | -1 | 0 | 0 | -1 | complex lifted from C4xS3 |
| ρ23 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | i | -i | -i | i | -1 | 0 | 0 | 1 | complex lifted from C4xS3 |
| ρ24 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | -i | i | i | -i | -1 | 0 | 0 | 1 | complex lifted from C4xS3 |
| ρ25 | 4 | 4 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ26 | 4 | 4 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from C2xF5 |
| ρ27 | 4 | -4 | 0 | 0 | 4 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -i | i | 1 | complex lifted from C4xF5 |
| ρ28 | 4 | -4 | 0 | 0 | 4 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | i | -i | 1 | complex lifted from C4xF5 |
| ρ29 | 8 | 8 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | orthogonal lifted from S3xF5 |
| ρ30 | 8 | -8 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | symplectic faithful, Schur index 2 |
►On 60 points
Generators in S60
(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)
(1 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 36 28 33)(26 35 29 32)(27 34 30 31)
(1 42 53 46 34)(2 37 54 47 35)(3 38 49 48 36)(4 39 50 43 31)(5 40 51 44 32)(6 41 52 45 33)(7 14 26 55 24)(8 15 27 56 19)(9 16 28 57 20)(10 17 29 58 21)(11 18 30 59 22)(12 13 25 60 23)
(1 56 4 59)(2 57 5 60)(3 58 6 55)(7 36 17 41)(8 31 18 42)(9 32 13 37)(10 33 14 38)(11 34 15 39)(12 35 16 40)(19 50 30 46)(20 51 25 47)(21 52 26 48)(22 53 27 43)(23 54 28 44)(24 49 29 45)
G:=sub<Sym(60)| (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31), (1,42,53,46,34)(2,37,54,47,35)(3,38,49,48,36)(4,39,50,43,31)(5,40,51,44,32)(6,41,52,45,33)(7,14,26,55,24)(8,15,27,56,19)(9,16,28,57,20)(10,17,29,58,21)(11,18,30,59,22)(12,13,25,60,23), (1,56,4,59)(2,57,5,60)(3,58,6,55)(7,36,17,41)(8,31,18,42)(9,32,13,37)(10,33,14,38)(11,34,15,39)(12,35,16,40)(19,50,30,46)(20,51,25,47)(21,52,26,48)(22,53,27,43)(23,54,28,44)(24,49,29,45)>;
G:=Group( (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31), (1,42,53,46,34)(2,37,54,47,35)(3,38,49,48,36)(4,39,50,43,31)(5,40,51,44,32)(6,41,52,45,33)(7,14,26,55,24)(8,15,27,56,19)(9,16,28,57,20)(10,17,29,58,21)(11,18,30,59,22)(12,13,25,60,23), (1,56,4,59)(2,57,5,60)(3,58,6,55)(7,36,17,41)(8,31,18,42)(9,32,13,37)(10,33,14,38)(11,34,15,39)(12,35,16,40)(19,50,30,46)(20,51,25,47)(21,52,26,48)(22,53,27,43)(23,54,28,44)(24,49,29,45) );
G=PermutationGroup([[(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)], [(1,59,4,56),(2,58,5,55),(3,57,6,60),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,48,16,45),(14,47,17,44),(15,46,18,43),(19,42,22,39),(20,41,23,38),(21,40,24,37),(25,36,28,33),(26,35,29,32),(27,34,30,31)], [(1,42,53,46,34),(2,37,54,47,35),(3,38,49,48,36),(4,39,50,43,31),(5,40,51,44,32),(6,41,52,45,33),(7,14,26,55,24),(8,15,27,56,19),(9,16,28,57,20),(10,17,29,58,21),(11,18,30,59,22),(12,13,25,60,23)], [(1,56,4,59),(2,57,5,60),(3,58,6,55),(7,36,17,41),(8,31,18,42),(9,32,13,37),(10,33,14,38),(11,34,15,39),(12,35,16,40),(19,50,30,46),(20,51,25,47),(21,52,26,48),(22,53,27,43),(23,54,28,44),(24,49,29,45)]])
Dic3xF5 is a maximal subgroup of
C4:F5:3S3 Dic6:5F5 C4xS3xF5 C22:F5.S3 C3:D4:F5
Dic3xF5 is a maximal quotient of C30.C42 C30.3C42 C30.4C42 D10.20D12 C30.M4(2)
Matrix representation of Dic3xF5 ►in GL6(F61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 11 | 0 | 0 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,11,0,0,0,0,0,0,0,11,0] >;
Dic3xF5 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times F_5 % in TeX
G:=Group("Dic3xF5"); // GroupNames label
G:=SmallGroup(240,95);
// by ID
G=gap.SmallGroup(240,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,490,3461,1745]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^5=d^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export