C2^4:F5

non-abelian, soluble, monomial

Aliases: C₂⁴⋊F₅, C₂⁴⋊C₅⋊C₄, C₂⁴⋊D₅.C₂, SmallGroup(320,1635)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂⁴⋊F₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂⁴⋊D₅ — C₂⁴⋊F₅

Lower central

C₂⁴⋊C₅ — C₂⁴⋊F₅

Upper central

C₁

Subgroups: 428 in 40 conjugacy classes, 5 normal (all characteristic)
C₁, C₂^[×3], C₄^[×3], C₂²^[×6], C₅, C₈, C₂×C₄^[×3], D₄^[×3], C₂³^[×3], D₅, C₂²⋊C₄^[×3], M₄(2), C₂×D₄^[×2], C₂⁴, F₅, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₂≀C₄, C₂⁴⋊C₅, C₂⁴⋊D₅, C₂⁴⋊F₅

Quotients:
C₁, C₂, C₄, F₅, C₂⁴⋊F₅

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁵=f⁴=1, fdf^-1=ab=ba, ac=ca, ebe^-1=fcf^-1=ad=da, eae^-1=faf^-1=d, bc=cb, bd=db, fbf^-1=acd, cd=dc, ece^-1=abd, ede^-1=abcd, fef^-1=e³ >

Permutation representations
►On 10 points - transitive group 10T24

Generators in S₁₀

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

(3 10)(4 6)

(3 10)(5 7)

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

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

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

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

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

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

G:=TransitiveGroup(10,24);

►On 10 points - transitive group 10T25

Generators in S₁₀

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

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

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

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

G:=TransitiveGroup(10,25);

►On 16 points: primitive - transitive group 16T711

Generators in S₁₆

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

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

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

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

(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)

(2 15 3 12)(4 14 6 13)(5 16)(7 11 9 10)

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

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

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

G:=TransitiveGroup(16,711);

►On 20 points - transitive group 20T77

Generators in S₂₀

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(20,77);

►On 20 points - transitive group 20T78

Generators in S₂₀

(2 11)(4 13)(9 18)(10 19)

(2 11)(3 12)(4 13)(5 14)(6 20)(9 18)

(1 15)(2 11)(6 20)(7 16)(9 18)(10 19)

(1 15)(3 12)(8 17)(9 18)

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

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

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

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

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

G:=TransitiveGroup(20,78);

►On 20 points - transitive group 20T79

Generators in S₂₀

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

(3 12)(4 13)(7 16)(8 17)

(3 12)(5 14)(7 16)(9 18)

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

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

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

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

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

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

G:=TransitiveGroup(20,79);

►On 20 points - transitive group 20T80

Generators in S₂₀

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(20,80);

►On 20 points - transitive group 20T83

Generators in S₂₀

(2 11)(4 13)(9 18)(10 19)

(2 11)(3 12)(4 13)(5 14)(6 20)(9 18)

(1 15)(2 11)(6 20)(7 16)(9 18)(10 19)

(1 15)(3 12)(8 17)(9 18)

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

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

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

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

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

G:=TransitiveGroup(20,83);

►On 20 points - transitive group 20T88

Generators in S₂₀

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

(3 12)(4 13)(7 16)(8 17)

(3 12)(5 14)(7 16)(9 18)

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

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

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

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

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

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

G:=TransitiveGroup(20,88);

Matrix representation ►G ⊆ 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

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

1	0	0	0	0
0	0	0	1	0
0	1	0	0	0
0	0	0	0	1
0	0	1	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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

Character table of C₂⁴⋊F₅

class	1	2A	2B	2C	4A	4B	4C	4D	5	8A	8B
size	1	5	10	20	20	40	40	40	64	40	40
ρ₁	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	-1	-1	1	-1	-1	linear of order 2
ρ₃	1	1	1	-1	-1	-1	-i	i	1	i	-i	linear of order 4
ρ₄	1	1	1	-1	-1	-1	i	-i	1	-i	i	linear of order 4
ρ₅	4	4	4	0	0	0	0	0	-1	0	0	orthogonal lifted from F₅
ρ₆	5	-3	1	1	1	-1	1	1	0	-1	-1	orthogonal faithful
ρ₇	5	-3	1	1	1	-1	-1	-1	0	1	1	orthogonal faithful
ρ₈	5	-3	1	-1	-1	1	-i	i	0	-i	i	complex faithful
ρ₉	5	-3	1	-1	-1	1	i	-i	0	i	-i	complex faithful
ρ₁₀	10	2	-2	2	-2	0	0	0	0	0	0	orthogonal faithful
ρ₁₁	10	2	-2	-2	2	0	0	0	0	0	0	orthogonal faithful

In GAP, Magma, Sage, TeX

C_2^4\rtimes F_5

% in TeX

G:=Group("C2^4:F5");

// GroupNames label

G:=SmallGroup(320,1635);

// by ID

G=gap.SmallGroup(320,1635);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,2,2,2,14,170,177,2803,850,1137,9104,1593,5045,4632,2329,986,2463,3695]);

// Polycyclic

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

// generators/relations

G = C₂⁴⋊F₅ order 320 = 2⁶·5

The semidirect product of C₂⁴ and F₅ acting faithfully

G = C24⋊F5 order 320 = 26·5

The semidirect product of C24 and F5 acting faithfully

G = C₂⁴⋊F₅ order 320 = 2⁶·5

The semidirect product of C₂⁴ and F₅ acting faithfully