C₅×#71296 #14293471

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#71296

Splittings

Hall	C₃×C₅ ⋊ #1707
Sylow	C₃ ⋊ C₅×#1707

Subgroups

	subgroup	G/H
center	C₂²×C₅	#940
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#494	C₂
socle, prime radical	C₂²×C₃×C₅	#127
hypercenter	C₅×#494	D₃
hypocenter	C₃	C₅×#1707
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#494

360 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 32 points:

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

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 19169);