C₅×#85543 #14511281

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

solvable

Aliases

direct product

C₅×#85543

Splittings

Fitting

C₅×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#585	D₃
hypercenter	C₅×#41	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

612 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236979);