C₅×#85406 #14511144

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#85406

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#550	D₃
prime radical	C₅×C₄⋊(#41⋊C₃)	C₂
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

404 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(7,11)(8,12)(14,41,29,27,30,42,19,25,32,16,17,40)(15,38,36,24,31,45,23,34,37,20,21,44)(18,26,33,43)(22,35,28,39);(6,8,9,12)(7,10,11,13);(6,7)(8,10)(9,11)(12,13)(14,15,19,23)(16,24,27,20)(17,28,30,22)(18,31,33,21)(25,35,41,39)(26,34,43,38)(29,37,32,36)(40,44,42,45)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236842);