C₃×C₅×#1338 #14274372

order	1920
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅×C₈ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₅×#1338

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

172 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 32
transitive degree	μ_t(G) = 1920

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

570 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 70);