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## G = D5×C32⋊C4order 360 = 23·32·5

### Direct product of D5 and C32⋊C4

Aliases: D5×C32⋊C4, C3⋊D151C4, C321(C4×D5), C3⋊S3.4D10, (C32×D5)⋊1C4, C32⋊Dic51C2, C53(C2×C32⋊C4), (C3×C15)⋊1(C2×C4), (C5×C32⋊C4)⋊2C2, (D5×C3⋊S3).1C2, (C5×C3⋊S3).1C22, SmallGroup(360,130)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — D5×C32⋊C4
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — D5×C3⋊S3 — D5×C32⋊C4
 Lower central C3×C15 — D5×C32⋊C4
 Upper central C1

Generators and relations for D5×C32⋊C4
G = < a,b,c,d,e | a5=b2=c3=d3=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Character table of D5×C32⋊C4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D 20A 20B 20C 20D size 1 5 9 45 4 4 9 9 45 45 2 2 20 20 18 18 8 8 8 8 18 18 18 18 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 i 1 1 -1 -1 -1 -1 1 1 1 1 i i -i -i linear of order 4 ρ6 1 -1 -1 1 1 1 i -i i -i 1 1 -1 -1 -1 -1 1 1 1 1 -i -i i i linear of order 4 ρ7 1 1 -1 -1 1 1 i -i -i i 1 1 1 1 -1 -1 1 1 1 1 -i -i i i linear of order 4 ρ8 1 1 -1 -1 1 1 -i i i -i 1 1 1 1 -1 -1 1 1 1 1 i i -i -i linear of order 4 ρ9 2 0 2 0 2 2 2 2 0 0 -1-√5/2 -1+√5/2 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 ρ10 2 0 2 0 2 2 -2 -2 0 0 -1-√5/2 -1+√5/2 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 0 2 0 2 2 -2 -2 0 0 -1+√5/2 -1-√5/2 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 ρ12 2 0 2 0 2 2 2 2 0 0 -1+√5/2 -1-√5/2 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 ρ13 2 0 -2 0 2 2 2i -2i 0 0 -1+√5/2 -1-√5/2 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 complex lifted from C4×D5 ρ14 2 0 -2 0 2 2 -2i 2i 0 0 -1+√5/2 -1-√5/2 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 complex lifted from C4×D5 ρ15 2 0 -2 0 2 2 -2i 2i 0 0 -1-√5/2 -1+√5/2 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 complex lifted from C4×D5 ρ16 2 0 -2 0 2 2 2i -2i 0 0 -1-√5/2 -1+√5/2 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 complex lifted from C4×D5 ρ17 4 -4 0 0 -2 1 0 0 0 0 4 4 2 -1 0 0 1 1 -2 -2 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ18 4 4 0 0 -2 1 0 0 0 0 4 4 -2 1 0 0 1 1 -2 -2 0 0 0 0 orthogonal lifted from C32⋊C4 ρ19 4 -4 0 0 1 -2 0 0 0 0 4 4 -1 2 0 0 -2 -2 1 1 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ20 4 4 0 0 1 -2 0 0 0 0 4 4 1 -2 0 0 -2 -2 1 1 0 0 0 0 orthogonal lifted from C32⋊C4 ρ21 8 0 0 0 -4 2 0 0 0 0 -2+2√5 -2-2√5 0 0 0 0 -1-√5/2 -1+√5/2 1+√5 1-√5 0 0 0 0 orthogonal faithful ρ22 8 0 0 0 -4 2 0 0 0 0 -2-2√5 -2+2√5 0 0 0 0 -1+√5/2 -1-√5/2 1-√5 1+√5 0 0 0 0 orthogonal faithful ρ23 8 0 0 0 2 -4 0 0 0 0 -2-2√5 -2+2√5 0 0 0 0 1-√5 1+√5 -1+√5/2 -1-√5/2 0 0 0 0 orthogonal faithful ρ24 8 0 0 0 2 -4 0 0 0 0 -2+2√5 -2-2√5 0 0 0 0 1+√5 1-√5 -1-√5/2 -1+√5/2 0 0 0 0 orthogonal faithful

Permutation representations of D5×C32⋊C4
On 30 points - transitive group 30T99
Generators in S30
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 26 11 21)(7 27 12 22)(8 28 13 23)(9 29 14 24)(10 30 15 25)

G:=sub<Sym(30)| (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26,11,21)(7,27,12,22)(8,28,13,23)(9,29,14,24)(10,30,15,25)>;

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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26,11,21)(7,27,12,22)(8,28,13,23)(9,29,14,24)(10,30,15,25) );

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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,26,11,21),(7,27,12,22),(8,28,13,23),(9,29,14,24),(10,30,15,25)]])

G:=TransitiveGroup(30,99);

Matrix representation of D5×C32⋊C4 in GL6(𝔽61)

 60 45 0 0 0 0 60 44 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
,
 1 0 0 0 0 0 1 60 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 0 60 0 0 0 0 1 60 0 0 0 0 0 0 0 60 0 0 0 0 1 60
,
 1 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 60 1 0 0 0 0 60 0
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(61))| [60,60,0,0,0,0,45,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

D5×C32⋊C4 in GAP, Magma, Sage, TeX

D_5\times C_3^2\rtimes C_4
% in TeX

G:=Group("D5xC3^2:C4");
// GroupNames label

G:=SmallGroup(360,130);
// by ID

G=gap.SmallGroup(360,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-5,31,489,111,490,376,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^3=d^3=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations

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