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PSL(2,𝔽₁₁) #10620343

order	660
isoclinism	isoclinic class of PSL(2,𝔽₁₁)
automorphism	Aut(G) = PGL(2,𝔽₁₁) (order 1320) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

projective special linear	PSL(2,𝔽₁₁)
projective symplectic	PSp(2,𝔽₁₁)
projective special unitary	PSU(2,𝔽₁₁)

Subgroups

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

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 11

A minimal faithful permutation representation — 2 generators on 11 points:

(1,5,11,9,10)(2,7,6,3,4);(1,2,8)(3,7,9)(5,6,10)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈
size	1	55	60	60	110	110	132	132
order	1	2	11	11	3	6	5	5
χ₁	1	1	1	1	1	1	1	1
χ₂	5	1	$\frac{- 1 + i \sqrt{11}}{2}$	$\frac{- 1 - i \sqrt{11}}{2}$	-1	1	0	0
χ₃	5	1	$\frac{- 1 - i \sqrt{11}}{2}$	$\frac{- 1 + i \sqrt{11}}{2}$	-1	1	0	0
χ₄	10	2	-1	-1	1	-1	0	0
χ₅	10	-2	-1	-1	1	1	0	0
χ₆	11	-1	0	0	-1	-1	1	1
χ₇	12	0	1	1	0	0	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$
χ₈	12	0	1	1	0	0	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of	PGL(2,𝔽₁₁), C₂×PSL(2,𝔽₁₁), C₃×PSL(2,𝔽₁₁), …
socle of	PGL(2,𝔽₁₁)
maximal subgroup of	PGL(2,𝔽₁₁), C₂×PSL(2,𝔽₁₁), C₃×PSL(2,𝔽₁₁), …
maximal normal subgroup of	PGL(2,𝔽₁₁), C₂×PSL(2,𝔽₁₁), C₃×PSL(2,𝔽₁₁), …
maximal quotient of	SL(2,𝔽₁₁), C₂×PSL(2,𝔽₁₁), C₃×PSL(2,𝔽₁₁), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(660, 13);