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SL(2,𝔽₅) #1195

order	120
isoclinism	isoclinic class of SL(2,𝔽₅)
automorphism	Aut(G) = S₅ (order 120) · Inn(G) = A₅ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = SO⁺(4,𝔽₅)
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

special linear	SL(2,𝔽₅)
symplectic	Sp(2,𝔽₅)
special unitary	SU(2,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, cosocle, hypercenter	C₂	A₅
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,4,8)(3,6,9,5)(7,12,13,17)(10,14,11,15)(16,20,21,24)(18,22,19,23);(1,3,7)(2,5,10)(4,9,13)(6,11,8)(12,16,20)(14,18,22)(15,19,23)(17,21,24)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉
size	1	1	12	12	12	12	20	20	30
order	1	2	5	5	10	10	3	6	4
χ₁	1	1	1	1	1	1	1	1	1
χ₂	2	-2	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	-1	1	0
χ₃	2	-2	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	-1	1	0
χ₄	3	3	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	0	0	-1
χ₅	3	3	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	0	0	-1
χ₆	4	4	-1	-1	-1	-1	1	1	0
χ₇	4	-4	-1	-1	1	1	1	-1	0
χ₈	5	5	0	0	0	0	-1	-1	1
χ₉	6	-6	1	1	-1	-1	0	0	0

Relationships

derived subgroup of	#6728, #6729, #6732, …
maximal subgroup of	#6728, #6729, #6732, …
maximal normal subgroup of	#6728, #6729, #6732, …
maximal quotient of	C₂×SL(2,𝔽₅), C₃×SL(2,𝔽₅), C₅×SL(2,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 5);