C₃×C₅×#1298 #14274332

order	1920
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₃×C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×C₅×#1298

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₃×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂²×C₃×C₅
socle	C₂²×C₃×C₅	#95
prime radical	C₂⁴×C₃×C₅	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

380 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 480

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

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

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, 30);