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G = F16  order 240 = 24·3·5

Frobenius group

metabelian, soluble, monomial, A-group

Aliases: F16, AGL1(𝔽16), C24⋊C15, C24⋊C5⋊C3, C22⋊A4⋊C5, SmallGroup(240,191)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — F16
Chief series
C1 — C24 — C24⋊C5 — F16
Lower central
C24 — F16
Upper central
C1

Generators and relations for F16
 G = < a,b,c,d,e | a2=b2=c2=d2=e15=1, ab=ba, ac=ca, ad=da, eae-1=bcd, ebe-1=bc=cb, ede-1=bd=db, cd=dc, ece-1=a >

C1
15C2
16C3
16C5
5C22
15C22
15C22
16C15
15C23
20A4
C24
C22⋊A4
C24⋊C5
F16

Character table of F16

 class 123A3B5A5B5C5D15A15B15C15D15E15F15G15H
 size 1151616161616161616161616161616
ρ11111111111111111    trivial
ρ211ζ32ζ31111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ311ζ3ζ321111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ41111ζ54ζ52ζ53ζ5ζ52ζ54ζ52ζ53ζ53ζ5ζ54ζ5    linear of order 5
ρ51111ζ53ζ54ζ5ζ52ζ54ζ53ζ54ζ5ζ5ζ52ζ53ζ52    linear of order 5
ρ61111ζ52ζ5ζ54ζ53ζ5ζ52ζ5ζ54ζ54ζ53ζ52ζ53    linear of order 5
ρ71111ζ5ζ53ζ52ζ54ζ53ζ5ζ53ζ52ζ52ζ54ζ5ζ54    linear of order 5
ρ811ζ3ζ32ζ5ζ53ζ52ζ54ζ3ζ53ζ32ζ5ζ32ζ53ζ32ζ52ζ3ζ52ζ3ζ54ζ3ζ5ζ32ζ54    linear of order 15
ρ911ζ32ζ3ζ52ζ5ζ54ζ53ζ32ζ5ζ3ζ52ζ3ζ5ζ3ζ54ζ32ζ54ζ32ζ53ζ32ζ52ζ3ζ53    linear of order 15
ρ1011ζ3ζ32ζ52ζ5ζ54ζ53ζ3ζ5ζ32ζ52ζ32ζ5ζ32ζ54ζ3ζ54ζ3ζ53ζ3ζ52ζ32ζ53    linear of order 15
ρ1111ζ32ζ3ζ53ζ54ζ5ζ52ζ32ζ54ζ3ζ53ζ3ζ54ζ3ζ5ζ32ζ5ζ32ζ52ζ32ζ53ζ3ζ52    linear of order 15
ρ1211ζ3ζ32ζ53ζ54ζ5ζ52ζ3ζ54ζ32ζ53ζ32ζ54ζ32ζ5ζ3ζ5ζ3ζ52ζ3ζ53ζ32ζ52    linear of order 15
ρ1311ζ32ζ3ζ5ζ53ζ52ζ54ζ32ζ53ζ3ζ5ζ3ζ53ζ3ζ52ζ32ζ52ζ32ζ54ζ32ζ5ζ3ζ54    linear of order 15
ρ1411ζ3ζ32ζ54ζ52ζ53ζ5ζ3ζ52ζ32ζ54ζ32ζ52ζ32ζ53ζ3ζ53ζ3ζ5ζ3ζ54ζ32ζ5    linear of order 15
ρ1511ζ32ζ3ζ54ζ52ζ53ζ5ζ32ζ52ζ3ζ54ζ3ζ52ζ3ζ53ζ32ζ53ζ32ζ5ζ32ζ54ζ3ζ5    linear of order 15
ρ1615-100000000000000    orthogonal faithful

Permutation representations of F16
►On 16 points: primitive, sharply doubly transitive - transitive group 16T447
Generators in S16
(1 14)(2 3)(4 9)(5 7)(6 12)(8 16)(10 13)(11 15)

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

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

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

(2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (1,14)(2,3)(4,9)(5,7)(6,12)(8,16)(10,13)(11,15), (1,4)(2,11)(3,15)(5,16)(6,13)(7,8)(9,14)(10,12), (1,15)(2,9)(3,4)(5,10)(6,8)(7,13)(11,14)(12,16), (1,8)(2,10)(3,13)(4,7)(5,9)(6,15)(11,12)(14,16), (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
 
G:=Group( (1,14)(2,3)(4,9)(5,7)(6,12)(8,16)(10,13)(11,15), (1,4)(2,11)(3,15)(5,16)(6,13)(7,8)(9,14)(10,12), (1,15)(2,9)(3,4)(5,10)(6,8)(7,13)(11,14)(12,16), (1,8)(2,10)(3,13)(4,7)(5,9)(6,15)(11,12)(14,16), (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(1,14),(2,3),(4,9),(5,7),(6,12),(8,16),(10,13),(11,15)], [(1,4),(2,11),(3,15),(5,16),(6,13),(7,8),(9,14),(10,12)], [(1,15),(2,9),(3,4),(5,10),(6,8),(7,13),(11,14),(12,16)], [(1,8),(2,10),(3,13),(4,7),(5,9),(6,15),(11,12),(14,16)], [(2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,447);
 

►On 20 points - transitive group 20T67
Generators in S20
(1 18)(2 14)(4 11)(5 12)(6 16)(7 17)(8 13)(9 19)

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

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

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

(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
 
G:=sub<Sym(20)| (1,18)(2,14)(4,11)(5,12)(6,16)(7,17)(8,13)(9,19), (1,8)(2,19)(4,16)(5,17)(6,11)(7,12)(9,14)(13,18), (1,13)(2,19)(3,15)(5,12)(7,17)(8,18)(9,14)(10,20), (1,8)(3,20)(4,6)(5,12)(7,17)(10,15)(11,16)(13,18), (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)>;
 
G:=Group( (1,18)(2,14)(4,11)(5,12)(6,16)(7,17)(8,13)(9,19), (1,8)(2,19)(4,16)(5,17)(6,11)(7,12)(9,14)(13,18), (1,13)(2,19)(3,15)(5,12)(7,17)(8,18)(9,14)(10,20), (1,8)(3,20)(4,6)(5,12)(7,17)(10,15)(11,16)(13,18), (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) );
 
G=PermutationGroup([[(1,18),(2,14),(4,11),(5,12),(6,16),(7,17),(8,13),(9,19)], [(1,8),(2,19),(4,16),(5,17),(6,11),(7,12),(9,14),(13,18)], [(1,13),(2,19),(3,15),(5,12),(7,17),(8,18),(9,14),(10,20)], [(1,8),(3,20),(4,6),(5,12),(7,17),(10,15),(11,16),(13,18)], [(1,2,3,4,5),(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)]])
 
G:=TransitiveGroup(20,67);
 

►On 30 points - transitive group 30T50
Generators in S30
(3 27)(6 30)(7 16)(9 18)(11 20)(12 21)(13 22)(14 23)

(1 25)(2 26)(3 27)(4 28)(8 17)(11 20)(12 21)(14 23)

(4 28)(7 16)(8 17)(10 19)(12 21)(13 22)(14 23)(15 24)

(1 25)(3 27)(5 29)(6 30)(7 16)(8 17)(12 21)(15 24)

(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)
 
G:=sub<Sym(30)| (3,27)(6,30)(7,16)(9,18)(11,20)(12,21)(13,22)(14,23), (1,25)(2,26)(3,27)(4,28)(8,17)(11,20)(12,21)(14,23), (4,28)(7,16)(8,17)(10,19)(12,21)(13,22)(14,23)(15,24), (1,25)(3,27)(5,29)(6,30)(7,16)(8,17)(12,21)(15,24), (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)>;
 
G:=Group( (3,27)(6,30)(7,16)(9,18)(11,20)(12,21)(13,22)(14,23), (1,25)(2,26)(3,27)(4,28)(8,17)(11,20)(12,21)(14,23), (4,28)(7,16)(8,17)(10,19)(12,21)(13,22)(14,23)(15,24), (1,25)(3,27)(5,29)(6,30)(7,16)(8,17)(12,21)(15,24), (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) );
 
G=PermutationGroup([[(3,27),(6,30),(7,16),(9,18),(11,20),(12,21),(13,22),(14,23)], [(1,25),(2,26),(3,27),(4,28),(8,17),(11,20),(12,21),(14,23)], [(4,28),(7,16),(8,17),(10,19),(12,21),(13,22),(14,23),(15,24)], [(1,25),(3,27),(5,29),(6,30),(7,16),(8,17),(12,21),(15,24)], [(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)]])
 
G:=TransitiveGroup(30,50);
 

F16 is a maximal subgroup of   F16⋊C2

Matrix representation of F16 ►in GL15(ℤ)

100000000000000
010000000000000
001000000000000
000001000000000
000-1-1-1000000000
000100000000000
000000010000000
000000100000000
000000-1-1-1000000
000000000010000
000000000100000
000000000-1-1-1000
000000000000001
000000000000-1-1-1
000000000000100
,
100000000000000
010000000000000
001000000000000
000010000000000
000100000000000
000-1-1-1000000000
000000-1-1-1000000
000000001000000
000000010000000
000000000-1-1-1000
000000000001000
000000000010000
000000000000010
000000000000100
000000000000-1-1-1
,
001000000000000
-1-1-1000000000000
100000000000000
000001000000000
000-1-1-1000000000
000100000000000
000000-1-1-1000000
000000001000000
000000010000000
000000000100000
000000000010000
000000000001000
000000000000-1-1-1
000000000000001
000000000000010
,
010000000000000
100000000000000
-1-1-1000000000000
000001000000000
000-1-1-1000000000
000100000000000
000000100000000
000000010000000
000000001000000
000000000001000
000000000-1-1-1000
000000000100000
000000000000010
000000000000100
000000000000-1-1-1
,
000000100000000
000000-1-1-1000000
000000010000000
000000000100000
000000000-1-1-1000
000000000010000
000000000000100
000000000000-1-1-1
000000000000010
100000000000000
-1-1-1000000000000
010000000000000
000100000000000
000-1-1-1000000000
000010000000000

G:=sub<GL(15,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1],[0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0],[0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0] >;
 

F16 in GAP, Magma, Sage, TeX 

F_{16}
 
% in TeX
 
G:=Group("F16");
 
// GroupNames label
 
G:=SmallGroup(240,191);
 
// by ID
 
G=gap.SmallGroup(240,191);
 
# by ID
 
G:=PCGroup([6,-3,-5,-2,2,2,2,2972,1358,5403,849,3154,2110,7565,911]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^15=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=b*c*d,e*b*e^-1=b*c=c*b,e*d*e^-1=b*d=d*b,c*d=d*c,e*c*e^-1=a>;
 
// generators/relations
 

Export

Subgroup lattice of F16 in TeX
Character table of F16 in TeX

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