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PSL(2,𝔽₁₇) #72209593

order	2448
isoclinism	isoclinic class of PSL(2,𝔽₁₇)
automorphism	Aut(G) = PGL(2,𝔽₁₇) (order 4896) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

projective special linear	PSL(2,𝔽₁₇)
projective symplectic	PSp(2,𝔽₁₇)
projective special unitary	PSU(2,𝔽₁₇)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	144	144	153	272	272	272	272	306	306	306
order	1	17	17	2	3	9	9	9	4	8	8
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	9	$\frac{1 - \sqrt{17}}{2}$	$\frac{1 + \sqrt{17}}{2}$	1	0	0	0	0	1	-1	-1
χ₃	9	$\frac{1 + \sqrt{17}}{2}$	$\frac{1 - \sqrt{17}}{2}$	1	0	0	0	0	1	-1	-1
χ₄	16	-1	-1	0	1	ζ⁴⁴−ζ⁵²+ζ¹¹⁶−ζ¹²⁴+ζ¹⁴⁰+ζ¹⁸⁸−ζ¹⁹⁶+ζ²¹²+ζ²⁶⁰−ζ²⁶⁸+ζ²⁸⁴+ζ³³²+ζ³⁵⁶+ζ⁴⁰⁴−ζ⁴¹²+ζ⁴²⁸−ζ⁴⁸⁴+ζ⁵⁰⁰+ζ⁵⁴⁸−ζ⁵⁵⁶+ζ⁵⁷²+ζ⁶²⁰−ζ⁶²⁸+ζ⁶⁴⁴+ζ⁶⁹²−ζ⁷⁰⁰+ζ⁷¹⁶+ζ⁷⁶⁴−ζ⁷⁷²+ζ⁷⁸⁸+ζ⁸³⁶−ζ⁸⁴⁴+ζ⁸⁶⁰+ζ⁹⁰⁸−ζ⁹¹⁶+ζ⁹³²+ζ⁹⁸⁰−ζ⁹⁸⁸+ζ¹⁰⁰⁴+ζ¹⁰⁵²−ζ¹⁰⁶⁰+ζ¹⁰⁷⁶+ζ¹¹²⁴−ζ¹¹³²+ζ¹¹⁴⁸+ζ¹¹⁹⁶−ζ¹²⁰⁴+ζ¹²²⁰	−ζ⁴−ζ⁷⁶−ζ¹⁴⁰−ζ¹⁴⁸−ζ²¹²−ζ²²⁰−ζ²⁸⁴−ζ²⁹²−ζ³⁵⁶−ζ³⁶⁴−ζ⁴²⁸−ζ⁴³⁶−ζ⁵⁰⁰−ζ⁵⁰⁸−ζ⁵⁷²−ζ⁵⁸⁰−ζ⁶⁴⁴−ζ⁶⁵²−ζ⁷¹⁶−ζ⁷²⁴−ζ⁷⁸⁸−ζ⁷⁹⁶−ζ⁸⁶⁰−ζ⁸⁶⁸−ζ⁹³²−ζ⁹⁴⁰−ζ¹⁰⁰⁴−ζ¹⁰¹²−ζ¹⁰⁷⁶−ζ¹⁰⁸⁴−ζ¹¹⁴⁸−ζ¹²²⁰	ζ⁴−ζ⁴⁴+ζ⁵²+ζ⁷⁶−ζ¹¹⁶+ζ¹²⁴+ζ¹⁴⁸−ζ¹⁸⁸+ζ¹⁹⁶+ζ²²⁰−ζ²⁶⁰+ζ²⁶⁸+ζ²⁹²−ζ³³²+ζ³⁶⁴−ζ⁴⁰⁴+ζ⁴¹²+ζ⁴³⁶+ζ⁴⁸⁴+ζ⁵⁰⁸−ζ⁵⁴⁸+ζ⁵⁵⁶+ζ⁵⁸⁰−ζ⁶²⁰+ζ⁶²⁸+ζ⁶⁵²−ζ⁶⁹²+ζ⁷⁰⁰+ζ⁷²⁴−ζ⁷⁶⁴+ζ⁷⁷²+ζ⁷⁹⁶−ζ⁸³⁶+ζ⁸⁴⁴+ζ⁸⁶⁸−ζ⁹⁰⁸+ζ⁹¹⁶+ζ⁹⁴⁰−ζ⁹⁸⁰+ζ⁹⁸⁸+ζ¹⁰¹²−ζ¹⁰⁵²+ζ¹⁰⁶⁰+ζ¹⁰⁸⁴−ζ¹¹²⁴+ζ¹¹³²−ζ¹¹⁹⁶+ζ¹²⁰⁴	0	0	0
χ₅	16	-1	-1	0	1	−ζ⁴−ζ⁷⁶−ζ¹⁴⁰−ζ¹⁴⁸−ζ²¹²−ζ²²⁰−ζ²⁸⁴−ζ²⁹²−ζ³⁵⁶−ζ³⁶⁴−ζ⁴²⁸−ζ⁴³⁶−ζ⁵⁰⁰−ζ⁵⁰⁸−ζ⁵⁷²−ζ⁵⁸⁰−ζ⁶⁴⁴−ζ⁶⁵²−ζ⁷¹⁶−ζ⁷²⁴−ζ⁷⁸⁸−ζ⁷⁹⁶−ζ⁸⁶⁰−ζ⁸⁶⁸−ζ⁹³²−ζ⁹⁴⁰−ζ¹⁰⁰⁴−ζ¹⁰¹²−ζ¹⁰⁷⁶−ζ¹⁰⁸⁴−ζ¹¹⁴⁸−ζ¹²²⁰	ζ⁴−ζ⁴⁴+ζ⁵²+ζ⁷⁶−ζ¹¹⁶+ζ¹²⁴+ζ¹⁴⁸−ζ¹⁸⁸+ζ¹⁹⁶+ζ²²⁰−ζ²⁶⁰+ζ²⁶⁸+ζ²⁹²−ζ³³²+ζ³⁶⁴−ζ⁴⁰⁴+ζ⁴¹²+ζ⁴³⁶+ζ⁴⁸⁴+ζ⁵⁰⁸−ζ⁵⁴⁸+ζ⁵⁵⁶+ζ⁵⁸⁰−ζ⁶²⁰+ζ⁶²⁸+ζ⁶⁵²−ζ⁶⁹²+ζ⁷⁰⁰+ζ⁷²⁴−ζ⁷⁶⁴+ζ⁷⁷²+ζ⁷⁹⁶−ζ⁸³⁶+ζ⁸⁴⁴+ζ⁸⁶⁸−ζ⁹⁰⁸+ζ⁹¹⁶+ζ⁹⁴⁰−ζ⁹⁸⁰+ζ⁹⁸⁸+ζ¹⁰¹²−ζ¹⁰⁵²+ζ¹⁰⁶⁰+ζ¹⁰⁸⁴−ζ¹¹²⁴+ζ¹¹³²−ζ¹¹⁹⁶+ζ¹²⁰⁴	ζ⁴⁴−ζ⁵²+ζ¹¹⁶−ζ¹²⁴+ζ¹⁴⁰+ζ¹⁸⁸−ζ¹⁹⁶+ζ²¹²+ζ²⁶⁰−ζ²⁶⁸+ζ²⁸⁴+ζ³³²+ζ³⁵⁶+ζ⁴⁰⁴−ζ⁴¹²+ζ⁴²⁸−ζ⁴⁸⁴+ζ⁵⁰⁰+ζ⁵⁴⁸−ζ⁵⁵⁶+ζ⁵⁷²+ζ⁶²⁰−ζ⁶²⁸+ζ⁶⁴⁴+ζ⁶⁹²−ζ⁷⁰⁰+ζ⁷¹⁶+ζ⁷⁶⁴−ζ⁷⁷²+ζ⁷⁸⁸+ζ⁸³⁶−ζ⁸⁴⁴+ζ⁸⁶⁰+ζ⁹⁰⁸−ζ⁹¹⁶+ζ⁹³²+ζ⁹⁸⁰−ζ⁹⁸⁸+ζ¹⁰⁰⁴+ζ¹⁰⁵²−ζ¹⁰⁶⁰+ζ¹⁰⁷⁶+ζ¹¹²⁴−ζ¹¹³²+ζ¹¹⁴⁸+ζ¹¹⁹⁶−ζ¹²⁰⁴+ζ¹²²⁰	0	0	0
χ₆	16	-1	-1	0	1	ζ⁴−ζ⁴⁴+ζ⁵²+ζ⁷⁶−ζ¹¹⁶+ζ¹²⁴+ζ¹⁴⁸−ζ¹⁸⁸+ζ¹⁹⁶+ζ²²⁰−ζ²⁶⁰+ζ²⁶⁸+ζ²⁹²−ζ³³²+ζ³⁶⁴−ζ⁴⁰⁴+ζ⁴¹²+ζ⁴³⁶+ζ⁴⁸⁴+ζ⁵⁰⁸−ζ⁵⁴⁸+ζ⁵⁵⁶+ζ⁵⁸⁰−ζ⁶²⁰+ζ⁶²⁸+ζ⁶⁵²−ζ⁶⁹²+ζ⁷⁰⁰+ζ⁷²⁴−ζ⁷⁶⁴+ζ⁷⁷²+ζ⁷⁹⁶−ζ⁸³⁶+ζ⁸⁴⁴+ζ⁸⁶⁸−ζ⁹⁰⁸+ζ⁹¹⁶+ζ⁹⁴⁰−ζ⁹⁸⁰+ζ⁹⁸⁸+ζ¹⁰¹²−ζ¹⁰⁵²+ζ¹⁰⁶⁰+ζ¹⁰⁸⁴−ζ¹¹²⁴+ζ¹¹³²−ζ¹¹⁹⁶+ζ¹²⁰⁴	ζ⁴⁴−ζ⁵²+ζ¹¹⁶−ζ¹²⁴+ζ¹⁴⁰+ζ¹⁸⁸−ζ¹⁹⁶+ζ²¹²+ζ²⁶⁰−ζ²⁶⁸+ζ²⁸⁴+ζ³³²+ζ³⁵⁶+ζ⁴⁰⁴−ζ⁴¹²+ζ⁴²⁸−ζ⁴⁸⁴+ζ⁵⁰⁰+ζ⁵⁴⁸−ζ⁵⁵⁶+ζ⁵⁷²+ζ⁶²⁰−ζ⁶²⁸+ζ⁶⁴⁴+ζ⁶⁹²−ζ⁷⁰⁰+ζ⁷¹⁶+ζ⁷⁶⁴−ζ⁷⁷²+ζ⁷⁸⁸+ζ⁸³⁶−ζ⁸⁴⁴+ζ⁸⁶⁰+ζ⁹⁰⁸−ζ⁹¹⁶+ζ⁹³²+ζ⁹⁸⁰−ζ⁹⁸⁸+ζ¹⁰⁰⁴+ζ¹⁰⁵²−ζ¹⁰⁶⁰+ζ¹⁰⁷⁶+ζ¹¹²⁴−ζ¹¹³²+ζ¹¹⁴⁸+ζ¹¹⁹⁶−ζ¹²⁰⁴+ζ¹²²⁰	−ζ⁴−ζ⁷⁶−ζ¹⁴⁰−ζ¹⁴⁸−ζ²¹²−ζ²²⁰−ζ²⁸⁴−ζ²⁹²−ζ³⁵⁶−ζ³⁶⁴−ζ⁴²⁸−ζ⁴³⁶−ζ⁵⁰⁰−ζ⁵⁰⁸−ζ⁵⁷²−ζ⁵⁸⁰−ζ⁶⁴⁴−ζ⁶⁵²−ζ⁷¹⁶−ζ⁷²⁴−ζ⁷⁸⁸−ζ⁷⁹⁶−ζ⁸⁶⁰−ζ⁸⁶⁸−ζ⁹³²−ζ⁹⁴⁰−ζ¹⁰⁰⁴−ζ¹⁰¹²−ζ¹⁰⁷⁶−ζ¹⁰⁸⁴−ζ¹¹⁴⁸−ζ¹²²⁰	0	0	0
χ₇	16	-1	-1	0	-2	1	1	1	0	0	0
χ₈	17	0	0	1	-1	-1	-1	-1	1	1	1
χ₉	18	1	1	2	0	0	0	0	-2	0	0
χ₁₀	18	1	1	-2	0	0	0	0	0	$\sqrt{2}$	$- \sqrt{2}$
χ₁₁	18	1	1	-2	0	0	0	0	0	$- \sqrt{2}$	$\sqrt{2}$

Here ζ is a primitive root of unity of order 1224 (ζ¹²²⁴ = 1).

Relationships

maximal subgroup of	PGL(2,𝔽₁₇)
maximal normal subgroup of	PGL(2,𝔽₁₇)
maximal quotient of	Sp(2,𝔽₁₇)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

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