SO(5,𝔽₃) #72225614

order	51840
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

special orthogonal

SO(5,𝔽₃)

Splittings

socle, derived

SU(4,𝔽₂) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	SU(4,𝔽₂)	C₂
LCS	G ▷ SU(4,𝔽₂)
UCS	C₁

350 conjugacy classes of subgroups; only the 3 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

(1,12)(2,18)(3,20)(4,17)(5,6)(7,26)(8,9)(10,19)(13,14)(15,24)(16,22)(21,23);(1,5,14,11,24,17,27,21,9,4)(2,23,16,15,8)(3,7,19,6,22)(10,13)(12,18,20,25,26)

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,12)(2,18)(3,20)(4,17)(5,6)(7,26)(8,9)(10,19)(13,14)(15,24)(16,22)(21,23), (1,5,14,11,24,17,27,21,9,4)(2,23,16,15,8)(3,7,19,6,22)(10,13)(12,18,20,25,26));