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PGL(2,𝔽₉) #10624295

order	720
isoclinism	isoclinic class of PGL(2,𝔽₉)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

projective general linear

PGL(2,𝔽₉)

Splittings

A₆ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	A₆	C₂
LCS	G ▷ A₆
UCS	C₁

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	36	45	72	72	72	72	80	90	90	90
order	1	2	2	5	5	10	10	3	4	8	8
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	-1	1	1	1	-1	-1	1	1	-1	-1
χ₃	8	2	0	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	-1	0	0	0
χ₄	8	2	0	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	-1	0	0	0
χ₅	8	-2	0	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	-1	0	0	0
χ₆	8	-2	0	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	-1	0	0	0
χ₇	9	1	1	-1	-1	1	1	0	1	-1	-1
χ₈	9	-1	1	-1	-1	-1	-1	0	1	1	1
χ₉	10	0	2	0	0	0	0	1	-2	0	0
χ₁₀	10	0	-2	0	0	0	0	1	0	$\sqrt{2}$	$- \sqrt{2}$
χ₁₁	10	0	-2	0	0	0	0	1	0	$- \sqrt{2}$	$\sqrt{2}$

Relationships

maximal subgroup of	PΓL(2,𝔽₉), C₂×PGL(2,𝔽₉), C₅×PGL(2,𝔽₉)
maximal normal subgroup of	PΓL(2,𝔽₉), C₂×PGL(2,𝔽₉), C₅×PGL(2,𝔽₉)
maximal quotient of	#13088275, #13088276, #13088277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 764);