A4:F5 - GroupNames

G = A₄⋊F₅ order 240 = 2⁴·3·5

The semidirect product of A₄ and F₅ acting via F₅/D₅=C₂

Aliases: A₄⋊F₅, D₅.S₄, C₅⋊(A₄⋊C₄), C₂²⋊(C₃⋊F₅), (C₅×A₄)⋊₁C₄, (C₂×C₁₀)⋊Dic₃, (D₅×A₄).₁C₂, (C₂²×D₅).S₃, SmallGroup(240,192)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×A₄ — A₄⋊F₅

Chief series

C₁ — C₂² — C₂×C₁₀ — C₅×A₄ — D₅×A₄ — A₄⋊F₅

Lower central

C₅×A₄ — A₄⋊F₅

Upper central

C₁

Generators and relations for A₄⋊F₅
G = < a,b,c,d,e | a²=b²=c³=d⁵=e⁴=1, cac^-1=eae^-1=ab=ba, ad=da, cbc^-1=a, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d³ >

C₁

₃C₂

₅C₂

₁₅C₂

₄C₃

C₅

C₂²

₁₅C₂²

₃₀C₄

₂₀C₆

D₅

₃C₁₀

₃D₅

₄C₁₅

₅C₂³

₁₅C₂×C₄

A₄

₂₀Dic₃

C₂×C₁₀

₃D₁₀

₆F₅

₄C₃×D₅

₁₅C₂²⋊C₄

₅C₂×A₄

C₂²×D₅

₃C₂×F₅

C₅×A₄

₄C₃⋊F₅

₅A₄⋊C₄

₃C₂²⋊F₅

D₅×A₄

A₄⋊F₅

Character table of A₄⋊F₅

class	1	2A	2B	2C	3	4A	4B	4C	4D	5	6	10	15A	15B
size	1	3	5	15	8	30	30	30	30	4	40	12	16	16
ρ₁	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	linear of order 2
ρ₃	1	1	-1	-1	1	-i	i	i	-i	1	-1	1	1	1	linear of order 4
ρ₄	1	1	-1	-1	1	i	-i	-i	i	1	-1	1	1	1	linear of order 4
ρ₅	2	2	2	2	-1	0	0	0	0	2	-1	2	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	-2	-2	-1	0	0	0	0	2	1	2	-1	-1	symplectic lifted from Dic₃, Schur index 2
ρ₇	3	-1	3	-1	0	1	-1	1	-1	3	0	-1	0	0	orthogonal lifted from S₄
ρ₈	3	-1	3	-1	0	-1	1	-1	1	3	0	-1	0	0	orthogonal lifted from S₄
ρ₉	3	-1	-3	1	0	-i	-i	i	i	3	0	-1	0	0	complex lifted from A₄⋊C₄
ρ₁₀	3	-1	-3	1	0	i	i	-i	-i	3	0	-1	0	0	complex lifted from A₄⋊C₄
ρ₁₁	4	4	0	0	4	0	0	0	0	-1	0	-1	-1	-1	orthogonal lifted from F₅
ρ₁₂	4	4	0	0	-2	0	0	0	0	-1	0	-1	^1-√-15/₂	^1+√-15/₂	complex lifted from C₃⋊F₅
ρ₁₃	4	4	0	0	-2	0	0	0	0	-1	0	-1	^1+√-15/₂	^1-√-15/₂	complex lifted from C₃⋊F₅
ρ₁₄	12	-4	0	0	0	0	0	0	0	-3	0	1	0	0	orthogonal faithful

Permutation representations of A₄⋊F₅
►On 20 points - transitive group 20T61

Generators in S₂₀

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

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

(6 11 18)(7 12 19)(8 13 20)(9 14 16)(10 15 17)

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

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

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

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

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

G:=TransitiveGroup(20,61);

►On 30 points - transitive group 30T53

Generators in S₃₀

(1 13)(2 14)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(10 30)

(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)

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

(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)

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

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

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

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

G:=TransitiveGroup(30,53);

►On 30 points - transitive group 30T64

Generators in S₃₀

(1 13)(2 14)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(10 30)

(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)

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

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

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

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

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

G:=TransitiveGroup(30,64);

A₄⋊F₅ is a maximal subgroup of F₅×S₄
A₄⋊F₅ is a maximal quotient of C₅⋊U₂(𝔽₃) D₁₀.S₄ Dic₅.S₄

Matrix representation of A₄⋊F₅ ►in GL₇(𝔽₆₁)

60	0	0	0	0	0	0
60	0	1	0	0	0	0
60	1	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

0	1	60	0	0	0	0
1	0	60	0	0	0	0
0	0	60	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

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

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	0	0	60
0	0	0	1	0	0	60
0	0	0	0	1	0	60
0	0	0	0	0	1	60

0	11	0	0	0	0	0
11	0	0	0	0	0	0
0	0	11	0	0	0	0
0	0	0	0	0	1	0
0	0	0	1	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0

G:=sub<GL(7,GF(61))| [60,60,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60],[0,11,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

A₄⋊F₅ in GAP, Magma, Sage, TeX

A_4\rtimes F_5

% in TeX

G:=Group("A4:F5");

// GroupNames label

G:=SmallGroup(240,192);

// by ID

G=gap.SmallGroup(240,192);

# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,2,12,146,867,585,3604,916,2165,1637]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of A₄⋊F₅ in TeX
Character table of A₄⋊F₅ in TeX

G = A4⋊F5 order 240 = 24·3·5

The semidirect product of A4 and F5 acting via F5/D5=C2

G = A₄⋊F₅ order 240 = 2⁴·3·5

The semidirect product of A₄ and F₅ acting via F₅/D₅=C₂