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## G = Dic3⋊F5order 240 = 24·3·5

### The semidirect product of Dic3 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — Dic3⋊F5
 Lower central C15 — C30 — Dic3⋊F5
 Upper central C1 — C2

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 >

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

Smallest permutation representation of Dic3⋊F5
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  Dic65F5  (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

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