C₅×#85442 #14511180

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#85442

Splittings

Fitting

C₂²×C₅×M₁₆ ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×S₄
commutator	C₂×A₄	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂²×C₅×M₁₆	D₃
socle	C₂³×C₅	C₂×C₄×D₃
prime radical	C₅×#5912	C₂
cosocle	C₄×A₄	C₂³×C₅
hypercenter	C₅×M₁₆	S₄
hypocenter	A₄	C₅×#131
LCS	G ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₄×C₅ ◁ C₅×M₁₆

540 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 25
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236878);