C₅×#85409 #14511147

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

solvable

Aliases

direct product

C₅×#85409

Splittings

Fitting

C₅×#550 ⋊ 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	C₅×#550	D₃
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

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

normal subgroups only

This group has 552 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) = 41
transitive degree	μ_t(G) = 320

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

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236845);