C₅×#68718 #14312461

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#68718

Splittings

Hall	C₃×C₅ ⋊ #2103
Sylow	C₃ ⋊ C₅×#2103

Subgroups

	subgroup	G/H
center	C₄×C₅	#875
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#338	C₂
socle	C₂×C₃×C₅	#400
prime radical	C₂³×C₃×C₅	C₂×Q₈
hypercenter	C₅×#338	D₃
hypocenter	C₃	C₅×#2103
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#338

296 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(6,7)(9,12,13,27,14,28,15,29)(10,17,18,34,19,25,30,39)(11,21,22,36,24,37,23,26)(16,31,20,33,32,40,35,38);(7,8)(9,11,15,22,14,24,13,23)(10,16,30,20,19,32,18,35)(12,26,29,21,28,36,27,37)(17,33,39,40,25,38,34,31);(9,10,13,18,14,19,15,30)(11,20,22,32,24,35,23,16)(12,25,27,39,28,17,29,34)(21,33,36,40,37,38,26,31)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 38159);