C₃×C₅×#2204 #14275238

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×#2204

Subgroups

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

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

normal subgroups only

This group has 472 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)(11,17)(12,15)(13,21)(14,18)(16,20)(19,24)(22,23);(10,15)(11,13)(12,20)(16,18)(17,23)(21,24);(9,11)(12,19)(13,14)(15,16)(17,21)(20,22)

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, 936);