metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊F5, D10.7D6, Dic15⋊2C4, D5.1Dic6, C15⋊(C4⋊C4), C3⋊2(C4⋊F5), (C2×F5).S3, (C3×D5).Q8, C5⋊(Dic3⋊C4), C2.5(S3×F5), C6.3(C2×F5), C10.3(C4×S3), C30.3(C2×C4), (C3×D5).2D4, (C6×F5).2C2, (C5×Dic3)⋊2C4, D5.1(C3⋊D4), (D5×Dic3).3C2, (C6×D5).7C22, (C2×C3⋊F5).2C2, SmallGroup(240,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊F5
G = < a,b,c,d | a6=c5=d4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c3 >
5C2
5C2
3C4
5C22
10C4
15C4
30C4
5C6
5C6
15C2×C4
15C2×C4
5Dic3
10Dic3
10C12
2F5
3Dic5
3C20
6F5
15C4⋊C4
Character table of Dic3⋊F5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 10 | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 30 | |
| size | 1 | 1 | 5 | 5 | 2 | 6 | 10 | 10 | 30 | 30 | 30 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | 1 | i | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | 1 | -i | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | -1 | i | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -1 | -i | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | -1 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | √3 | -√3 | √3 | -√3 | -1 | 0 | 0 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | -√3 | √3 | -√3 | √3 | -1 | 0 | 0 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 2 | -i | i | i | -i | -1 | 0 | 0 | -1 | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 2 | i | -i | -i | i | -1 | 0 | 0 | -1 | complex lifted from C4×S3 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | -√-3 | -√-3 | √-3 | √-3 | -1 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | √-3 | √-3 | -√-3 | -√-3 | -1 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ20 | 4 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×F5 |
| ρ21 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -√-5 | √-5 | 1 | complex lifted from C4⋊F5 |
| ρ22 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | √-5 | -√-5 | 1 | complex lifted from C4⋊F5 |
| ρ23 | 8 | 8 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | orthogonal lifted from S3×F5 |
| ρ24 | 8 | -8 | 0 | 0 | -4 | 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)(2 57)(3 58)(4 59)(5 60)(6 55)(7 33 14 41)(8 34 15 42)(9 35 16 37)(10 36 17 38)(11 31 18 39)(12 32 13 40)(19 53 27 46)(20 54 28 47)(21 49 29 48)(22 50 30 43)(23 51 25 44)(24 52 26 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)(2,57)(3,58)(4,59)(5,60)(6,55)(7,33,14,41)(8,34,15,42)(9,35,16,37)(10,36,17,38)(11,31,18,39)(12,32,13,40)(19,53,27,46)(20,54,28,47)(21,49,29,48)(22,50,30,43)(23,51,25,44)(24,52,26,45)>;
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)(2,57)(3,58)(4,59)(5,60)(6,55)(7,33,14,41)(8,34,15,42)(9,35,16,37)(10,36,17,38)(11,31,18,39)(12,32,13,40)(19,53,27,46)(20,54,28,47)(21,49,29,48)(22,50,30,43)(23,51,25,44)(24,52,26,45) );
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),(2,57),(3,58),(4,59),(5,60),(6,55),(7,33,14,41),(8,34,15,42),(9,35,16,37),(10,36,17,38),(11,31,18,39),(12,32,13,40),(19,53,27,46),(20,54,28,47),(21,49,29,48),(22,50,30,43),(23,51,25,44),(24,52,26,45)]])
Dic3⋊F5 is a maximal subgroup of
F5×Dic6 Dic6⋊5F5 (C4×S3)⋊F5 S3×C4⋊F5 C22⋊F5.S3 F5×C3⋊D4 C3⋊D4⋊F5
Dic3⋊F5 is a maximal quotient of Dic5.Dic6 Dic5.4Dic6 D10.Dic6 D10.2Dic6 D10.20D12 C30.4M4(2) Dic15⋊C8
Matrix representation of Dic3⋊F5 ►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 |
| 52 | 43 | 0 | 0 | 0 | 0 |
| 52 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 32 | 32 |
| 0 | 0 | 29 | 45 | 29 | 0 |
| 0 | 0 | 0 | 29 | 45 | 29 |
| 0 | 0 | 32 | 32 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 | 60 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 38 | 15 | 0 | 0 | 0 | 0 |
| 46 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 29 | 45 | 29 |
| 0 | 0 | 16 | 0 | 32 | 32 |
| 0 | 0 | 32 | 32 | 0 | 16 |
| 0 | 0 | 29 | 45 | 29 | 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],[52,52,0,0,0,0,43,9,0,0,0,0,0,0,16,29,0,32,0,0,0,45,29,32,0,0,32,29,45,0,0,0,32,0,29,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[38,46,0,0,0,0,15,23,0,0,0,0,0,0,0,16,32,29,0,0,29,0,32,45,0,0,45,32,0,29,0,0,29,32,16,0] >;
Dic3⋊F5 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes F_5 % in TeX
G:=Group("Dic3:F5"); // GroupNames label
G:=SmallGroup(240,97);
// by ID
G=gap.SmallGroup(240,97);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,490,3461,1745]);
// 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,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of Dic3⋊F5 in TeX
Character table of Dic3⋊F5 in TeX