C₃×C₅×#2205 #14275239

order	1920
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×C₅×#2205

Subgroups

	subgroup	G/H
center, socle	C₂×C₃×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₃×C₅
prime radical	C₃×C₅×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₅×Q₈ ◁ G

448 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

300 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 937);