metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊F5, D30⋊1C4, D10.6D6, D5.2D12, C5⋊(D6⋊C4), (C2×F5)⋊S3, C15⋊(C22⋊C4), (C6×F5)⋊1C2, (S3×C10)⋊1C4, C6.2(C2×F5), C2.4(S3×F5), C10.2(C4×S3), C30.2(C2×C4), (C3×D5).1D4, C3⋊1(C22⋊F5), D5.2(C3⋊D4), (C6×D5).6C22, (C2×C3⋊F5)⋊1C2, (C2×S3×D5).1C2, SmallGroup(240,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊F5
G = < a,b,c,d | a6=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c3 >
Subgroups: 392 in 68 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×F5, C2×F5, C22×D5, D6⋊C4, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C22⋊F5, C6×F5, C2×C3⋊F5, C2×S3×D5, D6⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C2×F5, D6⋊C4, C22⋊F5, S3×F5, D6⋊F5
Character table of D6⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 30 | |
| size | 1 | 1 | 5 | 5 | 6 | 30 | 2 | 10 | 10 | 30 | 30 | 4 | 2 | 10 | 10 | 4 | 12 | 12 | 10 | 10 | 10 | 10 | 8 | 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 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | 0 | 0 | √3 | -√3 | √3 | -√3 | -1 | 1 | orthogonal lifted from D12 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | 0 | 0 | -√3 | √3 | -√3 | √3 | -1 | 1 | orthogonal lifted from D12 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 2 | -1 | 1 | 1 | 2 | 0 | 0 | -i | -i | i | i | -1 | -1 | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 2 | -1 | 1 | 1 | 2 | 0 | 0 | i | i | -i | -i | -1 | -1 | complex lifted from C4×S3 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | 0 | 0 | √-3 | -√-3 | -√-3 | √-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | 0 | 0 | -√-3 | √-3 | √-3 | -√-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | 0 | 0 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ20 | 4 | 4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | -√5 | √5 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | √5 | -√5 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ23 | 8 | 8 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S3×F5 |
| ρ24 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal faithful, Schur index 2 |
►On 60 points
(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 6)(2 5)(3 4)(7 11)(8 10)(14 18)(15 17)(19 21)(22 24)(26 30)(27 29)(31 34)(32 33)(35 36)(37 38)(39 42)(40 41)(43 44)(45 48)(46 47)(49 50)(51 54)(52 53)(55 59)(56 58)
(1 38 53 44 36)(2 39 54 45 31)(3 40 49 46 32)(4 41 50 47 33)(5 42 51 48 34)(6 37 52 43 35)(7 21 29 14 55)(8 22 30 15 56)(9 23 25 16 57)(10 24 26 17 58)(11 19 27 18 59)(12 20 28 13 60)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 52 14 43)(8 53 15 44)(9 54 16 45)(10 49 17 46)(11 50 18 47)(12 51 13 48)(19 33 27 41)(20 34 28 42)(21 35 29 37)(22 36 30 38)(23 31 25 39)(24 32 26 40)
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), (1,6)(2,5)(3,4)(7,11)(8,10)(14,18)(15,17)(19,21)(22,24)(26,30)(27,29)(31,34)(32,33)(35,36)(37,38)(39,42)(40,41)(43,44)(45,48)(46,47)(49,50)(51,54)(52,53)(55,59)(56,58), (1,38,53,44,36)(2,39,54,45,31)(3,40,49,46,32)(4,41,50,47,33)(5,42,51,48,34)(6,37,52,43,35)(7,21,29,14,55)(8,22,30,15,56)(9,23,25,16,57)(10,24,26,17,58)(11,19,27,18,59)(12,20,28,13,60), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,52,14,43)(8,53,15,44)(9,54,16,45)(10,49,17,46)(11,50,18,47)(12,51,13,48)(19,33,27,41)(20,34,28,42)(21,35,29,37)(22,36,30,38)(23,31,25,39)(24,32,26,40)>;
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), (1,6)(2,5)(3,4)(7,11)(8,10)(14,18)(15,17)(19,21)(22,24)(26,30)(27,29)(31,34)(32,33)(35,36)(37,38)(39,42)(40,41)(43,44)(45,48)(46,47)(49,50)(51,54)(52,53)(55,59)(56,58), (1,38,53,44,36)(2,39,54,45,31)(3,40,49,46,32)(4,41,50,47,33)(5,42,51,48,34)(6,37,52,43,35)(7,21,29,14,55)(8,22,30,15,56)(9,23,25,16,57)(10,24,26,17,58)(11,19,27,18,59)(12,20,28,13,60), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,52,14,43)(8,53,15,44)(9,54,16,45)(10,49,17,46)(11,50,18,47)(12,51,13,48)(19,33,27,41)(20,34,28,42)(21,35,29,37)(22,36,30,38)(23,31,25,39)(24,32,26,40) );
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)], [(1,6),(2,5),(3,4),(7,11),(8,10),(14,18),(15,17),(19,21),(22,24),(26,30),(27,29),(31,34),(32,33),(35,36),(37,38),(39,42),(40,41),(43,44),(45,48),(46,47),(49,50),(51,54),(52,53),(55,59),(56,58)], [(1,38,53,44,36),(2,39,54,45,31),(3,40,49,46,32),(4,41,50,47,33),(5,42,51,48,34),(6,37,52,43,35),(7,21,29,14,55),(8,22,30,15,56),(9,23,25,16,57),(10,24,26,17,58),(11,19,27,18,59),(12,20,28,13,60)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,55),(7,52,14,43),(8,53,15,44),(9,54,16,45),(10,49,17,46),(11,50,18,47),(12,51,13,48),(19,33,27,41),(20,34,28,42),(21,35,29,37),(22,36,30,38),(23,31,25,39),(24,32,26,40)]])
D6⋊F5 is a maximal subgroup of
C4⋊F5⋊3S3 (C4×S3)⋊F5 F5×D12 D60⋊3C4 F5×C3⋊D4 S3×C22⋊F5 C3⋊D4⋊F5
D6⋊F5 is a maximal quotient of
D60⋊C4 D12⋊F5 Dic6⋊F5 Dic30⋊C4 D12⋊4F5 D12⋊2F5 D60⋊2C4 D60⋊5C4 D10.20D12 Dic5.22D12 D30⋊C8 D10.D12 D10.4D12 Dic5.D12 Dic5.4D12
Matrix representation of D6⋊F5 ►in GL6(𝔽61)
| 2 | 46 | 0 | 0 | 0 | 0 |
| 49 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 46 | 0 | 0 | 0 | 0 |
| 49 | 59 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 18 |
| 0 | 0 | 0 | 0 | 44 | 0 |
| 53 | 19 | 0 | 0 | 0 | 0 |
| 3 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 17 | 18 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
G:=sub<GL(6,GF(61))| [2,49,0,0,0,0,46,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,49,0,0,0,0,46,59,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,45,0,0,0,0,60,44,0,0,0,0,0,0,17,44,0,0,0,0,18,0],[53,3,0,0,0,0,19,8,0,0,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,1,0,0,0,0,0,0,1,0,0] >;
D6⋊F5 in GAP, Magma, Sage, TeX
D_6\rtimes F_5
% in TeX
G:=Group("D6:F5"); // GroupNames label
G:=SmallGroup(240,96);
// by ID
G=gap.SmallGroup(240,96);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,490,3461,1745]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^5=d^4=1,b*a*b=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
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