C₅×#71821 #14284010

order	1920
isoclinism	isoclinic class of #4537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#71821

Splittings

Hall	C₃×C₅ ⋊ #1526
Sylow	C₃ ⋊ C₅×#1526

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#217
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#533	D₃
hypocenter	C₃	C₅×#1526
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#533

452 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 24
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(6,7,8)(9,11,13,12)(10,14,15,16)(17,20,22,21)(18,23,19,24);(9,10)(11,14)(12,16)(13,15)(17,19,22,18)(20,24,21,23);(7,8)(10,14,15,16)(17,18,21,24,22,19,20,23)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 9708);