C₅×#67719 #14277737

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67719

Splittings

Hall	C₃×C₅ ⋊ #1328
Sylow	C₃ ⋊ C₅×#1328

Subgroups

	subgroup	G/H
center	C₄×C₅	#834
commutator	C₂³×C₃	C₂×C₅×C₈
Frattini	C₂×M₁₆	C₂×C₅×D₃
Fitting	C₃×C₅×#424	C₂
socle	C₂×C₃×C₅	#324
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×M₁₆	C₂²×C₅
hypercenter	C₅×#424	D₃
hypocenter	C₃	C₅×#1328
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#424

180 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 40 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 3435);