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SL(3,𝔽₃) #72209613

order	5616
isoclinism	isoclinic class of SL(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

projective general linear	PGL(3,𝔽₃)
projective special linear	PSL(3,𝔽₃)
special linear	SL(3,𝔽₃)

Subgroups

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

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,2)(4,5)(7,8)(9,11);(2,3,4)(5,6,7)(8,9,10)(11,12,13)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	104	117	432	432	432	432	624	702	702	702	936
order	1	3	2	13	13	13	13	3	4	8	8	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	12	3	4	-1	-1	-1	-1	0	0	0	0	1
χ₃	13	4	-3	0	0	0	0	1	1	-1	-1	0
χ₄	16	-2	0	−ζ⁴−ζ¹²−ζ³⁶	−ζ²⁰−ζ⁶⁰−ζ⁷⁶	−ζ²⁸−ζ⁴⁴−ζ⁸⁴	−ζ⁶⁸−ζ⁹²−ζ¹⁰⁰	1	0	0	0	0
χ₅	16	-2	0	−ζ²⁰−ζ⁶⁰−ζ⁷⁶	−ζ⁶⁸−ζ⁹²−ζ¹⁰⁰	−ζ⁴−ζ¹²−ζ³⁶	−ζ²⁸−ζ⁴⁴−ζ⁸⁴	1	0	0	0	0
χ₆	16	-2	0	−ζ²⁸−ζ⁴⁴−ζ⁸⁴	−ζ⁴−ζ¹²−ζ³⁶	−ζ⁶⁸−ζ⁹²−ζ¹⁰⁰	−ζ²⁰−ζ⁶⁰−ζ⁷⁶	1	0	0	0	0
χ₇	16	-2	0	−ζ⁶⁸−ζ⁹²−ζ¹⁰⁰	−ζ²⁸−ζ⁴⁴−ζ⁸⁴	−ζ²⁰−ζ⁶⁰−ζ⁷⁶	−ζ⁴−ζ¹²−ζ³⁶	1	0	0	0	0
χ₈	26	-1	2	0	0	0	0	-1	2	0	0	-1
χ₉	26	-1	-2	0	0	0	0	-1	0	$i \sqrt{2}$	$- i \sqrt{2}$	1
χ₁₀	26	-1	-2	0	0	0	0	-1	0	$- i \sqrt{2}$	$i \sqrt{2}$	1
χ₁₁	27	0	3	1	1	1	1	0	-1	-1	-1	0
χ₁₂	39	3	-1	0	0	0	0	0	-1	1	1	-1

Here ζ is a primitive root of unity of order 104 (ζ¹⁰⁴ = 1); i = ζ²⁶.

Relationships

derived subgroup of	C₂²×SL(3,𝔽₃), C₄×SL(3,𝔽₃), C₂³×SL(3,𝔽₃)
maximal subgroup of	C₂×SL(3,𝔽₃)
maximal normal subgroup of	C₂×SL(3,𝔽₃)
maximal quotient of	C₂×SL(3,𝔽₃)

GAP

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

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