C₅×#85665 #14511403

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85665

Splittings

Fitting, Hall	C₂²×C₅×#131 ⋊ C₃
Sylow	C₂²×#131 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₈	C₂²⋊A₄
commutator, hypocenter	C₂²×Q₈	C₃×C₄×C₅
Frattini	C₄	C₂×C₅×C₂²⋊A₄
Fitting	C₂²×C₅×#131	C₃
socle	C₂³×C₅	C₄×A₄
prime radical	C₅×#5948	C₂
cosocle	C₂²×#41	C₂×C₃×C₅
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₅×C₈

578 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 41
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(7,9,8)(10,11,12,16,15,19,22,29)(13,30,37,27,24,18,36,39,26,17,34,41,25,31,21,40,23,33,20,38,14,32,35,28);(6,9)(7,8);(10,24,15,23)(11,31,19,30)(12,35,22,34)(13,26,25,14)(16,39,29,38)(17,33,32,18)(20,37,36,21)(27,41,40,28)

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, 237101);