C₅×#85347 #14511085

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

solvable

Aliases

direct product

C₅×#85347

Splittings

Fitting

C₄×C₅×#41 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₄×C₅×#41	D₃
prime radical	C₂×C₅×#986	C₂
hypercenter	C₄²×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅

542 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 320

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236783);