C2xC2^4:D5 - GroupNames

G = C₂×C₂⁴⋊D₅ order 320 = 2⁶·5

Direct product of C₂ and C₂⁴⋊D₅

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₂⁴⋊D₅, C₂ ≀D₅, C₂⁵⋊D₅, C₂⁴⋊D₁₀, C₂⁴⋊C₅⋊C₂², (C₂×C₂⁴⋊C₅)⋊C₂, SmallGroup(320,1636)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂×C₂⁴⋊D₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂⁴⋊D₅ — C₂×C₂⁴⋊D₅

Lower central

C₂⁴⋊C₅ — C₂×C₂⁴⋊D₅

Upper central

C₁ — C₂

Generators and relations for C₂×C₂⁴⋊D₅
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f⁵=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg=bc=cb, bd=db, fcf^-1=gcg=be=eb, fbf^-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf^-1=bce, gdg=cde, fef^-1=bcde, gfg=f^-1 >

Subgroups: 1434 in 193 conjugacy classes, 9 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₂×C₄, D₄, C₂³, D₅, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, D₁₀, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²×D₄, C₂⁵, C₂×C₂²≀C₂, C₂⁴⋊C₅, C₂⁴⋊D₅, C₂×C₂⁴⋊C₅, C₂×C₂⁴⋊D₅
Quotients: C₁, C₂, C₂², D₅, D₁₀, C₂⁴⋊D₅, C₂×C₂⁴⋊D₅

Character table of C₂×C₂⁴⋊D₅

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	5A	5B	10A	10B
size	1	1	5	5	5	5	5	5	20	20	20	20	20	20	20	20	32	32	32	32
ρ₁	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	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	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	linear of order 2
ρ₅	2	-2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	^-1+√5/₂	^-1-√5/₂	^1-√5/₂	^1+√5/₂	orthogonal lifted from D₁₀
ρ₆	2	2	2	2	2	2	2	2	0	0	0	0	0	0	0	0	^-1-√5/₂	^-1+√5/₂	^-1-√5/₂	^-1+√5/₂	orthogonal lifted from D₅
ρ₇	2	-2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	^-1-√5/₂	^-1+√5/₂	^1+√5/₂	^1-√5/₂	orthogonal lifted from D₁₀
ρ₈	2	2	2	2	2	2	2	2	0	0	0	0	0	0	0	0	^-1+√5/₂	^-1-√5/₂	^-1+√5/₂	^-1-√5/₂	orthogonal lifted from D₅
ρ₉	5	-5	-1	1	1	-3	3	-1	-1	1	1	-1	1	-1	-1	1	0	0	0	0	orthogonal faithful
ρ₁₀	5	-5	3	1	-3	1	-1	-1	-1	1	-1	1	-1	1	-1	1	0	0	0	0	orthogonal faithful
ρ₁₁	5	5	1	-3	1	1	1	-3	1	1	-1	-1	-1	-1	1	1	0	0	0	0	orthogonal lifted from C₂⁴⋊D₅
ρ₁₂	5	5	1	-3	1	1	1	-3	-1	-1	1	1	1	1	-1	-1	0	0	0	0	orthogonal lifted from C₂⁴⋊D₅
ρ₁₃	5	-5	3	1	-3	1	-1	-1	1	-1	1	-1	1	-1	1	-1	0	0	0	0	orthogonal faithful
ρ₁₄	5	-5	-1	1	1	-3	3	-1	1	-1	-1	1	-1	1	1	-1	0	0	0	0	orthogonal faithful
ρ₁₅	5	5	1	1	1	-3	-3	1	-1	-1	-1	-1	1	1	1	1	0	0	0	0	orthogonal lifted from C₂⁴⋊D₅
ρ₁₆	5	5	-3	1	-3	1	1	1	-1	-1	1	1	-1	-1	1	1	0	0	0	0	orthogonal lifted from C₂⁴⋊D₅
ρ₁₇	5	-5	-1	-3	1	1	-1	3	1	-1	1	-1	-1	1	-1	1	0	0	0	0	orthogonal faithful
ρ₁₈	5	-5	-1	-3	1	1	-1	3	-1	1	-1	1	1	-1	1	-1	0	0	0	0	orthogonal faithful
ρ₁₉	5	5	-3	1	-3	1	1	1	1	1	-1	-1	1	1	-1	-1	0	0	0	0	orthogonal lifted from C₂⁴⋊D₅
ρ₂₀	5	5	1	1	1	-3	-3	1	1	1	1	1	-1	-1	-1	-1	0	0	0	0	orthogonal lifted from C₂⁴⋊D₅

Permutation representations of C₂×C₂⁴⋊D₅
►On 10 points - transitive group 10T23

Generators in S₁₀

(1 10)(2 6)(3 7)(4 8)(5 9)

(1 10)(2 6)(4 8)(5 9)

(3 7)(4 8)

(3 7)(5 9)

(1 10)(3 7)(4 8)(5 9)

(1 2 3 4 5)(6 7 8 9 10)

(1 9)(2 8)(3 7)(4 6)(5 10)

G:=sub<Sym(10)| (1,10)(2,6)(3,7)(4,8)(5,9), (1,10)(2,6)(4,8)(5,9), (3,7)(4,8), (3,7)(5,9), (1,10)(3,7)(4,8)(5,9), (1,2,3,4,5)(6,7,8,9,10), (1,9)(2,8)(3,7)(4,6)(5,10)>;

G:=Group( (1,10)(2,6)(3,7)(4,8)(5,9), (1,10)(2,6)(4,8)(5,9), (3,7)(4,8), (3,7)(5,9), (1,10)(3,7)(4,8)(5,9), (1,2,3,4,5)(6,7,8,9,10), (1,9)(2,8)(3,7)(4,6)(5,10) );

G=PermutationGroup([[(1,10),(2,6),(3,7),(4,8),(5,9)], [(1,10),(2,6),(4,8),(5,9)], [(3,7),(4,8)], [(3,7),(5,9)], [(1,10),(3,7),(4,8),(5,9)], [(1,2,3,4,5),(6,7,8,9,10)], [(1,9),(2,8),(3,7),(4,6),(5,10)]])

G:=TransitiveGroup(10,23);

►On 20 points - transitive group 20T71

Generators in S₂₀

(1 18)(2 19)(3 20)(4 16)(5 17)(6 14)(7 15)(8 11)(9 12)(10 13)

(1 13)(2 14)(4 11)(5 12)(6 19)(8 16)(9 17)(10 18)

(3 15)(4 11)(7 20)(8 16)

(3 15)(5 12)(7 20)(9 17)

(1 13)(3 15)(4 11)(5 12)(7 20)(8 16)(9 17)(10 18)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 12)(2 11)(3 15)(4 14)(5 13)(6 16)(7 20)(8 19)(9 18)(10 17)

G:=sub<Sym(20)| (1,18)(2,19)(3,20)(4,16)(5,17)(6,14)(7,15)(8,11)(9,12)(10,13), (1,13)(2,14)(4,11)(5,12)(6,19)(8,16)(9,17)(10,18), (3,15)(4,11)(7,20)(8,16), (3,15)(5,12)(7,20)(9,17), (1,13)(3,15)(4,11)(5,12)(7,20)(8,16)(9,17)(10,18), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,11)(3,15)(4,14)(5,13)(6,16)(7,20)(8,19)(9,18)(10,17)>;

G:=Group( (1,18)(2,19)(3,20)(4,16)(5,17)(6,14)(7,15)(8,11)(9,12)(10,13), (1,13)(2,14)(4,11)(5,12)(6,19)(8,16)(9,17)(10,18), (3,15)(4,11)(7,20)(8,16), (3,15)(5,12)(7,20)(9,17), (1,13)(3,15)(4,11)(5,12)(7,20)(8,16)(9,17)(10,18), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,11)(3,15)(4,14)(5,13)(6,16)(7,20)(8,19)(9,18)(10,17) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,16),(5,17),(6,14),(7,15),(8,11),(9,12),(10,13)], [(1,13),(2,14),(4,11),(5,12),(6,19),(8,16),(9,17),(10,18)], [(3,15),(4,11),(7,20),(8,16)], [(3,15),(5,12),(7,20),(9,17)], [(1,13),(3,15),(4,11),(5,12),(7,20),(8,16),(9,17),(10,18)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,16),(7,20),(8,19),(9,18),(10,17)]])

G:=TransitiveGroup(20,71);

►On 20 points - transitive group 20T73

Generators in S₂₀

(1 18)(2 19)(3 20)(4 16)(5 17)(6 14)(7 15)(8 11)(9 12)(10 13)

(1 13)(5 12)(9 17)(10 18)

(2 14)(5 12)(6 19)(9 17)

(1 13)(2 14)(3 15)(5 12)(6 19)(7 20)(9 17)(10 18)

(4 11)(5 12)(8 16)(9 17)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 8)(9 10)(11 14)(12 13)(16 19)(17 18)

G:=sub<Sym(20)| (1,18)(2,19)(3,20)(4,16)(5,17)(6,14)(7,15)(8,11)(9,12)(10,13), (1,13)(5,12)(9,17)(10,18), (2,14)(5,12)(6,19)(9,17), (1,13)(2,14)(3,15)(5,12)(6,19)(7,20)(9,17)(10,18), (4,11)(5,12)(8,16)(9,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,8)(9,10)(11,14)(12,13)(16,19)(17,18)>;

G:=Group( (1,18)(2,19)(3,20)(4,16)(5,17)(6,14)(7,15)(8,11)(9,12)(10,13), (1,13)(5,12)(9,17)(10,18), (2,14)(5,12)(6,19)(9,17), (1,13)(2,14)(3,15)(5,12)(6,19)(7,20)(9,17)(10,18), (4,11)(5,12)(8,16)(9,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,8)(9,10)(11,14)(12,13)(16,19)(17,18) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,16),(5,17),(6,14),(7,15),(8,11),(9,12),(10,13)], [(1,13),(5,12),(9,17),(10,18)], [(2,14),(5,12),(6,19),(9,17)], [(1,13),(2,14),(3,15),(5,12),(6,19),(7,20),(9,17),(10,18)], [(4,11),(5,12),(8,16),(9,17)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,8),(9,10),(11,14),(12,13),(16,19),(17,18)]])

G:=TransitiveGroup(20,73);

►On 20 points - transitive group 20T76

Generators in S₂₀

(1 18)(2 19)(3 20)(4 16)(5 17)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 6)(2 19)(4 16)(5 10)(7 12)(9 14)(11 18)(15 17)

(2 12)(3 20)(4 16)(5 15)(7 19)(8 13)(9 14)(10 17)

(1 11)(2 12)(3 8)(5 10)(6 18)(7 19)(13 20)(15 17)

(1 18)(3 20)(4 9)(5 10)(6 11)(8 13)(14 16)(15 17)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 15)(2 14)(3 13)(4 12)(5 11)(6 17)(7 16)(8 20)(9 19)(10 18)

G:=sub<Sym(20)| (1,18)(2,19)(3,20)(4,16)(5,17)(6,11)(7,12)(8,13)(9,14)(10,15), (1,6)(2,19)(4,16)(5,10)(7,12)(9,14)(11,18)(15,17), (2,12)(3,20)(4,16)(5,15)(7,19)(8,13)(9,14)(10,17), (1,11)(2,12)(3,8)(5,10)(6,18)(7,19)(13,20)(15,17), (1,18)(3,20)(4,9)(5,10)(6,11)(8,13)(14,16)(15,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,17)(7,16)(8,20)(9,19)(10,18)>;

G:=Group( (1,18)(2,19)(3,20)(4,16)(5,17)(6,11)(7,12)(8,13)(9,14)(10,15), (1,6)(2,19)(4,16)(5,10)(7,12)(9,14)(11,18)(15,17), (2,12)(3,20)(4,16)(5,15)(7,19)(8,13)(9,14)(10,17), (1,11)(2,12)(3,8)(5,10)(6,18)(7,19)(13,20)(15,17), (1,18)(3,20)(4,9)(5,10)(6,11)(8,13)(14,16)(15,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,17)(7,16)(8,20)(9,19)(10,18) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,16),(5,17),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,6),(2,19),(4,16),(5,10),(7,12),(9,14),(11,18),(15,17)], [(2,12),(3,20),(4,16),(5,15),(7,19),(8,13),(9,14),(10,17)], [(1,11),(2,12),(3,8),(5,10),(6,18),(7,19),(13,20),(15,17)], [(1,18),(3,20),(4,9),(5,10),(6,11),(8,13),(14,16),(15,17)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,17),(7,16),(8,20),(9,19),(10,18)]])

G:=TransitiveGroup(20,76);

►On 20 points - transitive group 20T81

Generators in S₂₀

(1 15)(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)(10 16)

(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)

(1 15)(2 11)(6 17)(10 16)

(1 15)(3 12)(7 18)(10 16)

(1 15)(2 11)(3 12)(4 13)(6 17)(7 18)(8 19)(10 16)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 16)(2 20)(3 19)(4 18)(5 17)(6 14)(7 13)(8 12)(9 11)(10 15)

G:=sub<Sym(20)| (1,15)(2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20)(10,16), (2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20), (1,15)(2,11)(6,17)(10,16), (1,15)(3,12)(7,18)(10,16), (1,15)(2,11)(3,12)(4,13)(6,17)(7,18)(8,19)(10,16), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,14)(7,13)(8,12)(9,11)(10,15)>;

G:=Group( (1,15)(2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20)(10,16), (2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20), (1,15)(2,11)(6,17)(10,16), (1,15)(3,12)(7,18)(10,16), (1,15)(2,11)(3,12)(4,13)(6,17)(7,18)(8,19)(10,16), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,14)(7,13)(8,12)(9,11)(10,15) );

G=PermutationGroup([[(1,15),(2,11),(3,12),(4,13),(5,14),(6,17),(7,18),(8,19),(9,20),(10,16)], [(2,11),(3,12),(4,13),(5,14),(6,17),(7,18),(8,19),(9,20)], [(1,15),(2,11),(6,17),(10,16)], [(1,15),(3,12),(7,18),(10,16)], [(1,15),(2,11),(3,12),(4,13),(6,17),(7,18),(8,19),(10,16)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,14),(7,13),(8,12),(9,11),(10,15)]])

G:=TransitiveGroup(20,81);

►On 20 points - transitive group 20T85

Generators in S₂₀

(1 20)(2 16)(3 17)(4 18)(5 19)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 15)(2 6)(4 8)(5 14)(9 19)(10 20)(11 16)(13 18)

(2 16)(3 7)(4 8)(5 19)(6 11)(9 14)(12 17)(13 18)

(1 20)(2 16)(3 12)(5 14)(6 11)(7 17)(9 19)(10 15)

(1 10)(3 7)(4 13)(5 14)(8 18)(9 19)(12 17)(15 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 8)(9 10)(11 13)(14 15)(16 18)(19 20)

G:=sub<Sym(20)| (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15), (1,15)(2,6)(4,8)(5,14)(9,19)(10,20)(11,16)(13,18), (2,16)(3,7)(4,8)(5,19)(6,11)(9,14)(12,17)(13,18), (1,20)(2,16)(3,12)(5,14)(6,11)(7,17)(9,19)(10,15), (1,10)(3,7)(4,13)(5,14)(8,18)(9,19)(12,17)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,8)(9,10)(11,13)(14,15)(16,18)(19,20)>;

G:=Group( (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15), (1,15)(2,6)(4,8)(5,14)(9,19)(10,20)(11,16)(13,18), (2,16)(3,7)(4,8)(5,19)(6,11)(9,14)(12,17)(13,18), (1,20)(2,16)(3,12)(5,14)(6,11)(7,17)(9,19)(10,15), (1,10)(3,7)(4,13)(5,14)(8,18)(9,19)(12,17)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,8)(9,10)(11,13)(14,15)(16,18)(19,20) );

G=PermutationGroup([[(1,20),(2,16),(3,17),(4,18),(5,19),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,15),(2,6),(4,8),(5,14),(9,19),(10,20),(11,16),(13,18)], [(2,16),(3,7),(4,8),(5,19),(6,11),(9,14),(12,17),(13,18)], [(1,20),(2,16),(3,12),(5,14),(6,11),(7,17),(9,19),(10,15)], [(1,10),(3,7),(4,13),(5,14),(8,18),(9,19),(12,17),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,8),(9,10),(11,13),(14,15),(16,18),(19,20)]])

G:=TransitiveGroup(20,85);

►On 20 points - transitive group 20T87

Generators in S₂₀

(1 20)(2 16)(3 17)(4 18)(5 19)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 15)(2 6)(4 13)(5 9)(8 18)(10 20)(11 16)(14 19)

(1 20)(2 16)(3 7)(4 13)(5 19)(6 11)(8 18)(9 14)(10 15)(12 17)

(3 12)(4 18)(5 9)(7 17)(8 13)(14 19)

(1 10)(3 12)(4 8)(5 14)(7 17)(9 19)(13 18)(15 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 13)(7 12)(8 11)(9 15)(10 14)(16 18)(19 20)

G:=sub<Sym(20)| (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15), (1,15)(2,6)(4,13)(5,9)(8,18)(10,20)(11,16)(14,19), (1,20)(2,16)(3,7)(4,13)(5,19)(6,11)(8,18)(9,14)(10,15)(12,17), (3,12)(4,18)(5,9)(7,17)(8,13)(14,19), (1,10)(3,12)(4,8)(5,14)(7,17)(9,19)(13,18)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,13)(7,12)(8,11)(9,15)(10,14)(16,18)(19,20)>;

G:=Group( (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15), (1,15)(2,6)(4,13)(5,9)(8,18)(10,20)(11,16)(14,19), (1,20)(2,16)(3,7)(4,13)(5,19)(6,11)(8,18)(9,14)(10,15)(12,17), (3,12)(4,18)(5,9)(7,17)(8,13)(14,19), (1,10)(3,12)(4,8)(5,14)(7,17)(9,19)(13,18)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,13)(7,12)(8,11)(9,15)(10,14)(16,18)(19,20) );

G=PermutationGroup([[(1,20),(2,16),(3,17),(4,18),(5,19),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,15),(2,6),(4,13),(5,9),(8,18),(10,20),(11,16),(14,19)], [(1,20),(2,16),(3,7),(4,13),(5,19),(6,11),(8,18),(9,14),(10,15),(12,17)], [(3,12),(4,18),(5,9),(7,17),(8,13),(14,19)], [(1,10),(3,12),(4,8),(5,14),(7,17),(9,19),(13,18),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,13),(7,12),(8,11),(9,15),(10,14),(16,18),(19,20)]])

G:=TransitiveGroup(20,87);

Polynomial with Galois group C₂×C₂⁴⋊D₅ over ℚ

action	f(x)	Disc(f)
10T23	x¹⁰+5x⁹-2x⁸-38x⁷-30x⁶+64x⁵+70x⁴-21x³-26x²+3x+1	3¹⁵·83·401⁴

Matrix representation of C₂×C₂⁴⋊D₅ ►in GL₅(ℤ)

-1	0	0	0	0
0	-1	0	0	0
0	0	-1	0	0
0	0	0	-1	0
0	0	0	0	-1

-1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	-1

1	0	0	0	0
0	-1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	-1

-1	0	0	0	0
0	-1	0	0	0
0	0	-1	0	0
0	0	0	1	0
0	0	0	0	-1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	-1	0
0	0	0	0	-1

0	0	0	0	1
1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0

0	0	0	0	-1
0	0	0	-1	0
0	0	-1	0	0
0	-1	0	0	0
-1	0	0	0	0

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,0] >;

C₂×C₂⁴⋊D₅ in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes D_5

% in TeX

G:=Group("C2xC2^4:D5");

// GroupNames label

G:=SmallGroup(320,1636);

// by ID

G=gap.SmallGroup(320,1636);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,2,2,2,338,1683,437,1068,9245,2539,4906,265]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^5=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,g*e*g=b*c=c*b,b*d=d*b,f*c*f^-1=g*c*g=b*e=e*b,f*b*f^-1=e,b*g=g*b,c*d=d*c,c*e=e*c,d*e=e*d,f*d*f^-1=b*c*e,g*d*g=c*d*e,f*e*f^-1=b*c*d*e,g*f*g=f^-1>;

// generators/relations

Export

Character table of C₂×C₂⁴⋊D₅ in TeX

G = C2×C24⋊D5 order 320 = 26·5

Direct product of C2 and C24⋊D5

G = C₂×C₂⁴⋊D₅ order 320 = 2⁶·5

Direct product of C₂ and C₂⁴⋊D₅