C₅×#85464 #14511202

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

Aliases

direct product

C₅×#85464

Splittings

Fitting

C₂²×C₅×Q₁₆ ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₅	D₄×S₄
commutator, cosocle	C₄×A₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂²×C₅×Q₁₆	D₃
socle	C₂³×C₅	D₃×D₄
prime radical	C₅×#5919	C₂
hypercenter	C₅×Q₁₆	S₄
hypocenter	A₄	C₅×#137
LCS	G ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₁₆

518 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 68 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)(10,12,14,18)(11,15,17,24)(13,22,23,20)(16,25,21,19);(10,11,14,17)(12,19,18,25)(13,21,23,16)(15,22,24,20);(7,8)(12,18)(15,24)(19,25)(20,22)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236900);