A4xF5 - GroupNames

G = A₄×F₅ order 240 = 2⁴·3·5

Direct product of A₄ and F₅

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

Aliases: A₄×F₅, C₅⋊(C₄×A₄), (C₂×C₁₀)⋊C₁₂, D₅.(C₂×A₄), (C₅×A₄)⋊₂C₄, C₂²⋊(C₃×F₅), (C₂²×F₅)⋊C₃, (D₅×A₄).₂C₂, (C₂²×D₅).C₆, SmallGroup(240,193)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — A₄×F₅

Chief series

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

Lower central

C₂×C₁₀ — 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=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

C₁

₃C₂

₅C₂

₁₅C₂

₄C₃

C₅

C₂²

₅C₄

₁₅C₄

₁₅C₂²

₂₀C₆

D₅

₃C₁₀

₃D₅

₄C₁₅

₅C₂³

₁₅C₂×C₄

A₄

₂₀C₁₂

C₂×C₁₀

F₅

₃D₁₀

₃F₅

₃D₁₀

₄C₃×D₅

₅C₂²×C₄

₅C₂×A₄

C₂²×D₅

₃C₂×F₅

C₅×A₄

₄C₃×F₅

₅C₄×A₄

C₂²×F₅

D₅×A₄

A₄×F₅

Character table of A₄×F₅

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	5	6A	6B	10	12A	12B	12C	12D	15A	15B
size	1	3	5	15	4	4	5	5	15	15	4	20	20	12	20	20	20	20	16	16
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	ζ₃²	ζ₃	-1	-1	-1	-1	1	ζ₃²	ζ₃	1	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	ζ₃	ζ₃²	linear of order 6
ρ₄	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₅	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₆	1	1	1	1	ζ₃	ζ₃²	-1	-1	-1	-1	1	ζ₃	ζ₃²	1	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	ζ₃²	ζ₃	linear of order 6
ρ₇	1	1	-1	-1	1	1	i	-i	-i	i	1	-1	-1	1	-i	-i	i	i	1	1	linear of order 4
ρ₈	1	1	-1	-1	1	1	-i	i	i	-i	1	-1	-1	1	i	i	-i	-i	1	1	linear of order 4
ρ₉	1	1	-1	-1	ζ₃	ζ₃²	i	-i	-i	i	1	ζ₆⁵	ζ₆	1	ζ₄³ζ₃²	ζ₄³ζ₃	ζ₄ζ₃²	ζ₄ζ₃	ζ₃²	ζ₃	linear of order 12
ρ₁₀	1	1	-1	-1	ζ₃²	ζ₃	-i	i	i	-i	1	ζ₆	ζ₆⁵	1	ζ₄ζ₃	ζ₄ζ₃²	ζ₄³ζ₃	ζ₄³ζ₃²	ζ₃	ζ₃²	linear of order 12
ρ₁₁	1	1	-1	-1	ζ₃	ζ₃²	-i	i	i	-i	1	ζ₆⁵	ζ₆	1	ζ₄ζ₃²	ζ₄ζ₃	ζ₄³ζ₃²	ζ₄³ζ₃	ζ₃²	ζ₃	linear of order 12
ρ₁₂	1	1	-1	-1	ζ₃²	ζ₃	i	-i	-i	i	1	ζ₆	ζ₆⁵	1	ζ₄³ζ₃	ζ₄³ζ₃²	ζ₄ζ₃	ζ₄ζ₃²	ζ₃	ζ₃²	linear of order 12
ρ₁₃	3	-1	3	-1	0	0	-3	-3	1	1	3	0	0	-1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₄	3	-1	3	-1	0	0	3	3	-1	-1	3	0	0	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₅	3	-1	-3	1	0	0	-3i	3i	-i	i	3	0	0	-1	0	0	0	0	0	0	complex lifted from C₄×A₄
ρ₁₆	3	-1	-3	1	0	0	3i	-3i	i	-i	3	0	0	-1	0	0	0	0	0	0	complex lifted from C₄×A₄
ρ₁₇	4	4	0	0	4	4	0	0	0	0	-1	0	0	-1	0	0	0	0	-1	-1	orthogonal lifted from F₅
ρ₁₈	4	4	0	0	-2+2√-3	-2-2√-3	0	0	0	0	-1	0	0	-1	0	0	0	0	ζ₆	ζ₆⁵	complex lifted from C₃×F₅
ρ₁₉	4	4	0	0	-2-2√-3	-2+2√-3	0	0	0	0	-1	0	0	-1	0	0	0	0	ζ₆⁵	ζ₆	complex lifted from C₃×F₅
ρ₂₀	12	-4	0	0	0	0	0	0	0	0	-3	0	0	1	0	0	0	0	0	0	orthogonal faithful

Permutation representations of A₄×F₅
►On 20 points - transitive group 20T68

Generators in S₂₀

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

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

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

(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)(12 13 15 14)(17 18 20 19)

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

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

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

G:=TransitiveGroup(20,68);

►On 30 points - transitive group 30T56

Generators in S₃₀

(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)

(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)

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

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

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

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

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

G:=TransitiveGroup(30,56);

►On 30 points - transitive group 30T57

Generators in S₃₀

(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)

(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)

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

(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)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)

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

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

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

G:=TransitiveGroup(30,57);

A₄×F₅ is a maximal quotient of SL₂(𝔽₃).F₅

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

1	5	0	0	0	0	0
0	60	0	0	0	0	0
48	59	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

1	0	5	0	0	0	0
48	60	59	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

1	0	0	0	0	0	0
48	60	60	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	60	60	60	60
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0

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

G:=sub<GL(7,GF(61))| [1,0,48,0,0,0,0,5,60,59,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,48,0,0,0,0,0,0,60,0,0,0,0,0,5,59,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,48,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,1,0,0,0,0,60,0,0,1,0,0,0,60,0,0,0],[60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,1,0,60] >;

A₄×F₅ in GAP, Magma, Sage, TeX

A_4\times F_5

% in TeX

G:=Group("A4xF5");

// GroupNames label

G:=SmallGroup(240,193);

// by ID

G=gap.SmallGroup(240,193);

# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-5,36,441,190,3461,1169]);

// Polycyclic

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

Direct product of A4 and F5

G = A₄×F₅ order 240 = 2⁴·3·5

Direct product of A₄ and F₅