C₅×#73122 #14510846

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

solvable

Aliases

direct product

C₅×#73122

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	C₂×C₅×#131	D₃
socle	C₂²×C₅	C₄×S₄
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₄⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 41
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(7,9)(10,32,37,16,13,31,22,27,26,11,36,41,15,17,35,29,25,33,12,40,24,19,20,39)(14,18,21,28,23,30,34,38);(6,7,8,9)(13,23)(14,25)(17,30)(18,32)(20,34)(21,36)(24,26)(27,38)(28,40)(31,33)(35,37)(39,41)

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, 236544);