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A₆ #67188

order	360
isoclinism	isoclinic class of A₆
automorphism	Aut(G) = PΓL(2,𝔽₉) (order 1440) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂×C₃

Aliases

alternating	A₆
projective special linear	PSL(2,𝔽₉)
projective symplectic	PSp(2,𝔽₉)
omega	Ω⁻(4,𝔽₃)
projective special unitary	PSU(2,𝔽₉)

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

minimal degree	μ(G) = 6
transitive degree	μ_t(G) = 6

A minimal faithful permutation representation — 2 generators on 6 points:

(2,3,4,5,6);(1,2,3)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇
size	1	40	40	45	72	72	90
order	1	3	3	2	5	5	4
χ₁	1	1	1	1	1	1	1
χ₂	5	2	-1	1	0	0	-1
χ₃	5	-1	2	1	0	0	-1
χ₄	8	-1	-1	0	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	0
χ₅	8	-1	-1	0	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	0
χ₆	9	0	0	1	-1	-1	1
χ₇	10	1	1	-2	0	0	0

Relationships

derived subgroup of	S₆, PGL(2,𝔽₉), #10624296, …
socle of	S₆, PGL(2,𝔽₉), #10624296, …
maximal subgroup of	S₆, PGL(2,𝔽₉), #10624296, …
maximal normal subgroup of	S₆, PGL(2,𝔽₉), #10624296, …
maximal quotient of	SL(2,𝔽₉), C₂×A₆, #11761914, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 118);