C₅×C₄⋊GU(2,𝔽₃) #14510794

order	1920
isoclinism	isoclinic class of #5386
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product	C₅×C₄⋊GU(2,𝔽₃)
unambiguous semidirect product	C₄⋊(C₅×GU(2,𝔽₃))

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×S₄
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₄×C₅×#41	D₃
socle	C₂²×C₅	C₂×SL(2,ℤ/4)
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₄²×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₄⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅

288 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 33
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2,3,4,5)(6,7,8,9)(10,20,33)(11,24,32)(12,25,21)(13,31,18)(14,28,30)(15,23,29)(16,17,22)(19,26,27);(7,9)(10,11,12,16,15,19,18,14)(13,21,20,29,25,31,23,33)(17,27,26,30,28,32,24,22)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236492);