D5xC3^2:C4 - GroupNames

G = D₅×C₃²⋊C₄ order 360 = 2³·3²·5

Direct product of D₅ and C₃²⋊C₄

direct product, metabelian, soluble, monomial, A-group

Aliases: D₅×C₃²⋊C₄, C₃⋊D₁₅⋊₁C₄, C₃²⋊₁(C₄×D₅), C₃⋊S₃.₄D₁₀, (C₃²×D₅)⋊₁C₄, C₃²⋊Dic₅⋊₁C₂, C₅⋊₃(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 >

C₁

₅C₂

₉C₂

₄₅C₂

₂C₃

C₅

₉C₄

₄₅C₄

₄₅C₂²

₆S₃

₁₀C₆

₃₀S₃

C₃²

D₅

₉C₁₀

₉D₅

₂C₁₅

₄₅C₂×C₄

₃₀D₆

C₃⋊S₃

₅C₃⋊S₃

₅C₃×C₆

₉Dic₅

₉C₂₀

₉D₁₀

₂C₃×D₅

₆D₁₅

₆C₅×S₃

₆D₁₅

₆C₅×S₃

C₃×C₁₅

C₃²⋊C₄

₅C₃²⋊C₄

₅C₂×C₃⋊S₃

₉C₄×D₅

₆S₃×D₅

C₃²×D₅

C₃⋊D₁₅

C₅×C₃⋊S₃

₅C₂×C₃²⋊C₄

D₅×C₃⋊S₃

C₃²⋊Dic₅

C₅×C₃²⋊C₄

D₅×C₃²⋊C₄

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

Export

Subgroup lattice of D₅×C₃²⋊C₄ in TeX
Character table of D₅×C₃²⋊C₄ in TeX

G = D5×C32⋊C4 order 360 = 23·32·5

Direct product of D5 and C32⋊C4

G = D₅×C₃²⋊C₄ order 360 = 2³·3²·5

Direct product of D₅ and C₃²⋊C₄