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G = F17order 272 = 24·17

Frobenius group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: F17, AGL1(𝔽17), C17⋊C16, D17.C8, C17⋊C8.C2, C17⋊C4.C4, Aut(D17), Hol(C17), SmallGroup(272,50)

Series: Derived Chief Lower central Upper central

C1C17 — F17
C1C17D17C17⋊C4C17⋊C8 — F17
C17 — F17
C1

Generators and relations for F17
 G = < a,b | a17=b16=1, bab-1=a6 >

17C2
17C4
17C8
17C16

Character table of F17

 class 124A4B8A8B8C8D16A16B16C16D16E16F16G16H17
 size 117171717171717171717171717171716
ρ111111111111111111    trivial
ρ211111111-1-1-1-1-1-1-1-11    linear of order 2
ρ31111-1-1-1-1i-i-i-i-iiii1    linear of order 4
ρ41111-1-1-1-1-iiiii-i-i-i1    linear of order 4
ρ511-1-1i-ii-iζ8ζ87ζ87ζ83ζ83ζ85ζ85ζ81    linear of order 8
ρ611-1-1-ii-iiζ83ζ85ζ85ζ8ζ8ζ87ζ87ζ831    linear of order 8
ρ711-1-1-ii-iiζ87ζ8ζ8ζ85ζ85ζ83ζ83ζ871    linear of order 8
ρ811-1-1i-ii-iζ85ζ83ζ83ζ87ζ87ζ8ζ8ζ851    linear of order 8
ρ91-1-iiζ1610ζ1614ζ162ζ166ζ165ζ1611ζ163ζ167ζ1615ζ16ζ169ζ16131    linear of order 16
ρ101-1i-iζ166ζ162ζ1614ζ1610ζ163ζ1613ζ165ζ16ζ169ζ167ζ1615ζ16111    linear of order 16
ρ111-1i-iζ166ζ162ζ1614ζ1610ζ1611ζ165ζ1613ζ169ζ16ζ1615ζ167ζ1631    linear of order 16
ρ121-1-iiζ1610ζ1614ζ162ζ166ζ1613ζ163ζ1611ζ1615ζ167ζ169ζ16ζ1651    linear of order 16
ρ131-1i-iζ1614ζ1610ζ166ζ162ζ167ζ169ζ16ζ1613ζ165ζ1611ζ163ζ16151    linear of order 16
ρ141-1i-iζ1614ζ1610ζ166ζ162ζ1615ζ16ζ169ζ165ζ1613ζ163ζ1611ζ1671    linear of order 16
ρ151-1-iiζ162ζ166ζ1610ζ1614ζ169ζ167ζ1615ζ163ζ1611ζ165ζ1613ζ161    linear of order 16
ρ161-1-iiζ162ζ166ζ1610ζ1614ζ16ζ1615ζ167ζ1611ζ163ζ1613ζ165ζ1691    linear of order 16
ρ1716000000000000000-1    orthogonal faithful

Permutation representations of F17
On 17 points: primitive, sharply doubly transitive - transitive group 17T5
Generators in S17
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)
(2 4 10 11 14 6 16 12 17 15 9 8 5 13 3 7)

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

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

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

G:=TransitiveGroup(17,5);

Matrix representation of F17 in GL16(ℤ)

0100000000000000
0010000000000000
0001000000000000
0000100000000000
0000010000000000
0000001000000000
0000000100000000
0000000010000000
0000000001000000
0000000000100000
0000000000010000
0000000000001000
0000000000000100
0000000000000010
0000000000000001
-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
,
1000000000000000
0000001000000000
0000000000001000
0100000000000000
0000000100000000
0000000000000100
0010000000000000
0000000010000000
0000000000000010
0001000000000000
0000000001000000
0000000000000001
0000100000000000
0000000000100000
-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
0000010000000000

G:=sub<GL(16,Integers())| [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,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0] >;

F17 in GAP, Magma, Sage, TeX

F_{17}
% in TeX

G:=Group("F17");
// GroupNames label

G:=SmallGroup(272,50);
// by ID

G=gap.SmallGroup(272,50);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,10,26,42,1204,1809,1314,819]);
// Polycyclic

G:=Group<a,b|a^17=b^16=1,b*a*b^-1=a^6>;
// generators/relations

Export

Subgroup lattice of F17 in TeX
Character table of F17 in TeX

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