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## G = D5×Dic3order 120 = 23·3·5

### Direct product of D5 and Dic3

Aliases: D5×Dic3, C10.1D6, C6.1D10, D10.2S3, Dic152C2, C30.1C22, C33(C4×D5), C154(C2×C4), (C3×D5)⋊1C4, C2.1(S3×D5), C52(C2×Dic3), (C6×D5).1C2, (C5×Dic3)⋊1C2, SmallGroup(120,8)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×Dic3
G = < a,b,c,d | a5=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Character table of D5×Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A 10B 15A 15B 20A 20B 20C 20D 30A 30B size 1 1 5 5 2 3 3 15 15 2 2 2 10 10 2 2 4 4 6 6 6 6 4 4 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 -i i i -i 1 1 -1 -1 1 -1 -1 1 1 i -i -i i -1 -1 linear of order 4 ρ6 1 -1 1 -1 1 -i i -i i 1 1 -1 1 -1 -1 -1 1 1 i -i -i i -1 -1 linear of order 4 ρ7 1 -1 1 -1 1 i -i i -i 1 1 -1 1 -1 -1 -1 1 1 -i i i -i -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 i -i -i i 1 1 -1 -1 1 -1 -1 1 1 -i i i -i -1 -1 linear of order 4 ρ9 2 2 -2 -2 -1 0 0 0 0 2 2 -1 1 1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ10 2 2 0 0 2 -2 -2 0 0 -1+√5/2 -1-√5/2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ11 2 2 0 0 2 2 2 0 0 -1-√5/2 -1+√5/2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ12 2 2 2 2 -1 0 0 0 0 2 2 -1 -1 -1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ13 2 2 0 0 2 -2 -2 0 0 -1-√5/2 -1+√5/2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ14 2 2 0 0 2 2 2 0 0 -1+√5/2 -1-√5/2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ15 2 -2 -2 2 -1 0 0 0 0 2 2 1 1 -1 -2 -2 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ16 2 -2 2 -2 -1 0 0 0 0 2 2 1 -1 1 -2 -2 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ17 2 -2 0 0 2 2i -2i 0 0 -1-√5/2 -1+√5/2 -2 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 1+√5/2 1-√5/2 complex lifted from C4×D5 ρ18 2 -2 0 0 2 -2i 2i 0 0 -1+√5/2 -1-√5/2 -2 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 1-√5/2 1+√5/2 complex lifted from C4×D5 ρ19 2 -2 0 0 2 2i -2i 0 0 -1+√5/2 -1-√5/2 -2 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 1-√5/2 1+√5/2 complex lifted from C4×D5 ρ20 2 -2 0 0 2 -2i 2i 0 0 -1-√5/2 -1+√5/2 -2 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 1+√5/2 1-√5/2 complex lifted from C4×D5 ρ21 4 4 0 0 -2 0 0 0 0 -1-√5 -1+√5 -2 0 0 -1-√5 -1+√5 1+√5/2 1-√5/2 0 0 0 0 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ22 4 4 0 0 -2 0 0 0 0 -1+√5 -1-√5 -2 0 0 -1+√5 -1-√5 1-√5/2 1+√5/2 0 0 0 0 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ23 4 -4 0 0 -2 0 0 0 0 -1-√5 -1+√5 2 0 0 1+√5 1-√5 1+√5/2 1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 symplectic faithful, Schur index 2 ρ24 4 -4 0 0 -2 0 0 0 0 -1+√5 -1-√5 2 0 0 1-√5 1+√5 1-√5/2 1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 symplectic faithful, Schur index 2

Smallest permutation representation of D5×Dic3
On 60 points
Generators in S60
(1 34 46 53 42)(2 35 47 54 37)(3 36 48 49 38)(4 31 43 50 39)(5 32 44 51 40)(6 33 45 52 41)(7 24 55 26 14)(8 19 56 27 15)(9 20 57 28 16)(10 21 58 29 17)(11 22 59 30 18)(12 23 60 25 13)
(1 42)(2 37)(3 38)(4 39)(5 40)(6 41)(7 26)(8 27)(9 28)(10 29)(11 30)(12 25)(19 56)(20 57)(21 58)(22 59)(23 60)(24 55)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)
(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)

G:=sub<Sym(60)| (1,34,46,53,42)(2,35,47,54,37)(3,36,48,49,38)(4,31,43,50,39)(5,32,44,51,40)(6,33,45,52,41)(7,24,55,26,14)(8,19,56,27,15)(9,20,57,28,16)(10,21,58,29,17)(11,22,59,30,18)(12,23,60,25,13), (1,42)(2,37)(3,38)(4,39)(5,40)(6,41)(7,26)(8,27)(9,28)(10,29)(11,30)(12,25)(19,56)(20,57)(21,58)(22,59)(23,60)(24,55)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49), (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)>;

G:=Group( (1,34,46,53,42)(2,35,47,54,37)(3,36,48,49,38)(4,31,43,50,39)(5,32,44,51,40)(6,33,45,52,41)(7,24,55,26,14)(8,19,56,27,15)(9,20,57,28,16)(10,21,58,29,17)(11,22,59,30,18)(12,23,60,25,13), (1,42)(2,37)(3,38)(4,39)(5,40)(6,41)(7,26)(8,27)(9,28)(10,29)(11,30)(12,25)(19,56)(20,57)(21,58)(22,59)(23,60)(24,55)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49), (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) );

G=PermutationGroup([[(1,34,46,53,42),(2,35,47,54,37),(3,36,48,49,38),(4,31,43,50,39),(5,32,44,51,40),(6,33,45,52,41),(7,24,55,26,14),(8,19,56,27,15),(9,20,57,28,16),(10,21,58,29,17),(11,22,59,30,18),(12,23,60,25,13)], [(1,42),(2,37),(3,38),(4,39),(5,40),(6,41),(7,26),(8,27),(9,28),(10,29),(11,30),(12,25),(19,56),(20,57),(21,58),(22,59),(23,60),(24,55),(31,50),(32,51),(33,52),(34,53),(35,54),(36,49)], [(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)]])

D5×Dic3 is a maximal subgroup of   Dic3⋊F5  D205S3  D20⋊S3  C4×S3×D5  Dic5.D6  C30.C23  D30.S3
D5×Dic3 is a maximal quotient of   C20.32D6  D10⋊Dic3  C30.Q8  D30.S3

Matrix representation of D5×Dic3 in GL4(𝔽61) generated by

 0 1 0 0 60 17 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 60 1 0 0 60 0
,
 50 0 0 0 0 50 0 0 0 0 15 23 0 0 38 46
G:=sub<GL(4,GF(61))| [0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,15,38,0,0,23,46] >;

D5×Dic3 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_3
% in TeX

G:=Group("D5xDic3");
// GroupNames label

G:=SmallGroup(120,8);
// by ID

G=gap.SmallGroup(120,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,26,168,2404]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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