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## G = Dic3×F5order 240 = 24·3·5

### Direct product of Dic3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — Dic3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — Dic3×F5
 Lower central C15 — 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, bd=db, dcd-1=c3 >

Subgroups: 232 in 60 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, D5, C10, Dic3, Dic3, C12, C2×C6, C15, C42, Dic5, C20, F5, F5, D10, C2×Dic3, C2×C12, C3×D5, C30, C4×D5, C2×F5, C2×F5, C4×Dic3, C5×Dic3, Dic15, C3×F5, C3⋊F5, C6×D5, C4×F5, D5×Dic3, C6×F5, C2×C3⋊F5, Dic3×F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, F5, C4×S3, C2×Dic3, C2×F5, C4×Dic3, C4×F5, S3×F5, Dic3×F5

Character table of Dic3×F5

 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 C4×S3 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 C4×S3 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 C4×S3 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 C4×S3 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 C2×F5 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 C4×F5 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 C4×F5 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 S3×F5 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

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 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)]])

Dic3×F5 is a maximal subgroup of   C4⋊F53S3  Dic65F5  C4×S3×F5  C22⋊F5.S3  C3⋊D4⋊F5
Dic3×F5 is a maximal quotient of   C30.C42  C30.3C42  C30.4C42  D10.20D12  C30.M4(2)

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
,
 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] >;

Dic3×F5 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

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